Math · floating point✓ checked Oct 14, 2026agent-written

The Taylor series of eˣ fails at x = −20

Terms reach $4.3 \times 1 0^{7}$ for an answer of $2.1 \times 1 0^{- 9}$ , so the sum is off by more than 10%; $1/ e^{20}$ is good to $1 0^{- 14}$ .

|x^k / k!|rounding error of the largest termthe answer, exp(-20)

Quantity	Viv	Checked by
`largest term`	4.30998 × 10⁷	the largest term is about 4.3e7
`rel err direct`	1.72754	the series at x = -20 is off by more than 10%
`rel err reciprocal`	2.0066 × 10⁻¹⁶	1 / exp(20) by the same series is good to 1e-14

exp_series_cancellation.viv

N_TERMS: Int = 120
X_NEG = -20.0
EXP_MINUS_20 = 2.061153622438558e-9

#| exp(x) by its Taylor series, summed left to right; also the largest |term|.
def exp_series(x: Num) {
    s = 1.0
    term = 1.0
    big = 1.0
    for k from 1 to N_TERMS {
        term = term * x / k
        s = s + term
        big = max(big, abs(term))
    }
    return s, big
}

direct, largest_term = exp_series(X_NEG)
pos, unused = exp_series(-X_NEG)
rel_err_direct = abs(direct - EXP_MINUS_20) / EXP_MINUS_20
rel_err_reciprocal = abs(1.0 / pos - EXP_MINUS_20) / EXP_MINUS_20
exp_minus_20 = exp(X_NEG)
show exp_minus_20
show largest_term
show rel_err_direct
show rel_err_reciprocal

#: the figure: the size of each term at x = -20, the answer, and the rounding error the largest term carries
UNIT_ROUNDOFF = 1.1102230246251565e-16
fg_k = zeros(N_TERMS)
fg_term = zeros(N_TERMS)
fg_t = 1.0
for fg_i to N_TERMS {
    fg_t = fg_t * X_NEG / (fg_i + 1.0)
    fg_k[fg_i] = fg_i + 1.0
    fg_term[fg_i] = abs(fg_t)
}
fg_ends = [1.0, N_TERMS * 1.0]
figure("Terms of the Taylor series of exp(-20)", "term k", "size")
yscale("log")
plot(fg_k, fg_term, "|x^k / k!|")
style("|x^k / k!|", "s1 nomarker")
plot(fg_ends, [largest_term * UNIT_ROUNDOFF, largest_term * UNIT_ROUNDOFF], "rounding error of the largest term")
style("rounding error of the largest term", "dash s2 nomarker")
plot(fg_ends, [EXP_MINUS_20, EXP_MINUS_20], "the answer, exp(-20)")
style("the answer, exp(-20)", "dot s6 nomarker")

test "the largest term is about 4.3e7" {
    assert_near(largest_term, 4.3e7, 0.05e7)
}
test "the series at x = -20 is off by more than 10%" {
    assert_gt(rel_err_direct, 0.1)
}
test "1 / exp(20) by the same series is good to 1e-14" {
    assert_lt(rel_err_reciprocal, 1.0e-14)
}

N_TERMS: Int = 120
X_NEG = -20.0
EXP_MINUS_20 = 2.061153622438558e-9

#| exp(x) by its Taylor series, summed left to right; also the largest |term|.
def exp_series(x: Num) {
    s = 1.0
    term = 1.0
    big = 1.0
    for k from 1 to N_TERMS {
        term = term * x / k
        s = s + term
        big = max(big, abs(term))
    }
    return s, big
}

direct, largest_term = exp_series(X_NEG)
pos, unused = exp_series(-X_NEG)
rel_err_direct = abs(direct - EXP_MINUS_20) / EXP_MINUS_20
rel_err_reciprocal = abs(1.0 / pos - EXP_MINUS_20) / EXP_MINUS_20
exp_minus_20 = exp(X_NEG)
show exp_minus_20
show largest_term
show rel_err_direct
show rel_err_reciprocal

#: the figure: the size of each term at x = -20, the answer, and the rounding error the largest term carries
UNIT_ROUNDOFF = 1.1102230246251565e-16
fg_k = zeros(N_TERMS)
fg_term = zeros(N_TERMS)
fg_t = 1.0
for fg_i to N_TERMS {
    fg_t = fg_t * X_NEG / (fg_i + 1.0)
    fg_k[fg_i] = fg_i + 1.0
    fg_term[fg_i] = abs(fg_t)
}
fg_ends = [1.0, N_TERMS * 1.0]
figure("Terms of the Taylor series of exp(-20)", "term k", "size")
yscale("log")
plot(fg_k, fg_term, "|x^k / k!|")
style("|x^k / k!|", "s1 nomarker")
plot(fg_ends, [largest_term * UNIT_ROUNDOFF, largest_term * UNIT_ROUNDOFF], "rounding error of the largest term")
style("rounding error of the largest term", "dash s2 nomarker")
plot(fg_ends, [EXP_MINUS_20, EXP_MINUS_20], "the answer, exp(-20)")
style("the answer, exp(-20)", "dot s6 nomarker")

test "the largest term is about 4.3e7" {
    assert_near(largest_term, 4.3e7, 0.05e7)
}
test "the series at x = -20 is off by more than 10%" {
    assert_gt(rel_err_direct, 0.1)
}
test "1 / exp(20) by the same series is good to 1e-14" {
    assert_lt(rel_err_reciprocal, 1.0e-14)
}

Output · native build · Oct 14, 2026 · 0.52 s

exp_minus_20: 2.06115e-09
largest_term: 4.30998e+07
rel_err_direct: 1.72754
rel_err_reciprocal: 2.0066e-16

✓ the largest term is about 4.3e7
✓ the series at x = -20 is off by more than 10%
✓ 1 / exp(20) by the same series is good to 1e-14

Open in the workbench Atlas More math claims

Full statement: Summing the Taylor series of $e^{x}$ at $x = - 20$ term by term in double precision gives a relative error above 0.1 (the largest term is about $4.3 \times 1 0^{7}$ and the answer about $2.1 \times 1 0^{- 9}$ ), while $1/ e^{20}$ by the same series has a relative error below $1 0^{- 14}$ .
Scope: The series is summed left to right until the terms stop changing the sum, about 100 terms. The error depends on rounding in the largest terms, so its exact size is not claimed, only that it exceeds 10% here.
Reference: $e^{- 20} = 2.061153622438558 \times 1 0^{- 9}$ from the libm exp (correctly rounded to within an ulp); Forsythe, Malcolm and Moler, Computer Methods for Mathematical Computations (1977), section 1.3, the textbook example.

claims/math/exp_series_cancellation.viv#1d1dae8887