The quadratic formula loses the small root

For $x^{2} - 1 0^{8} x + 1 = 0$ the textbook small root is off by 25%, while $c / q$ gets it to $1 0^{- 15}$ .

Small root of x^2 - bx + 1 by the textbook formula

(b - sqrt(b^2 - 4)) / 2

Quantity	Viv	Checked by
`rel err textbook`	0.254942	the textbook formula's small root is off by more than 10%
`rel err stable`	0	c / q gives the small root to 1e-15

quadratic_cancellation.viv

B = -1.0e8
C = 1.0
#: the small root to double precision: 1 / (the large root), with the large root 1e8 - 1e-8
SMALL_ROOT = 1.0e-8 * (1.0 + 1.0e-16)

disc = sqrt(B * B - 4.0 * C)
textbook = (-B - disc) / 2.0
big_root = (-B + disc) / 2.0
stable = C / big_root
rel_err_textbook = abs(textbook - SMALL_ROOT) / SMALL_ROOT
rel_err_stable = abs(stable - SMALL_ROOT) / SMALL_ROOT
show textbook
show stable
show rel_err_textbook
show rel_err_stable

#: the figure: the small root of x^2 - bx + 1 (b > 0) by both formulas as b grows, measured against c/q
N_SWEEP: Int = 15
fg_b = zeros(N_SWEEP)
fg_text = zeros(N_SWEEP)
for fg_k to N_SWEEP {
    fg_bk = pow(10.0, 1.0 + 0.5 * fg_k)
    fg_d = sqrt(fg_bk * fg_bk - 4.0 * C)
    fg_small_stable = C / ((fg_bk + fg_d) / 2.0)
    fg_b[fg_k] = fg_bk
    fg_text[fg_k] = abs((fg_bk - fg_d) / 2.0 - fg_small_stable) / fg_small_stable
}
figure("Small root of x^2 - bx + 1 by the textbook formula", "b", "relative error against c / q")
xscale("log")
yscale("log")
plot(fg_b, fg_text, "(b - sqrt(b^2 - 4)) / 2")
style("(b - sqrt(b^2 - 4)) / 2", "s2 circle")

test "the textbook formula's small root is off by more than 10%" {
    assert_gt(rel_err_textbook, 0.1)
}
test "c / q gives the small root to 1e-15" {
    assert_lt(rel_err_stable, 1.0e-15)
}

B = -1.0e8
C = 1.0
#: the small root to double precision: 1 / (the large root), with the large root 1e8 - 1e-8
SMALL_ROOT = 1.0e-8 * (1.0 + 1.0e-16)

disc = sqrt(B * B - 4.0 * C)
textbook = (-B - disc) / 2.0
big_root = (-B + disc) / 2.0
stable = C / big_root
rel_err_textbook = abs(textbook - SMALL_ROOT) / SMALL_ROOT
rel_err_stable = abs(stable - SMALL_ROOT) / SMALL_ROOT
show textbook
show stable
show rel_err_textbook
show rel_err_stable

#: the figure: the small root of x^2 - bx + 1 (b > 0) by both formulas as b grows, measured against c/q
N_SWEEP: Int = 15
fg_b = zeros(N_SWEEP)
fg_text = zeros(N_SWEEP)
for fg_k to N_SWEEP {
    fg_bk = pow(10.0, 1.0 + 0.5 * fg_k)
    fg_d = sqrt(fg_bk * fg_bk - 4.0 * C)
    fg_small_stable = C / ((fg_bk + fg_d) / 2.0)
    fg_b[fg_k] = fg_bk
    fg_text[fg_k] = abs((fg_bk - fg_d) / 2.0 - fg_small_stable) / fg_small_stable
}
figure("Small root of x^2 - bx + 1 by the textbook formula", "b", "relative error against c / q")
xscale("log")
yscale("log")
plot(fg_b, fg_text, "(b - sqrt(b^2 - 4)) / 2")
style("(b - sqrt(b^2 - 4)) / 2", "s2 circle")

test "the textbook formula's small root is off by more than 10%" {
    assert_gt(rel_err_textbook, 0.1)
}
test "c / q gives the small root to 1e-15" {
    assert_lt(rel_err_stable, 1.0e-15)
}

Output · native build · Oct 14, 2026 · 0.51 s

textbook: 7.45058e-09
stable: 1e-08
rel_err_textbook: 0.254942
rel_err_stable: 0

✓ the textbook formula's small root is off by more than 10%
✓ c / q gives the small root to 1e-15

Open in the workbench Atlas More math claims

Full statement: For $x^{2} - 1 0^{8} x + 1 = 0$ , the textbook formula $(- b - b^{2} - 4 c) /2$ gives the small root $1 0^{- 8}$ with a relative error above 0.1, while computing it as $c / q$ from the large root $q$ has a relative error below $1 0^{- 15}$ .
Scope: One equation, $b = - 1 0^{8}$ and $c = 1$ ; how much the textbook formula loses depends on $b^{2} / (4 c)$ .
Reference: The roots' product is c = 1 and their sum 10⁸, so the small root is $1/ (1 0^{8} - 1 0^{- 8} - \dots) = 1.0000000000000001 \times 1 0^{- 8}$ to double precision; Press et al., Numerical Recipes (3rd ed.), section 5.6, for the stable formula.

claims/math/quadratic_cancellation.viv#f73caacb6b