Average Error: 28.5 → 0.5
Time: 6.4s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(0 + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\left(0 + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r39378 = b;
        double r39379 = -r39378;
        double r39380 = r39378 * r39378;
        double r39381 = 4.0;
        double r39382 = a;
        double r39383 = r39381 * r39382;
        double r39384 = c;
        double r39385 = r39383 * r39384;
        double r39386 = r39380 - r39385;
        double r39387 = sqrt(r39386);
        double r39388 = r39379 + r39387;
        double r39389 = 2.0;
        double r39390 = r39389 * r39382;
        double r39391 = r39388 / r39390;
        return r39391;
}
↓
double f(double a, double b, double c) {
        double r39392 = 0.0;
        double r39393 = 4.0;
        double r39394 = a;
        double r39395 = c;
        double r39396 = r39394 * r39395;
        double r39397 = r39393 * r39396;
        double r39398 = r39392 + r39397;
        double r39399 = 1.0;
        double r39400 = b;
        double r39401 = -r39400;
        double r39402 = r39392 - r39397;
        double r39403 = fma(r39400, r39400, r39402);
        double r39404 = sqrt(r39403);
        double r39405 = r39401 - r39404;
        double r39406 = r39399 / r39405;
        double r39407 = r39398 * r39406;
        double r39408 = 2.0;
        double r39409 = r39408 * r39394;
        double r39410 = r39407 / r39409;
        return r39410;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.5

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.5

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied fma-neg0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)}}}}{2 \cdot a}\]

  7. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{0 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
  8. Using strategy rm

  9. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(0 + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a}\]

  10. Final simplification0.5

    \[\leadsto \frac{\left(0 + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, 0 - 4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))