Average Error: 28.5 → 0.5
Time: 10.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{4 \cdot \left(a \cdot c\right) + b \cdot \left(b - b\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right)}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r37373 = b;
        double r37374 = -r37373;
        double r37375 = r37373 * r37373;
        double r37376 = 4.0;
        double r37377 = a;
        double r37378 = r37376 * r37377;
        double r37379 = c;
        double r37380 = r37378 * r37379;
        double r37381 = r37375 - r37380;
        double r37382 = sqrt(r37381);
        double r37383 = r37374 + r37382;
        double r37384 = 2.0;
        double r37385 = r37384 * r37377;
        double r37386 = r37383 / r37385;
        return r37386;
}
↓
double f(double a, double b, double c) {
        double r37387 = 4.0;
        double r37388 = a;
        double r37389 = c;
        double r37390 = r37388 * r37389;
        double r37391 = r37387 * r37390;
        double r37392 = b;
        double r37393 = r37392 - r37392;
        double r37394 = r37392 * r37393;
        double r37395 = r37391 + r37394;
        double r37396 = -r37392;
        double r37397 = r37387 * r37388;
        double r37398 = r37389 * r37397;
        double r37399 = -r37398;
        double r37400 = fma(r37392, r37392, r37399);
        double r37401 = sqrt(r37400);
        double r37402 = r37396 - r37401;
        double r37403 = r37395 / r37402;
        double r37404 = 2.0;
        double r37405 = r37404 * r37388;
        double r37406 = r37403 / r37405;
        return r37406;
}

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

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

  6. Applied fma-neg0.5

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

  7. Simplified0.5

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

  9. Applied *-un-lft-identity0.5

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

  10. Applied sqrt-prod0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

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)))