Average Error: 28.2 → 0.3
Time: 2.6m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}} \cdot 2\]
double f(double a, double b, double c) {
        double r10618339 = b;
        double r10618340 = -r10618339;
        double r10618341 = r10618339 * r10618339;
        double r10618342 = 4.0;
        double r10618343 = a;
        double r10618344 = r10618342 * r10618343;
        double r10618345 = c;
        double r10618346 = r10618344 * r10618345;
        double r10618347 = r10618341 - r10618346;
        double r10618348 = sqrt(r10618347);
        double r10618349 = r10618340 + r10618348;
        double r10618350 = 2.0;
        double r10618351 = r10618350 * r10618343;
        double r10618352 = r10618349 / r10618351;
        return r10618352;
}
↓
double f(double a, double b, double c) {
        double r10618353 = c;
        double r10618354 = b;
        double r10618355 = -r10618354;
        double r10618356 = a;
        double r10618357 = r10618353 * r10618356;
        double r10618358 = -4.0;
        double r10618359 = r10618354 * r10618354;
        double r10618360 = fma(r10618357, r10618358, r10618359);
        double r10618361 = sqrt(r10618360);
        double r10618362 = r10618355 - r10618361;
        double r10618363 = r10618353 / r10618362;
        double r10618364 = 2.0;
        double r10618365 = r10618363 * r10618364;
        return r10618365;
}


\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}} \cdot 2

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.2

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

  3. Applied flip-+28.2

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

  4. Applied associate-/l/28.2

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

  5. Simplified0.4

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

  7. Applied associate-/r*0.3

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

  8. Simplified0.3

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

  10. Applied *-un-lft-identity0.3

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

  11. Applied times-frac0.3

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :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)))