Average Error: 28.5 → 0.6
Time: 21.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(3 \cdot a\right) \cdot c}{3 \cdot \left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\left(3 \cdot a\right) \cdot c}{3 \cdot \left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)\right)}
double f(double a, double b, double c) {
        double r70379 = b;
        double r70380 = -r70379;
        double r70381 = r70379 * r70379;
        double r70382 = 3.0;
        double r70383 = a;
        double r70384 = r70382 * r70383;
        double r70385 = c;
        double r70386 = r70384 * r70385;
        double r70387 = r70381 - r70386;
        double r70388 = sqrt(r70387);
        double r70389 = r70380 + r70388;
        double r70390 = r70389 / r70384;
        return r70390;
}
↓
double f(double a, double b, double c) {
        double r70391 = 3.0;
        double r70392 = a;
        double r70393 = r70391 * r70392;
        double r70394 = c;
        double r70395 = r70393 * r70394;
        double r70396 = b;
        double r70397 = -r70396;
        double r70398 = r70396 * r70396;
        double r70399 = r70392 * r70394;
        double r70400 = r70399 * r70391;
        double r70401 = r70398 - r70400;
        double r70402 = sqrt(r70401);
        double r70403 = r70397 - r70402;
        double r70404 = r70392 * r70403;
        double r70405 = r70391 * r70404;
        double r70406 = r70395 / r70405;
        return r70406;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.5

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.5

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

  5. Taylor expanded around 0 0.5

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]

  6. Simplified0.5

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}}}{3 \cdot a}\]
  7. Using strategy rm

  8. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(0 + \left(3 \cdot a\right) \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}{3 \cdot a}\]

  9. Applied associate-/l*0.5

    \[\leadsto \color{blue}{\frac{0 + \left(3 \cdot a\right) \cdot c}{\frac{3 \cdot a}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}}}}}\]

  10. Simplified0.4

    \[\leadsto \frac{0 + \left(3 \cdot a\right) \cdot c}{\color{blue}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)}}\]
  11. Using strategy rm

  12. Applied associate-*l*0.6

    \[\leadsto \frac{0 + \left(3 \cdot a\right) \cdot c}{\color{blue}{3 \cdot \left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)\right)}}\]

  13. Final simplification0.6

    \[\leadsto \frac{\left(3 \cdot a\right) \cdot c}{3 \cdot \left(a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3}\right)\right)}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))