Average Error: 28.6 → 0.3
Time: 18.3s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c \cdot 1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c \cdot 1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r75474 = b;
        double r75475 = -r75474;
        double r75476 = r75474 * r75474;
        double r75477 = 3.0;
        double r75478 = a;
        double r75479 = r75477 * r75478;
        double r75480 = c;
        double r75481 = r75479 * r75480;
        double r75482 = r75476 - r75481;
        double r75483 = sqrt(r75482);
        double r75484 = r75475 + r75483;
        double r75485 = r75484 / r75479;
        return r75485;
}
↓
double f(double a, double b, double c) {
        double r75486 = c;
        double r75487 = 1.0;
        double r75488 = r75486 * r75487;
        double r75489 = b;
        double r75490 = -r75489;
        double r75491 = r75489 * r75489;
        double r75492 = 3.0;
        double r75493 = a;
        double r75494 = r75492 * r75493;
        double r75495 = r75494 * r75486;
        double r75496 = r75491 - r75495;
        double r75497 = sqrt(r75496);
        double r75498 = r75490 - r75497;
        double r75499 = r75488 / r75498;
        return r75499;
}

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 28.6

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

  3. Applied flip-+28.6

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

  4. Simplified0.6

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

  6. Applied associate-/r*0.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}}\]

  7. Simplified0.5

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

  8. Taylor expanded around 0 0.4

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019291 
(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)))