Average Error: 28.9 → 0.5
Time: 5.4s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\frac{3 \cdot \left(a \cdot c\right)}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{\frac{3 \cdot \left(a \cdot c\right)}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}{a}
double f(double a, double b, double c) {
        double r112463 = b;
        double r112464 = -r112463;
        double r112465 = r112463 * r112463;
        double r112466 = 3.0;
        double r112467 = a;
        double r112468 = r112466 * r112467;
        double r112469 = c;
        double r112470 = r112468 * r112469;
        double r112471 = r112465 - r112470;
        double r112472 = sqrt(r112471);
        double r112473 = r112464 + r112472;
        double r112474 = r112473 / r112468;
        return r112474;
}
↓
double f(double a, double b, double c) {
        double r112475 = 3.0;
        double r112476 = a;
        double r112477 = c;
        double r112478 = r112476 * r112477;
        double r112479 = r112475 * r112478;
        double r112480 = r112479 / r112475;
        double r112481 = b;
        double r112482 = -r112481;
        double r112483 = r112481 * r112481;
        double r112484 = r112475 * r112476;
        double r112485 = r112484 * r112477;
        double r112486 = cbrt(r112485);
        double r112487 = r112486 * r112486;
        double r112488 = r112487 * r112486;
        double r112489 = r112483 - r112488;
        double r112490 = sqrt(r112489);
        double r112491 = r112482 - r112490;
        double r112492 = r112480 / r112491;
        double r112493 = r112492 / r112476;
        return r112493;
}

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 28.9

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

  3. Applied flip-+28.9

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

  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*0.6

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

  7. Simplified0.5

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

  9. Applied add-cube-cbrt0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2020027 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))