Average Error: 28.3 → 0.5
Time: 21.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \sqrt{c}\right) \cdot \sqrt{c}}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \sqrt{c}\right) \cdot \sqrt{c}}}}{3 \cdot a}
double f(double a, double b, double c) {
        double r68701 = b;
        double r68702 = -r68701;
        double r68703 = r68701 * r68701;
        double r68704 = 3.0;
        double r68705 = a;
        double r68706 = r68704 * r68705;
        double r68707 = c;
        double r68708 = r68706 * r68707;
        double r68709 = r68703 - r68708;
        double r68710 = sqrt(r68709);
        double r68711 = r68702 + r68710;
        double r68712 = r68711 / r68706;
        return r68712;
}
↓
double f(double a, double b, double c) {
        double r68713 = c;
        double r68714 = 3.0;
        double r68715 = a;
        double r68716 = r68714 * r68715;
        double r68717 = r68713 * r68716;
        double r68718 = b;
        double r68719 = -r68718;
        double r68720 = r68718 * r68718;
        double r68721 = sqrt(r68713);
        double r68722 = r68716 * r68721;
        double r68723 = r68722 * r68721;
        double r68724 = r68720 - r68723;
        double r68725 = sqrt(r68724);
        double r68726 = r68719 - r68725;
        double r68727 = r68717 / r68726;
        double r68728 = r68727 / r68716;
        return r68728;
}

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 28.3

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

  3. Applied flip-+28.3

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

  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. Using strategy rm

  6. Applied add-sqr-sqrt0.5

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

  7. Applied associate-*r*0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019209 
(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)))