Average Error: 28.6 → 0.3
Time: 19.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}^{3}}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right)}^{3}}}}
double f(double a, double b, double c) {
        double r86700 = b;
        double r86701 = -r86700;
        double r86702 = r86700 * r86700;
        double r86703 = 3.0;
        double r86704 = a;
        double r86705 = r86703 * r86704;
        double r86706 = c;
        double r86707 = r86705 * r86706;
        double r86708 = r86702 - r86707;
        double r86709 = sqrt(r86708);
        double r86710 = r86701 + r86709;
        double r86711 = r86710 / r86705;
        return r86711;
}
↓
double f(double a, double b, double c) {
        double r86712 = c;
        double r86713 = b;
        double r86714 = -r86713;
        double r86715 = 2.0;
        double r86716 = pow(r86713, r86715);
        double r86717 = 3.0;
        double r86718 = a;
        double r86719 = r86718 * r86712;
        double r86720 = r86717 * r86719;
        double r86721 = r86716 - r86720;
        double r86722 = 3.0;
        double r86723 = pow(r86721, r86722);
        double r86724 = cbrt(r86723);
        double r86725 = sqrt(r86724);
        double r86726 = r86714 - r86725;
        double r86727 = r86712 / r86726;
        return r86727;
}

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*0.6

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

  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-sqr-sqrt0.5

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

  10. Applied associate-*r*0.5

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

  12. Applied add-cbrt-cube0.5

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

  13. Simplified0.5

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019294 
(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)))