Average Error: 28.7 → 0.3
Time: 19.8s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}
double f(double a, double b, double c) {
        double r82799 = b;
        double r82800 = -r82799;
        double r82801 = r82799 * r82799;
        double r82802 = 3.0;
        double r82803 = a;
        double r82804 = r82802 * r82803;
        double r82805 = c;
        double r82806 = r82804 * r82805;
        double r82807 = r82801 - r82806;
        double r82808 = sqrt(r82807);
        double r82809 = r82800 + r82808;
        double r82810 = r82809 / r82804;
        return r82810;
}
↓
double f(double a, double b, double c) {
        double r82811 = c;
        double r82812 = b;
        double r82813 = r82812 * r82812;
        double r82814 = 3.0;
        double r82815 = a;
        double r82816 = r82814 * r82815;
        double r82817 = r82816 * r82811;
        double r82818 = r82813 - r82817;
        double r82819 = sqrt(r82818);
        double r82820 = r82812 + r82819;
        double r82821 = -r82820;
        double r82822 = r82811 / r82821;
        return r82822;
}

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 28.7

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

  3. Applied flip-+28.7

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

  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 add-cbrt-cube0.6

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

  7. Simplified0.6

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

  9. Applied *-un-lft-identity0.6

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{3}}}\right)}}}{3 \cdot a}\]

  10. Applied *-un-lft-identity0.6

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{3}}}\right)}}{3 \cdot a}\]

  11. Applied times-frac0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{3}}}}}}{3 \cdot a}\]

  12. Simplified0.6

    \[\leadsto \frac{\color{blue}{1} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{3}}}}}{3 \cdot a}\]

  13. Simplified0.5

    \[\leadsto \frac{1 \cdot \color{blue}{\frac{3 \cdot a}{\frac{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{c}}}}{3 \cdot a}\]
  14. Using strategy rm

  15. Applied clear-num0.4

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

  16. Simplified0.4

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(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)))