Average Error: 28.9 → 0.6
Time: 7.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3 \cdot \frac{\frac{a \cdot c}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3 \cdot \frac{\frac{a \cdot c}{3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}
double f(double a, double b, double c) {
        double r112997 = b;
        double r112998 = -r112997;
        double r112999 = r112997 * r112997;
        double r113000 = 3.0;
        double r113001 = a;
        double r113002 = r113000 * r113001;
        double r113003 = c;
        double r113004 = r113002 * r113003;
        double r113005 = r112999 - r113004;
        double r113006 = sqrt(r113005);
        double r113007 = r112998 + r113006;
        double r113008 = r113007 / r113002;
        return r113008;
}
↓
double f(double a, double b, double c) {
        double r113009 = 3.0;
        double r113010 = a;
        double r113011 = c;
        double r113012 = r113010 * r113011;
        double r113013 = r113012 / r113009;
        double r113014 = b;
        double r113015 = -r113014;
        double r113016 = r113014 * r113014;
        double r113017 = r113009 * r113010;
        double r113018 = r113017 * r113011;
        double r113019 = r113016 - r113018;
        double r113020 = sqrt(r113019);
        double r113021 = r113015 - r113020;
        double r113022 = r113013 / r113021;
        double r113023 = r113009 * r113022;
        double r113024 = r113023 / r113010;
        return r113024;
}

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 *-un-lft-identity0.5

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

  10. Applied *-un-lft-identity0.5

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

  11. Applied times-frac0.5

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

  12. Applied times-frac0.6

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

  13. Simplified0.6

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

  14. Final simplification0.6

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

Reproduce

herbie shell --seed 2019354 
(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)))