Average Error: 28.6 → 0.5
Time: 19.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(3 \cdot a\right) \cdot \frac{\frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(3 \cdot a\right) \cdot \frac{\frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3}}{a}
double f(double a, double b, double c) {
        double r98403 = b;
        double r98404 = -r98403;
        double r98405 = r98403 * r98403;
        double r98406 = 3.0;
        double r98407 = a;
        double r98408 = r98406 * r98407;
        double r98409 = c;
        double r98410 = r98408 * r98409;
        double r98411 = r98405 - r98410;
        double r98412 = sqrt(r98411);
        double r98413 = r98404 + r98412;
        double r98414 = r98413 / r98408;
        return r98414;
}
↓
double f(double a, double b, double c) {
        double r98415 = 3.0;
        double r98416 = a;
        double r98417 = r98415 * r98416;
        double r98418 = c;
        double r98419 = b;
        double r98420 = -r98419;
        double r98421 = r98419 * r98419;
        double r98422 = r98417 * r98418;
        double r98423 = r98421 - r98422;
        double r98424 = sqrt(r98423);
        double r98425 = r98420 - r98424;
        double r98426 = r98418 / r98425;
        double r98427 = r98426 / r98415;
        double r98428 = r98427 / r98416;
        double r98429 = r98417 * r98428;
        return r98429;
}

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

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

  6. Applied associate-/r*0.6

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

  7. Simplified0.6

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

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

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

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

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

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

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

  12. Applied times-frac0.5

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

  13. Applied times-frac0.5

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

  14. Applied times-frac0.5

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2019325 
(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)))