Average Error: 28.6 → 0.4
Time: 19.9s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}
double f(double a, double b, double c) {
        double r61559 = b;
        double r61560 = -r61559;
        double r61561 = r61559 * r61559;
        double r61562 = 3.0;
        double r61563 = a;
        double r61564 = r61562 * r61563;
        double r61565 = c;
        double r61566 = r61564 * r61565;
        double r61567 = r61561 - r61566;
        double r61568 = sqrt(r61567);
        double r61569 = r61560 + r61568;
        double r61570 = r61569 / r61564;
        return r61570;
}
↓
double f(double a, double b, double c) {
        double r61571 = 1.0;
        double r61572 = b;
        double r61573 = -r61572;
        double r61574 = r61572 * r61572;
        double r61575 = 3.0;
        double r61576 = a;
        double r61577 = r61575 * r61576;
        double r61578 = c;
        double r61579 = r61577 * r61578;
        double r61580 = r61574 - r61579;
        double r61581 = sqrt(r61580);
        double r61582 = r61573 - r61581;
        double r61583 = r61582 / r61578;
        double r61584 = r61571 / r61583;
        return r61584;
}

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 clear-num0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{0 + 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]

  7. Simplified0.5

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

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

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

  10. Applied times-frac0.6

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

  11. Applied add-sqr-sqrt0.6

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

  12. Applied times-frac0.5

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

  13. Applied associate-/l*0.5

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

  14. Simplified0.5

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

  16. Applied div-inv0.5

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

  17. Applied associate-/l*0.5

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

  18. Simplified0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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)))