Average Error: 28.8 → 0.5
Time: 12.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{a} \cdot \frac{a}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{a} \cdot \frac{a}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r94653 = b;
        double r94654 = -r94653;
        double r94655 = r94653 * r94653;
        double r94656 = 3.0;
        double r94657 = a;
        double r94658 = r94656 * r94657;
        double r94659 = c;
        double r94660 = r94658 * r94659;
        double r94661 = r94655 - r94660;
        double r94662 = sqrt(r94661);
        double r94663 = r94654 + r94662;
        double r94664 = r94663 / r94658;
        return r94664;
}
↓
double f(double a, double b, double c) {
        double r94665 = c;
        double r94666 = a;
        double r94667 = r94665 / r94666;
        double r94668 = b;
        double r94669 = -r94668;
        double r94670 = r94668 * r94668;
        double r94671 = 3.0;
        double r94672 = r94671 * r94666;
        double r94673 = r94672 * r94665;
        double r94674 = r94670 - r94673;
        double r94675 = sqrt(r94674);
        double r94676 = r94669 - r94675;
        double r94677 = r94666 / r94676;
        double r94678 = r94667 * r94677;
        return r94678;
}

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 28.8

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

  3. Applied flip-+28.8

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

  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 *-un-lft-identity0.6

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

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

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

  8. Applied times-frac0.6

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

  9. Simplified0.6

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

  10. Simplified0.6

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

  12. Applied associate-/r*0.6

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2019304 
(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)))