Average Error: 52.3 → 0.4
Time: 7.4s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \sqrt{c}\right) \cdot \sqrt{c}}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(\left(3 \cdot a\right) \cdot \sqrt{c}\right) \cdot \sqrt{c}}}}{3 \cdot a}
double f(double a, double b, double c) {
        double r102085 = b;
        double r102086 = -r102085;
        double r102087 = r102085 * r102085;
        double r102088 = 3.0;
        double r102089 = a;
        double r102090 = r102088 * r102089;
        double r102091 = c;
        double r102092 = r102090 * r102091;
        double r102093 = r102087 - r102092;
        double r102094 = sqrt(r102093);
        double r102095 = r102086 + r102094;
        double r102096 = r102095 / r102090;
        return r102096;
}
↓
double f(double a, double b, double c) {
        double r102097 = b;
        double r102098 = 2.0;
        double r102099 = pow(r102097, r102098);
        double r102100 = r102099 - r102099;
        double r102101 = 3.0;
        double r102102 = a;
        double r102103 = r102101 * r102102;
        double r102104 = c;
        double r102105 = r102103 * r102104;
        double r102106 = r102100 + r102105;
        double r102107 = -r102097;
        double r102108 = r102097 * r102097;
        double r102109 = sqrt(r102104);
        double r102110 = r102103 * r102109;
        double r102111 = r102110 * r102109;
        double r102112 = r102108 - r102111;
        double r102113 = sqrt(r102112);
        double r102114 = r102107 - r102113;
        double r102115 = r102106 / r102114;
        double r102116 = r102115 / r102103;
        return r102116;
}

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 52.3

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

  3. Applied flip-+52.3

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

  4. Simplified0.5

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

  6. Applied associate-*r*0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied associate-*r*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020033 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))