Average Error: 43.6 → 0.6
Time: 6.1s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{1}{1}}{\frac{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot \left(a \cdot c\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{1}{1}}{\frac{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot \left(a \cdot c\right)}}
double f(double a, double b, double c) {
        double r109167 = b;
        double r109168 = -r109167;
        double r109169 = r109167 * r109167;
        double r109170 = 3.0;
        double r109171 = a;
        double r109172 = r109170 * r109171;
        double r109173 = c;
        double r109174 = r109172 * r109173;
        double r109175 = r109169 - r109174;
        double r109176 = sqrt(r109175);
        double r109177 = r109168 + r109176;
        double r109178 = r109177 / r109172;
        return r109178;
}
↓
double f(double a, double b, double c) {
        double r109179 = 1.0;
        double r109180 = r109179 / r109179;
        double r109181 = 3.0;
        double r109182 = a;
        double r109183 = r109181 * r109182;
        double r109184 = b;
        double r109185 = -r109184;
        double r109186 = r109184 * r109184;
        double r109187 = c;
        double r109188 = r109183 * r109187;
        double r109189 = r109186 - r109188;
        double r109190 = sqrt(r109189);
        double r109191 = r109185 - r109190;
        double r109192 = r109183 * r109191;
        double r109193 = r109182 * r109187;
        double r109194 = r109181 * r109193;
        double r109195 = r109192 / r109194;
        double r109196 = r109180 / r109195;
        return r109196;
}

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 43.6

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

  3. Applied flip-+43.5

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

  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 *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

  9. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

  10. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  11. Applied associate-/l*0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{3 \cdot a}{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2020056 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))