Average Error: 6.3 → 1.3
Time: 20.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x - \frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x - \frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}
double f(double x, double y, double z, double t, double a) {
        double r18253057 = x;
        double r18253058 = y;
        double r18253059 = z;
        double r18253060 = t;
        double r18253061 = r18253059 - r18253060;
        double r18253062 = r18253058 * r18253061;
        double r18253063 = a;
        double r18253064 = r18253062 / r18253063;
        double r18253065 = r18253057 - r18253064;
        return r18253065;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r18253066 = x;
        double r18253067 = y;
        double r18253068 = a;
        double r18253069 = cbrt(r18253068);
        double r18253070 = z;
        double r18253071 = t;
        double r18253072 = r18253070 - r18253071;
        double r18253073 = cbrt(r18253072);
        double r18253074 = r18253069 / r18253073;
        double r18253075 = r18253074 * r18253074;
        double r18253076 = r18253067 / r18253075;
        double r18253077 = r18253073 / r18253069;
        double r18253078 = r18253076 * r18253077;
        double r18253079 = r18253066 - r18253078;
        return r18253079;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.3
Target0.8
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.3

    \[x - \frac{y \cdot \left(z - t\right)}{a}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt6.8

    \[\leadsto x - \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

  4. Applied times-frac3.2

    \[\leadsto x - \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity3.2

    \[\leadsto x - \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{\color{blue}{1 \cdot a}}}\]

  7. Applied cbrt-prod3.2

    \[\leadsto x - \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{a}}}\]

  8. Applied add-cube-cbrt3.3

    \[\leadsto x - \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}}\]

  9. Applied times-frac3.3

    \[\leadsto x - \frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\right)}\]

  10. Applied associate-*r*1.9

    \[\leadsto x - \color{blue}{\left(\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{1}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}}\]

  11. Simplified1.3

    \[\leadsto x - \color{blue}{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\]

  12. Final simplification1.3

    \[\leadsto x - \frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a}}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))