Average Error: 6.3 → 1.3
Time: 20.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x - \frac{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x - \frac{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}
double f(double x, double y, double z, double t, double a) {
        double r20116584 = x;
        double r20116585 = y;
        double r20116586 = z;
        double r20116587 = t;
        double r20116588 = r20116586 - r20116587;
        double r20116589 = r20116585 * r20116588;
        double r20116590 = a;
        double r20116591 = r20116589 / r20116590;
        double r20116592 = r20116584 - r20116591;
        return r20116592;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r20116593 = x;
        double r20116594 = y;
        double r20116595 = a;
        double r20116596 = cbrt(r20116595);
        double r20116597 = z;
        double r20116598 = t;
        double r20116599 = r20116597 - r20116598;
        double r20116600 = cbrt(r20116599);
        double r20116601 = r20116596 / r20116600;
        double r20116602 = r20116601 * r20116601;
        double r20116603 = r20116594 / r20116602;
        double r20116604 = r20116603 / r20116601;
        double r20116605 = r20116593 - r20116604;
        return r20116605;
}

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 associate-/l*5.3

    \[\leadsto x - \color{blue}{\frac{y}{\frac{a}{z - t}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt5.7

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

  6. Applied add-cube-cbrt5.9

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

  7. Applied times-frac5.9

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

  8. Applied associate-/r*1.3

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

  9. Simplified1.3

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

  10. Final simplification1.3

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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)))