Average Error: 6.1 → 1.2
Time: 45.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{a}}}{\frac{\sqrt[3]{\sqrt[3]{a}}}{z - t}}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{a}}}{\frac{\sqrt[3]{\sqrt[3]{a}}}{z - t}}
double f(double x, double y, double z, double t, double a) {
        double r21768695 = x;
        double r21768696 = y;
        double r21768697 = z;
        double r21768698 = t;
        double r21768699 = r21768697 - r21768698;
        double r21768700 = r21768696 * r21768699;
        double r21768701 = a;
        double r21768702 = r21768700 / r21768701;
        double r21768703 = r21768695 - r21768702;
        return r21768703;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r21768704 = x;
        double r21768705 = y;
        double r21768706 = cbrt(r21768705);
        double r21768707 = r21768706 * r21768706;
        double r21768708 = a;
        double r21768709 = cbrt(r21768708);
        double r21768710 = r21768709 * r21768709;
        double r21768711 = cbrt(r21768710);
        double r21768712 = r21768711 * r21768709;
        double r21768713 = r21768707 / r21768712;
        double r21768714 = r21768706 / r21768709;
        double r21768715 = cbrt(r21768709);
        double r21768716 = z;
        double r21768717 = t;
        double r21768718 = r21768716 - r21768717;
        double r21768719 = r21768715 / r21768718;
        double r21768720 = r21768714 / r21768719;
        double r21768721 = r21768713 * r21768720;
        double r21768722 = r21768704 - r21768721;
        return r21768722;
}

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

Original6.1
Target0.7
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  3. Applied associate-/l*5.5

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

  5. Applied *-un-lft-identity5.5

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

  6. Applied add-cube-cbrt6.0

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

  7. Applied times-frac6.0

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

  8. Applied associate-/r*2.8

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

  9. Simplified2.8

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

  11. Applied *-un-lft-identity2.8

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

  12. Applied add-cube-cbrt2.9

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

  13. Applied cbrt-prod2.9

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

  14. Applied times-frac2.9

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

  15. Applied add-cube-cbrt3.0

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

  16. Applied times-frac3.0

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

  17. Applied times-frac1.2

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

  18. Simplified1.2

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

  19. Final simplification1.2

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

Reproduce

herbie shell --seed 2019171 
(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)))