Average Error: 6.2 → 2.3
Time: 5.4s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x - \left(\sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x - \left(\sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}\right)}
double f(double x, double y, double z, double t, double a) {
        double r377474 = x;
        double r377475 = y;
        double r377476 = z;
        double r377477 = t;
        double r377478 = r377476 - r377477;
        double r377479 = r377475 * r377478;
        double r377480 = a;
        double r377481 = r377479 / r377480;
        double r377482 = r377474 - r377481;
        return r377482;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r377483 = x;
        double r377484 = y;
        double r377485 = cbrt(r377484);
        double r377486 = r377485 * r377485;
        double r377487 = a;
        double r377488 = cbrt(r377487);
        double r377489 = r377486 / r377488;
        double r377490 = r377485 / r377488;
        double r377491 = z;
        double r377492 = t;
        double r377493 = r377491 - r377492;
        double r377494 = r377493 / r377488;
        double r377495 = r377490 * r377494;
        double r377496 = r377489 * r377495;
        double r377497 = cbrt(r377496);
        double r377498 = r377497 * r377497;
        double r377499 = r377498 * r377497;
        double r377500 = r377483 - r377499;
        return r377500;
}

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

Original6.2
Target0.7
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  3. Applied add-cube-cbrt6.6

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

  4. Applied times-frac3.1

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

  6. Applied add-cube-cbrt3.2

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

  7. Applied times-frac3.2

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

  8. Applied associate-*l*2.2

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

  10. Applied add-cube-cbrt2.3

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

  11. Final simplification2.3

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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