Average Error: 6.3 → 0.9
Time: 15.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} + x\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} + x
double f(double x, double y, double z, double t, double a) {
        double r16704436 = x;
        double r16704437 = y;
        double r16704438 = z;
        double r16704439 = t;
        double r16704440 = r16704438 - r16704439;
        double r16704441 = r16704437 * r16704440;
        double r16704442 = a;
        double r16704443 = r16704441 / r16704442;
        double r16704444 = r16704436 + r16704443;
        return r16704444;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r16704445 = z;
        double r16704446 = t;
        double r16704447 = r16704445 - r16704446;
        double r16704448 = y;
        double r16704449 = cbrt(r16704448);
        double r16704450 = a;
        double r16704451 = cbrt(r16704450);
        double r16704452 = r16704449 / r16704451;
        double r16704453 = r16704447 * r16704452;
        double r16704454 = r16704449 * r16704449;
        double r16704455 = r16704451 * r16704451;
        double r16704456 = r16704454 / r16704455;
        double r16704457 = r16704453 * r16704456;
        double r16704458 = x;
        double r16704459 = r16704457 + r16704458;
        return r16704459;
}

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
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  2. Taylor expanded around 0 6.3

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

  3. Simplified2.6

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

  5. Applied add-cube-cbrt3.1

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

  6. Applied add-cube-cbrt3.3

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

  7. Applied times-frac3.3

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

  8. Applied associate-*l*0.9

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

  9. Final simplification0.9

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :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)))