Average Error: 5.9 → 0.9
Time: 23.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)\right)\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r15872425 = x;
        double r15872426 = y;
        double r15872427 = z;
        double r15872428 = t;
        double r15872429 = r15872427 - r15872428;
        double r15872430 = r15872426 * r15872429;
        double r15872431 = a;
        double r15872432 = r15872430 / r15872431;
        double r15872433 = r15872425 + r15872432;
        return r15872433;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r15872434 = x;
        double r15872435 = y;
        double r15872436 = cbrt(r15872435);
        double r15872437 = a;
        double r15872438 = cbrt(r15872437);
        double r15872439 = r15872436 / r15872438;
        double r15872440 = z;
        double r15872441 = t;
        double r15872442 = r15872440 - r15872441;
        double r15872443 = r15872442 * r15872439;
        double r15872444 = r15872439 * r15872443;
        double r15872445 = r15872439 * r15872444;
        double r15872446 = r15872434 + r15872445;
        return r15872446;
}

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

Original5.9
Target0.8
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 5.9

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

  2. Simplified2.6

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

  4. Applied fma-udef2.6

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

  6. Applied add-cube-cbrt3.1

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

  7. Applied add-cube-cbrt3.2

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

  8. Applied times-frac3.2

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

  9. Applied associate-*r*1.0

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

  10. Simplified0.9

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

  11. Final simplification0.9

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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)))