Average Error: 6.2 → 1.0
Time: 18.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)
double f(double x, double y, double z, double t, double a) {
        double r13054371 = x;
        double r13054372 = y;
        double r13054373 = z;
        double r13054374 = t;
        double r13054375 = r13054373 - r13054374;
        double r13054376 = r13054372 * r13054375;
        double r13054377 = a;
        double r13054378 = r13054376 / r13054377;
        double r13054379 = r13054371 + r13054378;
        return r13054379;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13054380 = x;
        double r13054381 = z;
        double r13054382 = t;
        double r13054383 = r13054381 - r13054382;
        double r13054384 = y;
        double r13054385 = cbrt(r13054384);
        double r13054386 = a;
        double r13054387 = cbrt(r13054386);
        double r13054388 = r13054385 / r13054387;
        double r13054389 = r13054383 * r13054388;
        double r13054390 = r13054388 * r13054388;
        double r13054391 = r13054389 * r13054390;
        double r13054392 = r13054380 + r13054391;
        return r13054392;
}

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
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.2

    \[x + \frac{y \cdot \left(z - t\right)}{a}\]
  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-cbrt5.9

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

  7. Applied times-frac5.9

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

  8. Applied add-cube-cbrt6.1

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

  9. Applied times-frac2.2

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

  10. Simplified2.2

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

  11. Simplified1.0

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

  12. Final simplification1.0

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

Reproduce

herbie shell --seed 2019169 +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)))