Average Error: 5.9 → 1.0
Time: 22.8s
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 r12335115 = x;
        double r12335116 = y;
        double r12335117 = z;
        double r12335118 = t;
        double r12335119 = r12335117 - r12335118;
        double r12335120 = r12335116 * r12335119;
        double r12335121 = a;
        double r12335122 = r12335120 / r12335121;
        double r12335123 = r12335115 - r12335122;
        return r12335123;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r12335124 = x;
        double r12335125 = z;
        double r12335126 = t;
        double r12335127 = r12335125 - r12335126;
        double r12335128 = y;
        double r12335129 = cbrt(r12335128);
        double r12335130 = a;
        double r12335131 = cbrt(r12335130);
        double r12335132 = r12335129 / r12335131;
        double r12335133 = r12335127 * r12335132;
        double r12335134 = r12335132 * r12335132;
        double r12335135 = r12335133 * r12335134;
        double r12335136 = r12335124 - r12335135;
        return r12335136;
}

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
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 5.9

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

  3. Applied associate-/l*5.7

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

  5. Applied *-un-lft-identity5.7

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

  6. Applied add-cube-cbrt6.2

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

  7. Applied times-frac6.2

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

  8. Applied add-cube-cbrt6.3

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

  9. Applied times-frac2.1

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

  10. Simplified2.1

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

  11. Simplified1.0

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

  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 2019172 +o rules:numerics
(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)))