Average Error: 5.9 → 0.9
Time: 22.8s
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 r14111127 = x;
        double r14111128 = y;
        double r14111129 = z;
        double r14111130 = t;
        double r14111131 = r14111129 - r14111130;
        double r14111132 = r14111128 * r14111131;
        double r14111133 = a;
        double r14111134 = r14111132 / r14111133;
        double r14111135 = r14111127 + r14111134;
        return r14111135;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14111136 = x;
        double r14111137 = y;
        double r14111138 = cbrt(r14111137);
        double r14111139 = a;
        double r14111140 = cbrt(r14111139);
        double r14111141 = r14111138 / r14111140;
        double r14111142 = z;
        double r14111143 = t;
        double r14111144 = r14111142 - r14111143;
        double r14111145 = r14111144 * r14111141;
        double r14111146 = r14111141 * r14111145;
        double r14111147 = r14111141 * r14111146;
        double r14111148 = r14111136 + r14111147;
        return r14111148;
}

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)))