Average Error: 6.2 → 0.8
Time: 19.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)
double f(double x, double y, double z, double t, double a) {
        double r16940079 = x;
        double r16940080 = y;
        double r16940081 = z;
        double r16940082 = t;
        double r16940083 = r16940081 - r16940082;
        double r16940084 = r16940080 * r16940083;
        double r16940085 = a;
        double r16940086 = r16940084 / r16940085;
        double r16940087 = r16940079 + r16940086;
        return r16940087;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r16940088 = x;
        double r16940089 = y;
        double r16940090 = cbrt(r16940089);
        double r16940091 = a;
        double r16940092 = cbrt(r16940091);
        double r16940093 = r16940090 / r16940092;
        double r16940094 = z;
        double r16940095 = t;
        double r16940096 = r16940094 - r16940095;
        double r16940097 = r16940096 * r16940093;
        double r16940098 = r16940097 * r16940093;
        double r16940099 = r16940093 * r16940098;
        double r16940100 = r16940088 + r16940099;
        return r16940100;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  2. Taylor expanded around 0 6.2

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

  3. Simplified2.5

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

  5. Applied add-cube-cbrt3.1

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

  6. Applied add-cube-cbrt3.2

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

  7. Applied times-frac3.2

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

  8. Applied associate-*r*1.0

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

  9. Simplified0.8

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

  10. Final simplification0.8

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

Reproduce

herbie shell --seed 2019169 
(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)))