Average Error: 6.3 → 1.5
Time: 16.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}
double f(double x, double y, double z, double t, double a) {
        double r13873829 = x;
        double r13873830 = y;
        double r13873831 = z;
        double r13873832 = t;
        double r13873833 = r13873831 - r13873832;
        double r13873834 = r13873830 * r13873833;
        double r13873835 = a;
        double r13873836 = r13873834 / r13873835;
        double r13873837 = r13873829 + r13873836;
        return r13873837;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13873838 = x;
        double r13873839 = y;
        double r13873840 = a;
        double r13873841 = cbrt(r13873840);
        double r13873842 = z;
        double r13873843 = t;
        double r13873844 = r13873842 - r13873843;
        double r13873845 = cbrt(r13873844);
        double r13873846 = r13873841 / r13873845;
        double r13873847 = r13873846 * r13873846;
        double r13873848 = r13873839 / r13873847;
        double r13873849 = r13873848 / r13873846;
        double r13873850 = r13873838 + r13873849;
        return r13873850;
}

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

Original6.3
Target0.6
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied associate-/l*5.8

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

  5. Applied add-cube-cbrt6.2

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

  6. Applied add-cube-cbrt6.4

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

  7. Applied times-frac6.3

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

  8. Applied associate-/r*1.5

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

  9. Simplified1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019174 +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)))