Average Error: 6.0 → 2.3
Time: 5.0s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{z - t}{\sqrt[3]{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{y}}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}}\right)\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{z - t}{\sqrt[3]{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{y}}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}}\right)
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + (((double) (y * ((double) (z - t)))) / a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (x + ((double) ((((double) (((double) cbrt(y)) * ((double) cbrt(y)))) / ((double) cbrt(a))) * ((double) ((((double) (z - t)) / ((double) cbrt(((double) cbrt(a))))) * (((double) cbrt(y)) / ((double) pow(((double) cbrt(((double) cbrt(a)))), 5.0)))))))));
}

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

Original6.0
Target0.7
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.0

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

  3. Applied add-cube-cbrt6.4

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

  4. Applied times-frac3.2

    \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z - t}{\sqrt[3]{a}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt3.3

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

  7. Applied *-un-lft-identity3.3

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

  8. Applied times-frac3.3

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

  9. Applied associate-*r*3.0

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

  10. Simplified3.2

    \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{a} \cdot {\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}}} \cdot \frac{z - t}{\sqrt[3]{\sqrt[3]{a}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt3.2

    \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{a} \cdot {\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}} \cdot \frac{z - t}{\sqrt[3]{\sqrt[3]{a}}}\]

  13. Applied times-frac3.2

    \[\leadsto x + \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}}\right)} \cdot \frac{z - t}{\sqrt[3]{\sqrt[3]{a}}}\]

  14. Applied associate-*l*2.3

    \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}} \cdot \frac{z - t}{\sqrt[3]{\sqrt[3]{a}}}\right)}\]

  15. Simplified2.3

    \[\leadsto x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \color{blue}{\left(\frac{z - t}{\sqrt[3]{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{y}}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}}\right)}\]

  16. Final simplification2.3

    \[\leadsto x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(\frac{z - t}{\sqrt[3]{\sqrt[3]{a}}} \cdot \frac{\sqrt[3]{y}}{{\left(\sqrt[3]{\sqrt[3]{a}}\right)}^{5}}\right)\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :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)))