Average Error: 6.1 → 1.1
Time: 9.5s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x - \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}}}{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}} \cdot \frac{\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}}{\sqrt[3]{\sqrt[3]{a}}}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x - \frac{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a}}}{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}} \cdot \frac{\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}}{\sqrt[3]{\sqrt[3]{a}}}
(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))
↓
(FPCore (x y z t a)
 :precision binary64
 (-
  x
  (*
   (/ (/ (* (cbrt y) (cbrt y)) (cbrt a)) (* (cbrt (cbrt a)) (cbrt (cbrt a))))
   (/ (* (- z t) (/ (cbrt y) (cbrt a))) (cbrt (cbrt a))))))
double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}
↓
double code(double x, double y, double z, double t, double a) {
	return x - ((((cbrt(y) * cbrt(y)) / cbrt(a)) / (cbrt(cbrt(a)) * cbrt(cbrt(a)))) * (((z - t) * (cbrt(y) / cbrt(a))) / cbrt(cbrt(a))));
}

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

Original6.1
Target0.7
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  3. Applied add-cube-cbrt_binary64_86386.6

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

  4. Applied associate-/r*_binary64_85476.6

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

  5. Simplified3.1

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

  7. Applied add-cube-cbrt_binary64_86383.3

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

  8. Applied times-frac_binary64_86093.3

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

  9. Applied associate-*l*_binary64_85442.2

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

  10. Simplified2.2

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

  12. Applied add-cube-cbrt_binary64_86382.3

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

  13. Applied times-frac_binary64_86091.1

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

  14. Final simplification1.1

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

Reproduce

herbie shell --seed 2021098 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))