Average Error: 5.8 → 0.9
Time: 11.6s
Precision: binary64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(t - z\right)\right)\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \left(t - z\right)\right)
(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 a)))
   (* (/ (cbrt y) (cbrt a)) (- t z)))))
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(a))) * ((cbrt(y) / cbrt(a)) * (t - z)));
}

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

Original5.8
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 5.8

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

  2. Simplified2.2

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

  4. Applied add-cube-cbrt_binary64_48872.7

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

  5. Applied add-cube-cbrt_binary64_48872.8

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

  6. Applied times-frac_binary64_48582.8

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

  7. Applied associate-*l*_binary64_47930.9

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

  8. Final simplification0.9

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

Reproduce

herbie shell --seed 2021176 
(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)))