Average Error: 12.4 → 1.8
Time: 3.0s
Precision: binary64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[x - \left(\left(x \cdot \sqrt[3]{z}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
x - \left(\left(x \cdot \sqrt[3]{z}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}
(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))
↓
(FPCore (x y z)
 :precision binary64
 (-
  x
  (*
   (* (* x (cbrt z)) (/ (cbrt z) (* (cbrt y) (cbrt y))))
   (/ (cbrt z) (cbrt y)))))
double code(double x, double y, double z) {
	return (((double) (x * ((double) (y - z)))) / y);
}
↓
double code(double x, double y, double z) {
	return ((double) (x - ((double) (((double) (((double) (x * ((double) cbrt(z)))) * (((double) cbrt(z)) / ((double) (((double) cbrt(y)) * ((double) cbrt(y))))))) * (((double) cbrt(z)) / ((double) cbrt(y)))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.4
Target3.3
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Initial program 12.4

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

  2. Simplified3.6

    \[\leadsto \color{blue}{x - x \cdot \frac{z}{y}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt4.0

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

  5. Applied add-cube-cbrt4.0

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

  6. Applied times-frac4.0

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

  7. Applied associate-*r*1.1

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

  8. Simplified1.1

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

  10. Applied associate-*r*1.8

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

  11. Final simplification1.8

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))