Average Error: 12.4 → 1.1
Time: 2.6s
Precision: binary64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}
(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (/ (* (cbrt (- y z)) (cbrt (- y z))) (* (cbrt y) (cbrt y)))
  (/ x (/ (cbrt y) (cbrt (- y z))))))
double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}
↓
double code(double x, double y, double z) {
	return ((cbrt(y - z) * cbrt(y - z)) / (cbrt(y) * cbrt(y))) * (x / (cbrt(y) / cbrt(y - z)));
}

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
Target3.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Initial program 12.4

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

  3. Applied associate-/l*_binary643.2

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

  5. Applied add-cube-cbrt_binary644.4

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

  6. Applied add-cube-cbrt_binary643.7

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

  7. Applied times-frac_binary643.7

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

  8. Applied *-un-lft-identity_binary643.7

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]

  9. Applied times-frac_binary641.1

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}} \cdot \frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}\]

  10. Simplified1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2020224 
(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))