Average Error: 7.3 → 1.9
Time: 6.4s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{\frac{2}{y - t}} \cdot \sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\left(x \cdot \frac{\sqrt[3]{\frac{2}{y - t}} \cdot \sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}
double code(double x, double y, double z, double t) {
	return (((double) (x * 2.0)) / ((double) (((double) (y * z)) - ((double) (t * z)))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (x * (((double) (((double) cbrt((2.0 / ((double) (y - t))))) * ((double) cbrt((2.0 / ((double) (y - t))))))) / ((double) (((double) cbrt(z)) * ((double) cbrt(z))))))) * (((double) cbrt((2.0 / ((double) (y - t))))) / ((double) cbrt(z)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.3
Target2.2
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 7.3

    \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

  2. Simplified6.3

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

  4. Applied div-inv6.5

    \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot \frac{1}{z \cdot \left(y - t\right)}}\]
  5. Using strategy rm

  6. Applied associate-*l*6.5

    \[\leadsto \color{blue}{x \cdot \left(2 \cdot \frac{1}{z \cdot \left(y - t\right)}\right)}\]

  7. Simplified6.0

    \[\leadsto x \cdot \color{blue}{\frac{\frac{2}{y - t}}{z}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt6.7

    \[\leadsto x \cdot \frac{\frac{2}{y - t}}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

  10. Applied add-cube-cbrt6.8

    \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{y - t}} \cdot \sqrt[3]{\frac{2}{y - t}}\right) \cdot \sqrt[3]{\frac{2}{y - t}}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

  11. Applied times-frac6.8

    \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{\frac{2}{y - t}} \cdot \sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}\right)}\]

  12. Applied associate-*r*1.9

    \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{\frac{2}{y - t}} \cdot \sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}}\]

  13. Final simplification1.9

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{\frac{2}{y - t}} \cdot \sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))