Average Error: 12.1 → 1.1
Time: 3.8s
Precision: binary64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\left(x \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}\right)\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{t - z}}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\left(x \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}}\right)\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{t - z}}
double code(double x, double y, double z, double t) {
	return (((double) (x * ((double) (y - z)))) / ((double) (t - z)));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (((double) (x * ((double) (((double) cbrt(((double) (y - z)))) * (((double) cbrt(((double) (y - z)))) / ((double) (((double) cbrt(((double) (t - z)))) * ((double) cbrt(((double) (t - z))))))))))) * (((double) cbrt(((double) (y - z)))) / ((double) cbrt(((double) (t - z)))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

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Target

Original12.1
Target2.0
Herbie1.1
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Initial program 12.1

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

  2. Simplified2.0

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

  4. Applied add-cube-cbrt3.1

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

  5. Applied add-cube-cbrt2.8

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

  6. Applied times-frac2.8

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

  7. Applied associate-*r*1.0

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

  8. Simplified1.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (/ x (/ (- t z) (- y z)))

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