Average Error: 11.9 → 1.0
Time: 12.6s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}} \cdot \frac{x}{\frac{\sqrt[3]{t - z}}{\sqrt[3]{y - z}}}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{t - z} \cdot \sqrt[3]{t - z}} \cdot \frac{x}{\frac{\sqrt[3]{t - z}}{\sqrt[3]{y - z}}}
double f(double x, double y, double z, double t) {
        double r26360322 = x;
        double r26360323 = y;
        double r26360324 = z;
        double r26360325 = r26360323 - r26360324;
        double r26360326 = r26360322 * r26360325;
        double r26360327 = t;
        double r26360328 = r26360327 - r26360324;
        double r26360329 = r26360326 / r26360328;
        return r26360329;
}
↓
double f(double x, double y, double z, double t) {
        double r26360330 = y;
        double r26360331 = z;
        double r26360332 = r26360330 - r26360331;
        double r26360333 = cbrt(r26360332);
        double r26360334 = r26360333 * r26360333;
        double r26360335 = t;
        double r26360336 = r26360335 - r26360331;
        double r26360337 = cbrt(r26360336);
        double r26360338 = r26360337 * r26360337;
        double r26360339 = r26360334 / r26360338;
        double r26360340 = x;
        double r26360341 = r26360337 / r26360333;
        double r26360342 = r26360340 / r26360341;
        double r26360343 = r26360339 * r26360342;
        return r26360343;
}

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

Original11.9
Target2.0
Herbie1.0
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Initial program 11.9

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

  3. Applied associate-/l*2.0

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

  5. Applied add-cube-cbrt3.1

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

  6. Applied add-cube-cbrt2.8

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

  7. Applied times-frac2.8

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

  8. Applied *-un-lft-identity2.8

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

  9. Applied times-frac1.0

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

  10. Simplified1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"

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

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