Average Error: 11.2 → 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 r457499 = x;
        double r457500 = y;
        double r457501 = z;
        double r457502 = r457500 - r457501;
        double r457503 = r457499 * r457502;
        double r457504 = t;
        double r457505 = r457504 - r457501;
        double r457506 = r457503 / r457505;
        return r457506;
}
↓
double f(double x, double y, double z, double t) {
        double r457507 = y;
        double r457508 = z;
        double r457509 = r457507 - r457508;
        double r457510 = cbrt(r457509);
        double r457511 = r457510 * r457510;
        double r457512 = t;
        double r457513 = r457512 - r457508;
        double r457514 = cbrt(r457513);
        double r457515 = r457514 * r457514;
        double r457516 = r457511 / r457515;
        double r457517 = x;
        double r457518 = r457514 / r457510;
        double r457519 = r457517 / r457518;
        double r457520 = r457516 * r457519;
        return r457520;
}

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

Original11.2
Target2.0
Herbie1.0
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Initial program 11.2

    \[\frac{x \cdot \left(y - z\right)}{t - z}\]
  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-cbrt3.0

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

  6. Applied add-cube-cbrt2.7

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

  7. Applied times-frac2.7

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

  8. Applied *-un-lft-identity2.7

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

  9. Applied times-frac1.1

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

  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 2019194 +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)))