Average Error: 11.2 → 1.1
Time: 3.2s
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 r647691 = x;
        double r647692 = y;
        double r647693 = z;
        double r647694 = r647692 - r647693;
        double r647695 = r647691 * r647694;
        double r647696 = t;
        double r647697 = r647696 - r647693;
        double r647698 = r647695 / r647697;
        return r647698;
}
↓
double f(double x, double y, double z, double t) {
        double r647699 = y;
        double r647700 = z;
        double r647701 = r647699 - r647700;
        double r647702 = cbrt(r647701);
        double r647703 = r647702 * r647702;
        double r647704 = t;
        double r647705 = r647704 - r647700;
        double r647706 = cbrt(r647705);
        double r647707 = r647706 * r647706;
        double r647708 = r647703 / r647707;
        double r647709 = x;
        double r647710 = r647706 / r647702;
        double r647711 = r647709 / r647710;
        double r647712 = r647708 * r647711;
        return r647712;
}

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
Target2.2
Herbie1.1
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Initial program 11.2

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

  3. Applied associate-/l*2.2

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

  5. Applied add-cube-cbrt3.2

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

  6. Applied add-cube-cbrt2.9

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

  7. Applied times-frac2.9

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

  8. Applied *-un-lft-identity2.9

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

  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. Simplified1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2020049 
(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)))