Average Error: 10.8 → 1.9
Time: 27.1s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1082063369415516 \cdot 10^{-245}:\\ \;\;\;\;x \cdot \left(\frac{y}{t - z} - \frac{z}{t - z}\right)\\ \mathbf{elif}\;z \le 1.5991111356517593 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{t - z} \cdot \left(\sqrt[3]{x} \cdot \left(y - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.1082063369415516 \cdot 10^{-245}:\\
\;\;\;\;x \cdot \left(\frac{y}{t - z} - \frac{z}{t - z}\right)\\

\mathbf{elif}\;z \le 1.5991111356517593 \cdot 10^{-135}:\\
\;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{t - z} \cdot \left(\sqrt[3]{x} \cdot \left(y - z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r28420047 = x;
        double r28420048 = y;
        double r28420049 = z;
        double r28420050 = r28420048 - r28420049;
        double r28420051 = r28420047 * r28420050;
        double r28420052 = t;
        double r28420053 = r28420052 - r28420049;
        double r28420054 = r28420051 / r28420053;
        return r28420054;
}
↓
double f(double x, double y, double z, double t) {
        double r28420055 = z;
        double r28420056 = -1.1082063369415516e-245;
        bool r28420057 = r28420055 <= r28420056;
        double r28420058 = x;
        double r28420059 = y;
        double r28420060 = t;
        double r28420061 = r28420060 - r28420055;
        double r28420062 = r28420059 / r28420061;
        double r28420063 = r28420055 / r28420061;
        double r28420064 = r28420062 - r28420063;
        double r28420065 = r28420058 * r28420064;
        double r28420066 = 1.5991111356517593e-135;
        bool r28420067 = r28420055 <= r28420066;
        double r28420068 = cbrt(r28420058);
        double r28420069 = r28420068 * r28420068;
        double r28420070 = r28420069 / r28420061;
        double r28420071 = r28420059 - r28420055;
        double r28420072 = r28420068 * r28420071;
        double r28420073 = r28420070 * r28420072;
        double r28420074 = r28420061 / r28420071;
        double r28420075 = r28420058 / r28420074;
        double r28420076 = r28420067 ? r28420073 : r28420075;
        double r28420077 = r28420057 ? r28420065 : r28420076;
        return r28420077;
}

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

Original10.8
Target2.1
Herbie1.9
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1082063369415516e-245

    1. Initial program 11.3

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

    3. Applied *-un-lft-identity11.3

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

    4. Applied times-frac1.7

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

    5. Simplified1.7

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

    7. Applied div-sub1.7

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

    if -1.1082063369415516e-245 < z < 1.5991111356517593e-135

    1. Initial program 5.9

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

    3. Applied associate-/l*5.2

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

    5. Applied div-inv5.2

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

    6. Applied add-cube-cbrt6.0

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

    7. Applied times-frac4.2

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

    8. Simplified4.2

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

    if 1.5991111356517593e-135 < z

    1. Initial program 12.8

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

    3. Applied associate-/l*0.9

      \[\leadsto \color{blue}{\frac{x}{\frac{t - z}{y - z}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1082063369415516 \cdot 10^{-245}:\\ \;\;\;\;x \cdot \left(\frac{y}{t - z} - \frac{z}{t - z}\right)\\ \mathbf{elif}\;z \le 1.5991111356517593 \cdot 10^{-135}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{t - z} \cdot \left(\sqrt[3]{x} \cdot \left(y - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{t - z}{y - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))