Average Error: 11.8 → 2.4
Time: 12.6s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{t - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.2131459224267867 \cdot 10^{-6} \lor \neg \left(z \le 1.363066750823439 \cdot 10^{-188}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t - z}}{\frac{\frac{1}{y - z}}{x}}\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{t - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.2131459224267867 \cdot 10^{-6} \lor \neg \left(z \le 1.363066750823439 \cdot 10^{-188}\right):\\
\;\;\;\;x \cdot \frac{y - z}{t - z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{t - z}}{\frac{\frac{1}{y - z}}{x}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r653860 = x;
        double r653861 = y;
        double r653862 = z;
        double r653863 = r653861 - r653862;
        double r653864 = r653860 * r653863;
        double r653865 = t;
        double r653866 = r653865 - r653862;
        double r653867 = r653864 / r653866;
        return r653867;
}
↓
double f(double x, double y, double z, double t) {
        double r653868 = z;
        double r653869 = -1.2131459224267867e-06;
        bool r653870 = r653868 <= r653869;
        double r653871 = 1.3630667508234389e-188;
        bool r653872 = r653868 <= r653871;
        double r653873 = !r653872;
        bool r653874 = r653870 || r653873;
        double r653875 = x;
        double r653876 = y;
        double r653877 = r653876 - r653868;
        double r653878 = t;
        double r653879 = r653878 - r653868;
        double r653880 = r653877 / r653879;
        double r653881 = r653875 * r653880;
        double r653882 = 1.0;
        double r653883 = r653882 / r653879;
        double r653884 = r653882 / r653877;
        double r653885 = r653884 / r653875;
        double r653886 = r653883 / r653885;
        double r653887 = r653874 ? r653881 : r653886;
        return r653887;
}

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

Original11.8
Target2.4
Herbie2.4
\[\frac{x}{\frac{t - z}{y - z}}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2131459224267867e-06 or 1.3630667508234389e-188 < z

    1. Initial program 14.7

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

    3. Applied *-un-lft-identity14.7

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

    4. Applied times-frac0.9

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

    5. Simplified0.9

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

    if -1.2131459224267867e-06 < z < 1.3630667508234389e-188

    1. Initial program 5.7

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

    3. Applied associate-/l*5.5

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

    5. Applied clear-num5.8

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{t - z}{y - z}}{x}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity5.8

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

    8. Applied div-inv5.8

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

    9. Applied times-frac5.9

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

    10. Applied associate-/r*5.7

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

    11. Simplified5.7

      \[\leadsto \frac{\color{blue}{\frac{1}{t - z}}}{\frac{\frac{1}{y - z}}{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2131459224267867 \cdot 10^{-6} \lor \neg \left(z \le 1.363066750823439 \cdot 10^{-188}\right):\\ \;\;\;\;x \cdot \frac{y - z}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t - z}}{\frac{\frac{1}{y - z}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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)))