Average Error: 11.8 → 2.4
Time: 11.3s
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 r256947 = x;
        double r256948 = y;
        double r256949 = z;
        double r256950 = r256948 - r256949;
        double r256951 = r256947 * r256950;
        double r256952 = t;
        double r256953 = r256952 - r256949;
        double r256954 = r256951 / r256953;
        return r256954;
}
↓
double f(double x, double y, double z, double t) {
        double r256955 = z;
        double r256956 = -1.2131459224267867e-06;
        bool r256957 = r256955 <= r256956;
        double r256958 = 1.3630667508234389e-188;
        bool r256959 = r256955 <= r256958;
        double r256960 = !r256959;
        bool r256961 = r256957 || r256960;
        double r256962 = x;
        double r256963 = y;
        double r256964 = r256963 - r256955;
        double r256965 = t;
        double r256966 = r256965 - r256955;
        double r256967 = r256964 / r256966;
        double r256968 = r256962 * r256967;
        double r256969 = 1.0;
        double r256970 = r256969 / r256966;
        double r256971 = r256969 / r256964;
        double r256972 = r256971 / r256962;
        double r256973 = r256970 / r256972;
        double r256974 = r256961 ? r256968 : r256973;
        return r256974;
}

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)))