Average Error: 6.7 → 2.5
Time: 12.2s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.20052350072620737692343564222038722076 \cdot 10^{-21} \lor \neg \left(z \le 7.10776943103653364138280811307393783598 \cdot 10^{-114}\right):\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot \left(y - t\right)}{x \cdot 2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.20052350072620737692343564222038722076 \cdot 10^{-21} \lor \neg \left(z \le 7.10776943103653364138280811307393783598 \cdot 10^{-114}\right):\\
\;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{z \cdot \left(y - t\right)}{x \cdot 2}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r444895 = x;
        double r444896 = 2.0;
        double r444897 = r444895 * r444896;
        double r444898 = y;
        double r444899 = z;
        double r444900 = r444898 * r444899;
        double r444901 = t;
        double r444902 = r444901 * r444899;
        double r444903 = r444900 - r444902;
        double r444904 = r444897 / r444903;
        return r444904;
}
↓
double f(double x, double y, double z, double t) {
        double r444905 = z;
        double r444906 = -2.2005235007262074e-21;
        bool r444907 = r444905 <= r444906;
        double r444908 = 7.107769431036534e-114;
        bool r444909 = r444905 <= r444908;
        double r444910 = !r444909;
        bool r444911 = r444907 || r444910;
        double r444912 = x;
        double r444913 = 2.0;
        double r444914 = r444912 * r444913;
        double r444915 = r444914 / r444905;
        double r444916 = y;
        double r444917 = t;
        double r444918 = r444916 - r444917;
        double r444919 = r444915 / r444918;
        double r444920 = 1.0;
        double r444921 = r444905 * r444918;
        double r444922 = r444921 / r444914;
        double r444923 = r444920 / r444922;
        double r444924 = r444911 ? r444919 : r444923;
        return r444924;
}

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

Original6.7
Target2.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.2005235007262074e-21 or 7.107769431036534e-114 < z

    1. Initial program 8.8

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified7.1

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

    4. Applied associate-/r*2.1

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}}\]

    if -2.2005235007262074e-21 < z < 7.107769431036534e-114

    1. Initial program 2.9

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified2.9

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

    4. Applied clear-num3.3

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.20052350072620737692343564222038722076 \cdot 10^{-21} \lor \neg \left(z \le 7.10776943103653364138280811307393783598 \cdot 10^{-114}\right):\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot \left(y - t\right)}{x \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))