Average Error: 4.4 → 1.3
Time: 18.3s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1.0 - z} = -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 1.4180564209083267 \cdot 10^{+296}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\frac{1}{1.0 - z}}{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1.0 - z} = -\infty:\\
\;\;\;\;\frac{y \cdot x}{z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 1.4180564209083267 \cdot 10^{+296}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\frac{1}{1.0 - z}}{\frac{1}{t}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{y \cdot x}{z}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r22048920 = x;
        double r22048921 = y;
        double r22048922 = z;
        double r22048923 = r22048921 / r22048922;
        double r22048924 = t;
        double r22048925 = 1.0;
        double r22048926 = r22048925 - r22048922;
        double r22048927 = r22048924 / r22048926;
        double r22048928 = r22048923 - r22048927;
        double r22048929 = r22048920 * r22048928;
        return r22048929;
}
↓
double f(double x, double y, double z, double t) {
        double r22048930 = y;
        double r22048931 = z;
        double r22048932 = r22048930 / r22048931;
        double r22048933 = t;
        double r22048934 = 1.0;
        double r22048935 = r22048934 - r22048931;
        double r22048936 = r22048933 / r22048935;
        double r22048937 = r22048932 - r22048936;
        double r22048938 = -inf.0;
        bool r22048939 = r22048937 <= r22048938;
        double r22048940 = x;
        double r22048941 = r22048930 * r22048940;
        double r22048942 = r22048941 / r22048931;
        double r22048943 = 1.4180564209083267e+296;
        bool r22048944 = r22048937 <= r22048943;
        double r22048945 = 1.0;
        double r22048946 = r22048945 / r22048935;
        double r22048947 = r22048945 / r22048933;
        double r22048948 = r22048946 / r22048947;
        double r22048949 = r22048932 - r22048948;
        double r22048950 = r22048940 * r22048949;
        double r22048951 = r22048944 ? r22048950 : r22048942;
        double r22048952 = r22048939 ? r22048942 : r22048951;
        return r22048952;
}

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

Original4.4
Target4.3
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \lt -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \lt 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1.0 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 1.4180564209083267e+296 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 55.6

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]

    2. Taylor expanded around 0 2.4

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < 1.4180564209083267e+296

    1. Initial program 1.1

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]
    2. Using strategy rm

    3. Applied clear-num1.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\frac{1.0 - z}{t}}}\right)\]
    4. Using strategy rm

    5. Applied div-inv1.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{1}{\color{blue}{\left(1.0 - z\right) \cdot \frac{1}{t}}}\right)\]

    6. Applied associate-/r*1.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{\frac{1}{1.0 - z}}{\frac{1}{t}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1.0 - z} = -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 1.4180564209083267 \cdot 10^{+296}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\frac{1}{1.0 - z}}{\frac{1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))