Average Error: 4.7 → 3.0
Time: 5.7s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.40937757321820234 \cdot 10^{261}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.40937757321820234 \cdot 10^{261}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r471876 = x;
        double r471877 = y;
        double r471878 = z;
        double r471879 = r471877 / r471878;
        double r471880 = t;
        double r471881 = 1.0;
        double r471882 = r471881 - r471878;
        double r471883 = r471880 / r471882;
        double r471884 = r471879 - r471883;
        double r471885 = r471876 * r471884;
        return r471885;
}
↓
double f(double x, double y, double z, double t) {
        double r471886 = y;
        double r471887 = z;
        double r471888 = r471886 / r471887;
        double r471889 = t;
        double r471890 = 1.0;
        double r471891 = r471890 - r471887;
        double r471892 = r471889 / r471891;
        double r471893 = r471888 - r471892;
        double r471894 = 2.4093775732182023e+261;
        bool r471895 = r471893 <= r471894;
        double r471896 = x;
        double r471897 = r471896 * r471893;
        double r471898 = r471896 / r471887;
        double r471899 = r471886 * r471891;
        double r471900 = r471887 * r471889;
        double r471901 = r471899 - r471900;
        double r471902 = r471901 / r471891;
        double r471903 = r471898 * r471902;
        double r471904 = r471895 ? r471897 : r471903;
        return r471904;
}

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

Original4.7
Target4.4
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < 2.4093775732182023e+261

    1. Initial program 3.1

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

    if 2.4093775732182023e+261 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 33.9

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
    2. Using strategy rm

    3. Applied frac-sub35.3

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

    4. Applied associate-*r/1.7

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

    6. Applied times-frac1.8

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.40937757321820234 \cdot 10^{261}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y \cdot \left(1 - z\right) - z \cdot t}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

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

  (* x (- (/ y z) (/ t (- 1 z)))))