Average Error: 4.4 → 1.2
Time: 26.6s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1.0 - z} = -\infty:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 1.4180564209083267 \cdot 10^{+296}:\\ \;\;\;\;\left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]
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}:\\
\;\;\;\;\left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right) \cdot x\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r19086089 = x;
        double r19086090 = y;
        double r19086091 = z;
        double r19086092 = r19086090 / r19086091;
        double r19086093 = t;
        double r19086094 = 1.0;
        double r19086095 = r19086094 - r19086091;
        double r19086096 = r19086093 / r19086095;
        double r19086097 = r19086092 - r19086096;
        double r19086098 = r19086089 * r19086097;
        return r19086098;
}
↓
double f(double x, double y, double z, double t) {
        double r19086099 = y;
        double r19086100 = z;
        double r19086101 = r19086099 / r19086100;
        double r19086102 = t;
        double r19086103 = 1.0;
        double r19086104 = r19086103 - r19086100;
        double r19086105 = r19086102 / r19086104;
        double r19086106 = r19086101 - r19086105;
        double r19086107 = -inf.0;
        bool r19086108 = r19086106 <= r19086107;
        double r19086109 = x;
        double r19086110 = r19086099 * r19086109;
        double r19086111 = r19086110 / r19086100;
        double r19086112 = 1.4180564209083267e+296;
        bool r19086113 = r19086106 <= r19086112;
        double r19086114 = 1.0;
        double r19086115 = r19086114 / r19086104;
        double r19086116 = r19086102 * r19086115;
        double r19086117 = r19086101 - r19086116;
        double r19086118 = r19086117 * r19086109;
        double r19086119 = r19086113 ? r19086118 : r19086111;
        double r19086120 = r19086108 ? r19086111 : r19086119;
        return r19086120;
}

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
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \lt -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \lt 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1.0 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right)\\ \end{array}\]

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 div-inv1.2

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

  4. Final simplification1.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))))