Average Error: 4.4 → 1.3
Time: 17.2s
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 r17717231 = x;
        double r17717232 = y;
        double r17717233 = z;
        double r17717234 = r17717232 / r17717233;
        double r17717235 = t;
        double r17717236 = 1.0;
        double r17717237 = r17717236 - r17717233;
        double r17717238 = r17717235 / r17717237;
        double r17717239 = r17717234 - r17717238;
        double r17717240 = r17717231 * r17717239;
        return r17717240;
}
↓
double f(double x, double y, double z, double t) {
        double r17717241 = y;
        double r17717242 = z;
        double r17717243 = r17717241 / r17717242;
        double r17717244 = t;
        double r17717245 = 1.0;
        double r17717246 = r17717245 - r17717242;
        double r17717247 = r17717244 / r17717246;
        double r17717248 = r17717243 - r17717247;
        double r17717249 = -inf.0;
        bool r17717250 = r17717248 <= r17717249;
        double r17717251 = x;
        double r17717252 = r17717241 * r17717251;
        double r17717253 = r17717252 / r17717242;
        double r17717254 = 1.4180564209083267e+296;
        bool r17717255 = r17717248 <= r17717254;
        double r17717256 = 1.0;
        double r17717257 = r17717256 / r17717246;
        double r17717258 = r17717256 / r17717244;
        double r17717259 = r17717257 / r17717258;
        double r17717260 = r17717243 - r17717259;
        double r17717261 = r17717251 * r17717260;
        double r17717262 = r17717255 ? r17717261 : r17717253;
        double r17717263 = r17717250 ? r17717253 : r17717262;
        return r17717263;
}

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)))))