Average Error: 4.4 → 4.4
Time: 6.1s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.77373656942475791 \cdot 10^{-180} \lor \neg \left(t \le 6.2185368398920625 \cdot 10^{-38}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(x \cdot \left(-t\right)\right) \cdot \frac{1}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.77373656942475791 \cdot 10^{-180} \lor \neg \left(t \le 6.2185368398920625 \cdot 10^{-38}\right):\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r396075 = x;
        double r396076 = y;
        double r396077 = z;
        double r396078 = r396076 / r396077;
        double r396079 = t;
        double r396080 = 1.0;
        double r396081 = r396080 - r396077;
        double r396082 = r396079 / r396081;
        double r396083 = r396078 - r396082;
        double r396084 = r396075 * r396083;
        return r396084;
}
↓
double f(double x, double y, double z, double t) {
        double r396085 = t;
        double r396086 = -1.773736569424758e-180;
        bool r396087 = r396085 <= r396086;
        double r396088 = 6.218536839892062e-38;
        bool r396089 = r396085 <= r396088;
        double r396090 = !r396089;
        bool r396091 = r396087 || r396090;
        double r396092 = x;
        double r396093 = y;
        double r396094 = r396092 * r396093;
        double r396095 = 1.0;
        double r396096 = z;
        double r396097 = r396095 / r396096;
        double r396098 = r396094 * r396097;
        double r396099 = 1.0;
        double r396100 = r396099 - r396096;
        double r396101 = r396085 / r396100;
        double r396102 = -r396101;
        double r396103 = r396092 * r396102;
        double r396104 = r396098 + r396103;
        double r396105 = r396093 / r396096;
        double r396106 = r396092 * r396105;
        double r396107 = -r396085;
        double r396108 = r396092 * r396107;
        double r396109 = r396095 / r396100;
        double r396110 = r396108 * r396109;
        double r396111 = r396106 + r396110;
        double r396112 = r396091 ? r396104 : r396111;
        return r396112;
}

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
Target4.1
Herbie4.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.773736569424758e-180 or 6.218536839892062e-38 < t

    1. Initial program 3.7

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

    3. Applied sub-neg3.7

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

    4. Applied distribute-lft-in3.7

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

    6. Applied div-inv3.8

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

    7. Applied associate-*r*4.4

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

    if -1.773736569424758e-180 < t < 6.218536839892062e-38

    1. Initial program 5.7

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

    3. Applied sub-neg5.7

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

    4. Applied distribute-lft-in5.7

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

    6. Applied div-inv5.7

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

    7. Applied distribute-lft-neg-in5.7

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

    8. Applied associate-*r*4.5

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

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.77373656942475791 \cdot 10^{-180} \lor \neg \left(t \le 6.2185368398920625 \cdot 10^{-38}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(x \cdot \left(-t\right)\right) \cdot \frac{1}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))))