Average Error: 4.6 → 2.8
Time: 7.1s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -4.6819217472818379 \cdot 10^{190}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.57561547149504369 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -4.6819217472818379 \cdot 10^{190}:\\
\;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.57561547149504369 \cdot 10^{-171}:\\
\;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r655772 = x;
        double r655773 = y;
        double r655774 = z;
        double r655775 = r655773 / r655774;
        double r655776 = t;
        double r655777 = 1.0;
        double r655778 = r655777 - r655774;
        double r655779 = r655776 / r655778;
        double r655780 = r655775 - r655779;
        double r655781 = r655772 * r655780;
        return r655781;
}
↓
double f(double x, double y, double z, double t) {
        double r655782 = y;
        double r655783 = z;
        double r655784 = r655782 / r655783;
        double r655785 = t;
        double r655786 = 1.0;
        double r655787 = r655786 - r655783;
        double r655788 = r655785 / r655787;
        double r655789 = r655784 - r655788;
        double r655790 = -4.681921747281838e+190;
        bool r655791 = r655789 <= r655790;
        double r655792 = x;
        double r655793 = r655792 * r655782;
        double r655794 = r655793 / r655783;
        double r655795 = -r655785;
        double r655796 = r655792 * r655795;
        double r655797 = r655796 / r655787;
        double r655798 = r655794 + r655797;
        double r655799 = -2.5756154714950437e-171;
        bool r655800 = r655789 <= r655799;
        double r655801 = 1.0;
        double r655802 = r655787 / r655785;
        double r655803 = r655801 / r655802;
        double r655804 = r655784 - r655803;
        double r655805 = r655792 * r655804;
        double r655806 = r655801 / r655783;
        double r655807 = -r655788;
        double r655808 = r655792 * r655807;
        double r655809 = fma(r655793, r655806, r655808);
        double r655810 = r655800 ? r655805 : r655809;
        double r655811 = r655791 ? r655798 : r655810;
        return r655811;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target4.4
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -4.681921747281838e+190

    1. Initial program 18.4

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

    3. Applied div-inv18.4

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

    4. Applied fma-neg18.4

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

    6. Applied fma-udef18.4

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

    7. Applied distribute-lft-in18.4

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

    8. Simplified0.8

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

    10. Applied distribute-neg-frac0.8

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

    11. Applied associate-*r/1.2

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

    if -4.681921747281838e+190 < (- (/ y z) (/ t (- 1.0 z))) < -2.5756154714950437e-171

    1. Initial program 0.2

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

    3. Applied clear-num0.3

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

    if -2.5756154714950437e-171 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 4.7

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

    3. Applied div-inv4.7

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

    4. Applied fma-neg4.7

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

    6. Applied fma-udef4.7

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

    7. Applied distribute-lft-in4.7

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

    8. Simplified4.4

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

    10. Applied div-inv4.4

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

    11. Applied fma-def4.4

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

  4. Final simplification2.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -4.6819217472818379 \cdot 10^{190}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -2.57561547149504369 \cdot 10^{-171}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))))