Average Error: 4.5 → 1.4
Time: 7.1s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right) \cdot x\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\
\;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right) \cdot x\\

\end{array}
double f(double x, double y, double z, double t) {
        double r490814 = x;
        double r490815 = y;
        double r490816 = z;
        double r490817 = r490815 / r490816;
        double r490818 = t;
        double r490819 = 1.0;
        double r490820 = r490819 - r490816;
        double r490821 = r490818 / r490820;
        double r490822 = r490817 - r490821;
        double r490823 = r490814 * r490822;
        return r490823;
}
↓
double f(double x, double y, double z, double t) {
        double r490824 = y;
        double r490825 = z;
        double r490826 = r490824 / r490825;
        double r490827 = t;
        double r490828 = 1.0;
        double r490829 = r490828 - r490825;
        double r490830 = r490827 / r490829;
        double r490831 = r490826 - r490830;
        double r490832 = -inf.0;
        bool r490833 = r490831 <= r490832;
        double r490834 = 1.1070795943172787e+304;
        bool r490835 = r490831 <= r490834;
        double r490836 = !r490835;
        bool r490837 = r490833 || r490836;
        double r490838 = x;
        double r490839 = r490824 * r490829;
        double r490840 = r490825 * r490827;
        double r490841 = r490839 - r490840;
        double r490842 = r490838 * r490841;
        double r490843 = r490825 * r490829;
        double r490844 = r490842 / r490843;
        double r490845 = 1.0;
        double r490846 = cbrt(r490829);
        double r490847 = r490827 / r490846;
        double r490848 = r490846 * r490846;
        double r490849 = r490845 / r490848;
        double r490850 = r490847 * r490849;
        double r490851 = -r490850;
        double r490852 = fma(r490845, r490826, r490851);
        double r490853 = r490838 * r490852;
        double r490854 = -r490847;
        double r490855 = r490854 + r490847;
        double r490856 = r490849 * r490855;
        double r490857 = r490856 * r490838;
        double r490858 = r490853 + r490857;
        double r490859 = r490837 ? r490844 : r490858;
        return r490859;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.5
Target4.2
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 1.1070795943172787e+304 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 61.8

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

    3. Applied frac-sub61.8

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

    4. Applied associate-*r/0.2

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

    if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < 1.1070795943172787e+304

    1. Initial program 1.2

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

    3. Applied add-cube-cbrt1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}}\right)\]

    4. Applied *-un-lft-identity1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}\right)\]

    5. Applied times-frac1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}}\right)\]

    6. Applied add-sqr-sqrt28.9

      \[\leadsto x \cdot \left(\color{blue}{\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}} - \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\]

    7. Applied prod-diff28.9

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{y}{z}}, \sqrt{\frac{y}{z}}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right)}\]

    8. Applied distribute-lft-in28.9

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\sqrt{\frac{y}{z}}, \sqrt{\frac{y}{z}}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + x \cdot \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)}\]

    9. Simplified1.5

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)} + x \cdot \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\]

    10. Simplified1.5

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \left(\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))