Average Error: 4.7 → 3.7
Time: 7.1s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.9812160670364881 \cdot 10^{269}:\\ \;\;\;\;x \cdot \left(\left(\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{y}{z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y}{z}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{y}{z}}}\right) - \frac{t}{1 - z}\right) + \frac{1}{1 - z} \cdot \left(\left(-t\right) + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.9812160670364881 \cdot 10^{269}:\\
\;\;\;\;x \cdot \left(\left(\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{y}{z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y}{z}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{y}{z}}}\right) - \frac{t}{1 - z}\right) + \frac{1}{1 - z} \cdot \left(\left(-t\right) + t\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r339666 = x;
        double r339667 = y;
        double r339668 = z;
        double r339669 = r339667 / r339668;
        double r339670 = t;
        double r339671 = 1.0;
        double r339672 = r339671 - r339668;
        double r339673 = r339670 / r339672;
        double r339674 = r339669 - r339673;
        double r339675 = r339666 * r339674;
        return r339675;
}
↓
double f(double x, double y, double z, double t) {
        double r339676 = y;
        double r339677 = z;
        double r339678 = r339676 / r339677;
        double r339679 = t;
        double r339680 = 1.0;
        double r339681 = r339680 - r339677;
        double r339682 = r339679 / r339681;
        double r339683 = r339678 - r339682;
        double r339684 = 1.981216067036488e+269;
        bool r339685 = r339683 <= r339684;
        double r339686 = x;
        double r339687 = cbrt(r339678);
        double r339688 = r339687 * r339687;
        double r339689 = cbrt(r339687);
        double r339690 = r339689 * r339689;
        double r339691 = r339690 * r339689;
        double r339692 = r339688 * r339691;
        double r339693 = r339692 - r339682;
        double r339694 = 1.0;
        double r339695 = r339694 / r339681;
        double r339696 = -r339679;
        double r339697 = r339696 + r339679;
        double r339698 = r339695 * r339697;
        double r339699 = r339693 + r339698;
        double r339700 = r339686 * r339699;
        double r339701 = r339676 * r339681;
        double r339702 = r339677 * r339679;
        double r339703 = r339701 - r339702;
        double r339704 = r339686 * r339703;
        double r339705 = r339677 * r339681;
        double r339706 = r339704 / r339705;
        double r339707 = r339685 ? r339700 : r339706;
        return r339707;
}

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

Original4.7
Target4.2
Herbie3.7
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < 1.981216067036488e+269

    1. Initial program 3.1

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

    3. Applied div-inv3.2

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

    5. Applied add-cube-cbrt3.7

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

    6. Applied prod-diff3.7

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

    7. Simplified3.6

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

    8. Simplified3.6

      \[\leadsto x \cdot \left(\left(\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right) \cdot \sqrt[3]{\frac{y}{z}} - \frac{t}{1 - z}\right) + \color{blue}{\frac{1}{1 - z} \cdot \left(\left(-t\right) + t\right)}\right)\]
    9. Using strategy rm

    10. Applied add-cube-cbrt3.8

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

    if 1.981216067036488e+269 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 36.8

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

    3. Applied frac-sub38.0

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

    4. Applied associate-*r/1.5

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.9812160670364881 \cdot 10^{269}:\\ \;\;\;\;x \cdot \left(\left(\left(\sqrt[3]{\frac{y}{z}} \cdot \sqrt[3]{\frac{y}{z}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{y}{z}}} \cdot \sqrt[3]{\sqrt[3]{\frac{y}{z}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{y}{z}}}\right) - \frac{t}{1 - z}\right) + \frac{1}{1 - z} \cdot \left(\left(-t\right) + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))))