Average Error: 4.8 → 4.2
Time: 4.4s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -10610978712379951100 \lor \neg \left(t \le 1.730478846643417 \cdot 10^{99}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right) \cdot \frac{\sqrt[3]{y}}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;t \le -10610978712379951100 \lor \neg \left(t \le 1.730478846643417 \cdot 10^{99}\right):\\
\;\;\;\;\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right) \cdot \frac{\sqrt[3]{y}}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r380733 = x;
        double r380734 = y;
        double r380735 = z;
        double r380736 = r380734 / r380735;
        double r380737 = t;
        double r380738 = 1.0;
        double r380739 = r380738 - r380735;
        double r380740 = r380737 / r380739;
        double r380741 = r380736 - r380740;
        double r380742 = r380733 * r380741;
        return r380742;
}
↓
double f(double x, double y, double z, double t) {
        double r380743 = t;
        double r380744 = -1.0610978712379951e+19;
        bool r380745 = r380743 <= r380744;
        double r380746 = 1.730478846643417e+99;
        bool r380747 = r380743 <= r380746;
        double r380748 = !r380747;
        bool r380749 = r380745 || r380748;
        double r380750 = y;
        double r380751 = cbrt(r380750);
        double r380752 = r380751 * r380751;
        double r380753 = x;
        double r380754 = r380752 * r380753;
        double r380755 = z;
        double r380756 = r380751 / r380755;
        double r380757 = r380754 * r380756;
        double r380758 = 1.0;
        double r380759 = r380758 - r380755;
        double r380760 = r380743 / r380759;
        double r380761 = -r380760;
        double r380762 = r380753 * r380761;
        double r380763 = r380757 + r380762;
        double r380764 = r380750 / r380755;
        double r380765 = r380753 * r380764;
        double r380766 = -r380743;
        double r380767 = r380753 * r380766;
        double r380768 = r380767 / r380759;
        double r380769 = r380765 + r380768;
        double r380770 = r380749 ? r380763 : r380769;
        return r380770;
}

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

Original4.8
Target4.4
Herbie4.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.0610978712379951e+19 or 1.730478846643417e+99 < t

    1. Initial program 4.1

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

    3. Applied sub-neg4.1

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

    4. Applied distribute-lft-in4.1

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

    6. Applied *-un-lft-identity4.1

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

    7. Applied add-cube-cbrt4.3

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

    8. Applied times-frac4.3

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

    9. Applied associate-*r*3.4

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

    10. Simplified3.4

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

    if -1.0610978712379951e+19 < t < 1.730478846643417e+99

    1. Initial program 5.2

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

    3. Applied sub-neg5.2

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

    4. Applied distribute-lft-in5.2

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

    6. Applied distribute-neg-frac5.2

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

    7. Applied associate-*r/4.6

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -10610978712379951100 \lor \neg \left(t \le 1.730478846643417 \cdot 10^{99}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot x\right) \cdot \frac{\sqrt[3]{y}}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))