Average Error: 4.5 → 4.0
Time: 20.0s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.112516848695828332647411301833499628185 \cdot 10^{-104} \lor \neg \left(x \le 4.150754646520127175787942828061912974483 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} + \frac{t \cdot \left(-x\right)}{1 - z}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.112516848695828332647411301833499628185 \cdot 10^{-104} \lor \neg \left(x \le 4.150754646520127175787942828061912974483 \cdot 10^{-156}\right):\\
\;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r246759 = x;
        double r246760 = y;
        double r246761 = z;
        double r246762 = r246760 / r246761;
        double r246763 = t;
        double r246764 = 1.0;
        double r246765 = r246764 - r246761;
        double r246766 = r246763 / r246765;
        double r246767 = r246762 - r246766;
        double r246768 = r246759 * r246767;
        return r246768;
}
↓
double f(double x, double y, double z, double t) {
        double r246769 = x;
        double r246770 = -5.1125168486958283e-104;
        bool r246771 = r246769 <= r246770;
        double r246772 = 4.150754646520127e-156;
        bool r246773 = r246769 <= r246772;
        double r246774 = !r246773;
        bool r246775 = r246771 || r246774;
        double r246776 = y;
        double r246777 = z;
        double r246778 = r246776 / r246777;
        double r246779 = t;
        double r246780 = 1.0;
        double r246781 = 1.0;
        double r246782 = r246781 - r246777;
        double r246783 = r246780 / r246782;
        double r246784 = r246779 * r246783;
        double r246785 = r246778 - r246784;
        double r246786 = r246769 * r246785;
        double r246787 = r246769 / r246777;
        double r246788 = r246776 * r246787;
        double r246789 = -r246769;
        double r246790 = r246779 * r246789;
        double r246791 = r246790 / r246782;
        double r246792 = r246788 + r246791;
        double r246793 = r246775 ? r246786 : r246792;
        return r246793;
}

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

Original4.5
Target4.1
Herbie4.0
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.1125168486958283e-104 or 4.150754646520127e-156 < x

    1. Initial program 3.0

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

    3. Applied div-inv3.0

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

    if -5.1125168486958283e-104 < x < 4.150754646520127e-156

    1. Initial program 7.2

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

    3. Applied *-un-lft-identity7.2

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

    4. Applied add-cube-cbrt7.5

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

    5. Applied times-frac7.5

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

    6. Simplified7.5

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

    8. Applied sub-neg7.5

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

    9. Applied distribute-lft-in7.5

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

    10. Simplified6.0

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

    11. Simplified5.7

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

    13. Applied associate-*l/5.7

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.112516848695828332647411301833499628185 \cdot 10^{-104} \lor \neg \left(x \le 4.150754646520127175787942828061912974483 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z} + \frac{t \cdot \left(-x\right)}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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)))) -7.62322630331204244e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.41339449277023022e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))