Average Error: 4.7 → 2.4
Time: 17.8s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.273245738592455952784286915322615210697 \cdot 10^{61}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.308282208361249760659009261849846214049 \cdot 10^{177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(-\frac{x}{\frac{1 - z}{t}}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.273245738592455952784286915322615210697 \cdot 10^{61}:\\
\;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\

\mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.308282208361249760659009261849846214049 \cdot 10^{177}:\\
\;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r359601 = x;
        double r359602 = y;
        double r359603 = z;
        double r359604 = r359602 / r359603;
        double r359605 = t;
        double r359606 = 1.0;
        double r359607 = r359606 - r359603;
        double r359608 = r359605 / r359607;
        double r359609 = r359604 - r359608;
        double r359610 = r359601 * r359609;
        return r359610;
}
↓
double f(double x, double y, double z, double t) {
        double r359611 = y;
        double r359612 = z;
        double r359613 = r359611 / r359612;
        double r359614 = t;
        double r359615 = 1.0;
        double r359616 = r359615 - r359612;
        double r359617 = r359614 / r359616;
        double r359618 = r359613 - r359617;
        double r359619 = -1.273245738592456e+61;
        bool r359620 = r359618 <= r359619;
        double r359621 = x;
        double r359622 = r359621 * r359611;
        double r359623 = r359622 / r359612;
        double r359624 = -r359614;
        double r359625 = r359621 * r359624;
        double r359626 = r359625 / r359616;
        double r359627 = r359623 + r359626;
        double r359628 = 7.30828220836125e+177;
        bool r359629 = r359618 <= r359628;
        double r359630 = r359612 / r359611;
        double r359631 = r359621 / r359630;
        double r359632 = -r359617;
        double r359633 = r359621 * r359632;
        double r359634 = r359631 + r359633;
        double r359635 = r359616 / r359614;
        double r359636 = r359621 / r359635;
        double r359637 = -r359636;
        double r359638 = r359623 + r359637;
        double r359639 = r359629 ? r359634 : r359638;
        double r359640 = r359620 ? r359627 : r359639;
        return r359640;
}

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
Target4.5
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ y z) (/ t (- 1.0 z))) < -1.273245738592456e+61

    1. Initial program 8.6

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

    3. Applied sub-neg8.6

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

    4. Applied distribute-lft-in8.6

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

    5. Simplified3.5

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

    7. Applied distribute-neg-frac3.5

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

    8. Applied associate-*r/5.1

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

    if -1.273245738592456e+61 < (- (/ y z) (/ t (- 1.0 z))) < 7.30828220836125e+177

    1. Initial program 1.6

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

    3. Applied sub-neg1.6

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

    4. Applied distribute-lft-in1.6

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

    5. Simplified6.8

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

    7. Applied clear-num7.0

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

    9. Applied *-un-lft-identity7.0

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

    10. Applied add-cube-cbrt7.0

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

    11. Applied times-frac7.0

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

    12. Simplified7.0

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

    13. Simplified1.7

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

    if 7.30828220836125e+177 < (- (/ y z) (/ t (- 1.0 z)))

    1. Initial program 16.4

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

    3. Applied sub-neg16.4

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

    4. Applied distribute-lft-in16.4

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

    5. Simplified1.3

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

    7. Applied clear-num1.3

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

    9. Applied *-un-lft-identity1.3

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

    10. Applied associate-*l*1.3

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

    11. Simplified1.4

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.273245738592455952784286915322615210697 \cdot 10^{61}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.308282208361249760659009261849846214049 \cdot 10^{177}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(-\frac{x}{\frac{1 - z}{t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))))