Average Error: 5.0 → 5.2
Time: 13.6s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.296999623703697827095896691621829118589 \cdot 10^{-254}:\\ \;\;\;\;\left(\frac{y}{\sqrt[3]{z}} \cdot x\right) \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} + \frac{-x}{\frac{1 - z}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y + \frac{-x}{\frac{1 - z}{t}}\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.296999623703697827095896691621829118589 \cdot 10^{-254}:\\
\;\;\;\;\left(\frac{y}{\sqrt[3]{z}} \cdot x\right) \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} + \frac{-x}{\frac{1 - z}{t}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r325881 = x;
        double r325882 = y;
        double r325883 = z;
        double r325884 = r325882 / r325883;
        double r325885 = t;
        double r325886 = 1.0;
        double r325887 = r325886 - r325883;
        double r325888 = r325885 / r325887;
        double r325889 = r325884 - r325888;
        double r325890 = r325881 * r325889;
        return r325890;
}
↓
double f(double x, double y, double z, double t) {
        double r325891 = z;
        double r325892 = -1.2969996237036978e-254;
        bool r325893 = r325891 <= r325892;
        double r325894 = y;
        double r325895 = cbrt(r325891);
        double r325896 = r325894 / r325895;
        double r325897 = x;
        double r325898 = r325896 * r325897;
        double r325899 = 1.0;
        double r325900 = r325895 * r325895;
        double r325901 = r325899 / r325900;
        double r325902 = r325898 * r325901;
        double r325903 = -r325897;
        double r325904 = 1.0;
        double r325905 = r325904 - r325891;
        double r325906 = t;
        double r325907 = r325905 / r325906;
        double r325908 = r325903 / r325907;
        double r325909 = r325902 + r325908;
        double r325910 = r325897 / r325891;
        double r325911 = r325910 * r325894;
        double r325912 = r325911 + r325908;
        double r325913 = r325893 ? r325909 : r325912;
        return r325913;
}

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

Original5.0
Target4.4
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2969996237036978e-254

    1. Initial program 4.7

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

    3. Applied clear-num4.8

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

    5. Applied sub-neg4.8

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

    6. Applied distribute-lft-in4.8

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

    7. Simplified4.8

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

    8. Simplified4.8

      \[\leadsto \frac{y}{z} \cdot x + \color{blue}{\frac{x \cdot -1}{\frac{1 - z}{t}}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt5.3

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

    11. Applied *-un-lft-identity5.3

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

    12. Applied times-frac5.3

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

    13. Applied associate-*l*4.2

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

    14. Simplified4.2

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

    if -1.2969996237036978e-254 < z

    1. Initial program 5.3

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

    3. Applied clear-num5.4

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

    5. Applied sub-neg5.4

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

    6. Applied distribute-lft-in5.4

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

    7. Simplified5.4

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

    8. Simplified5.4

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

    9. Taylor expanded around 0 6.1

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

    10. Simplified6.1

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.296999623703697827095896691621829118589 \cdot 10^{-254}:\\ \;\;\;\;\left(\frac{y}{\sqrt[3]{z}} \cdot x\right) \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} + \frac{-x}{\frac{1 - z}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y + \frac{-x}{\frac{1 - z}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))