Average Error: 4.9 → 4.4
Time: 15.2s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.888119603159991972673513497308156068823 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;t \le 5.068989222201373875751898664751595704249 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{z}} \cdot \left(\frac{x}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right) + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.888119603159991972673513497308156068823 \cdot 10^{-295}:\\
\;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\

\mathbf{elif}\;t \le 5.068989222201373875751898664751595704249 \cdot 10^{-49}:\\
\;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt[3]{z}} \cdot \left(\frac{x}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right) + x \cdot \left(-\frac{t}{1 - z}\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r274901 = x;
        double r274902 = y;
        double r274903 = z;
        double r274904 = r274902 / r274903;
        double r274905 = t;
        double r274906 = 1.0;
        double r274907 = r274906 - r274903;
        double r274908 = r274905 / r274907;
        double r274909 = r274904 - r274908;
        double r274910 = r274901 * r274909;
        return r274910;
}
↓
double f(double x, double y, double z, double t) {
        double r274911 = t;
        double r274912 = -2.888119603159992e-295;
        bool r274913 = r274911 <= r274912;
        double r274914 = x;
        double r274915 = y;
        double r274916 = r274914 * r274915;
        double r274917 = z;
        double r274918 = r274916 / r274917;
        double r274919 = 1.0;
        double r274920 = r274919 - r274917;
        double r274921 = r274911 / r274920;
        double r274922 = -r274921;
        double r274923 = r274914 * r274922;
        double r274924 = r274918 + r274923;
        double r274925 = 5.068989222201374e-49;
        bool r274926 = r274911 <= r274925;
        double r274927 = cbrt(r274917);
        double r274928 = r274927 * r274927;
        double r274929 = r274914 / r274928;
        double r274930 = r274915 / r274927;
        double r274931 = r274929 * r274930;
        double r274932 = r274931 + r274923;
        double r274933 = 1.0;
        double r274934 = r274933 / r274927;
        double r274935 = r274914 / r274927;
        double r274936 = r274935 * r274930;
        double r274937 = r274934 * r274936;
        double r274938 = r274937 + r274923;
        double r274939 = r274926 ? r274932 : r274938;
        double r274940 = r274913 ? r274924 : r274939;
        return r274940;
}

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

Original4.9
Target4.5
Herbie4.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -2.888119603159992e-295

    1. Initial program 5.1

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

    3. Applied sub-neg5.1

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

    4. Applied distribute-lft-in5.1

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

    6. Applied associate-*r/4.9

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

    if -2.888119603159992e-295 < t < 5.068989222201374e-49

    1. Initial program 6.2

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

    3. Applied sub-neg6.2

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

    4. Applied distribute-lft-in6.2

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

    6. Applied add-cube-cbrt6.9

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

    7. Applied *-un-lft-identity6.9

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

    8. Applied times-frac6.9

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

    9. Applied associate-*r*5.9

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

    10. Simplified5.9

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

    if 5.068989222201374e-49 < t

    1. Initial program 3.6

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

    3. Applied sub-neg3.6

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

    4. Applied distribute-lft-in3.6

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

    6. Applied add-cube-cbrt3.9

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

    7. Applied *-un-lft-identity3.9

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

    8. Applied times-frac3.9

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

    9. Applied associate-*r*2.8

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

    10. Simplified2.8

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

    12. Applied *-un-lft-identity2.8

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

    13. Applied times-frac2.8

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

    14. Applied associate-*l*2.3

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

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.888119603159991972673513497308156068823 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;t \le 5.068989222201373875751898664751595704249 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{z}} \cdot \left(\frac{x}{\sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right) + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))))