Average Error: 4.0 → 2.5
Time: 13.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.9179404843656426 \cdot 10^{-38} \lor \neg \left(t \le 4.29643249951236581 \cdot 10^{-254}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -3.9179404843656426 \cdot 10^{-38} \lor \neg \left(t \le 4.29643249951236581 \cdot 10^{-254}\right):\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r353875 = x;
        double r353876 = y;
        double r353877 = 2.0;
        double r353878 = z;
        double r353879 = t;
        double r353880 = a;
        double r353881 = r353879 + r353880;
        double r353882 = sqrt(r353881);
        double r353883 = r353878 * r353882;
        double r353884 = r353883 / r353879;
        double r353885 = b;
        double r353886 = c;
        double r353887 = r353885 - r353886;
        double r353888 = 5.0;
        double r353889 = 6.0;
        double r353890 = r353888 / r353889;
        double r353891 = r353880 + r353890;
        double r353892 = 3.0;
        double r353893 = r353879 * r353892;
        double r353894 = r353877 / r353893;
        double r353895 = r353891 - r353894;
        double r353896 = r353887 * r353895;
        double r353897 = r353884 - r353896;
        double r353898 = r353877 * r353897;
        double r353899 = exp(r353898);
        double r353900 = r353876 * r353899;
        double r353901 = r353875 + r353900;
        double r353902 = r353875 / r353901;
        return r353902;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r353903 = t;
        double r353904 = -3.9179404843656426e-38;
        bool r353905 = r353903 <= r353904;
        double r353906 = 4.296432499512366e-254;
        bool r353907 = r353903 <= r353906;
        double r353908 = !r353907;
        bool r353909 = r353905 || r353908;
        double r353910 = x;
        double r353911 = y;
        double r353912 = 2.0;
        double r353913 = z;
        double r353914 = 1.0;
        double r353915 = r353913 / r353914;
        double r353916 = a;
        double r353917 = r353903 + r353916;
        double r353918 = sqrt(r353917);
        double r353919 = r353918 / r353903;
        double r353920 = b;
        double r353921 = c;
        double r353922 = r353920 - r353921;
        double r353923 = 5.0;
        double r353924 = 6.0;
        double r353925 = r353923 / r353924;
        double r353926 = r353916 + r353925;
        double r353927 = 3.0;
        double r353928 = r353903 * r353927;
        double r353929 = r353912 / r353928;
        double r353930 = r353926 - r353929;
        double r353931 = r353922 * r353930;
        double r353932 = -r353931;
        double r353933 = fma(r353915, r353919, r353932);
        double r353934 = r353912 * r353933;
        double r353935 = exp(r353934);
        double r353936 = r353911 * r353935;
        double r353937 = r353910 + r353936;
        double r353938 = r353910 / r353937;
        double r353939 = r353913 * r353918;
        double r353940 = r353916 - r353925;
        double r353941 = r353940 * r353928;
        double r353942 = r353939 * r353941;
        double r353943 = r353916 * r353916;
        double r353944 = r353925 * r353925;
        double r353945 = r353943 - r353944;
        double r353946 = r353945 * r353928;
        double r353947 = r353940 * r353912;
        double r353948 = r353946 - r353947;
        double r353949 = r353922 * r353948;
        double r353950 = r353903 * r353949;
        double r353951 = r353942 - r353950;
        double r353952 = r353903 * r353941;
        double r353953 = r353951 / r353952;
        double r353954 = r353912 * r353953;
        double r353955 = exp(r353954);
        double r353956 = r353911 * r353955;
        double r353957 = r353910 + r353956;
        double r353958 = r353910 / r353957;
        double r353959 = r353909 ? r353938 : r353958;
        return r353959;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original4.0
Target3.0
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.9179404843656426e-38 or 4.296432499512366e-254 < t

    1. Initial program 3.2

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity3.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    4. Applied times-frac1.7

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    5. Applied fma-neg0.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}}\]

    if -3.9179404843656426e-38 < t < 4.296432499512366e-254

    1. Initial program 7.2

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
    2. Using strategy rm

    3. Applied flip-+10.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

    4. Applied frac-sub10.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

    5. Applied associate-*r/10.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

    6. Applied frac-sub8.2

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.9179404843656426 \cdot 10^{-38} \lor \neg \left(t \le 4.29643249951236581 \cdot 10^{-254}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2 (/ (- (* (* z (sqrt (+ t a))) (* (* 3 t) (- a (/ 5 6)))) (* (- (* (+ (/ 5 6) a) (* 3 t)) 2) (* (- a (/ 5 6)) (* (- b c) t)))) (* (* (* t t) 3) (- a (/ 5 6))))))))) (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3))))))))))))

  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))