Average Error: 2.0 → 0.5
Time: 20.4s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 2.031077836212711156393861827249665598026 \cdot 10^{-178}:\\ \;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\left({a}^{1}\right)}^{1} \cdot \left(e^{\left(\sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot \sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right) \cdot \sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot y\right)}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;a \le 2.031077836212711156393861827249665598026 \cdot 10^{-178}:\\
\;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{{\left({a}^{1}\right)}^{1} \cdot \left(e^{\left(\sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot \sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right) \cdot \sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot y\right)}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r108945 = x;
        double r108946 = y;
        double r108947 = z;
        double r108948 = log(r108947);
        double r108949 = r108946 * r108948;
        double r108950 = t;
        double r108951 = 1.0;
        double r108952 = r108950 - r108951;
        double r108953 = a;
        double r108954 = log(r108953);
        double r108955 = r108952 * r108954;
        double r108956 = r108949 + r108955;
        double r108957 = b;
        double r108958 = r108956 - r108957;
        double r108959 = exp(r108958);
        double r108960 = r108945 * r108959;
        double r108961 = r108960 / r108946;
        return r108961;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r108962 = a;
        double r108963 = 2.0310778362127112e-178;
        bool r108964 = r108962 <= r108963;
        double r108965 = 1.0;
        double r108966 = 1.0;
        double r108967 = pow(r108962, r108966);
        double r108968 = r108965 / r108967;
        double r108969 = pow(r108968, r108966);
        double r108970 = x;
        double r108971 = z;
        double r108972 = r108965 / r108971;
        double r108973 = log(r108972);
        double r108974 = y;
        double r108975 = r108973 * r108974;
        double r108976 = r108965 / r108962;
        double r108977 = log(r108976);
        double r108978 = t;
        double r108979 = r108977 * r108978;
        double r108980 = b;
        double r108981 = r108979 + r108980;
        double r108982 = r108975 + r108981;
        double r108983 = exp(r108982);
        double r108984 = r108983 * r108974;
        double r108985 = r108970 / r108984;
        double r108986 = r108969 * r108985;
        double r108987 = pow(r108967, r108966);
        double r108988 = cbrt(r108982);
        double r108989 = r108988 * r108988;
        double r108990 = r108989 * r108988;
        double r108991 = exp(r108990);
        double r108992 = r108991 * r108974;
        double r108993 = r108987 * r108992;
        double r108994 = r108970 / r108993;
        double r108995 = r108964 ? r108986 : r108994;
        return r108995;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if a < 2.0310778362127112e-178

    1. Initial program 0.7

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.1

      \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
    4. Using strategy rm

    5. Applied associate-/l*3.9

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}\]

    6. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}}\]

    if 2.0310778362127112e-178 < a

    1. Initial program 2.3

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

    2. Taylor expanded around inf 2.4

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

    3. Simplified1.6

      \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
    4. Using strategy rm

    5. Applied associate-/l*0.6

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}\]

    6. Taylor expanded around inf 0.6

      \[\leadsto \frac{x}{\color{blue}{{\left({a}^{1}\right)}^{1} \cdot \left(e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y\right)}}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 2.031077836212711156393861827249665598026 \cdot 10^{-178}:\\ \;\;\;\;{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{\left({a}^{1}\right)}^{1} \cdot \left(e^{\left(\sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot \sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right) \cdot \sqrt[3]{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))