Average Error: 2.0 → 0.6
Time: 14.9s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02164264538042179 \cdot 10^{-94} \lor \neg \left(x \le 1.1903560079159271 \cdot 10^{-222}\right):\\ \;\;\;\;\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\sqrt[3]{{\left(\mathsf{fma}\left(\log \left(\frac{1}{z}\right), y, \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)\right)}^{3}}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{{a}^{1}}\right)}^{1}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{z}\right), y, \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \frac{x}{y}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02164264538042179 \cdot 10^{-94} \lor \neg \left(x \le 1.1903560079159271 \cdot 10^{-222}\right):\\
\;\;\;\;\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\sqrt[3]{{\left(\mathsf{fma}\left(\log \left(\frac{1}{z}\right), y, \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)\right)}^{3}}}}}{y}\\

\mathbf{else}:\\
\;\;\;\;\frac{{\left(\frac{1}{{a}^{1}}\right)}^{1}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{z}\right), y, \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \frac{x}{y}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r98098 = x;
        double r98099 = y;
        double r98100 = z;
        double r98101 = log(r98100);
        double r98102 = r98099 * r98101;
        double r98103 = t;
        double r98104 = 1.0;
        double r98105 = r98103 - r98104;
        double r98106 = a;
        double r98107 = log(r98106);
        double r98108 = r98105 * r98107;
        double r98109 = r98102 + r98108;
        double r98110 = b;
        double r98111 = r98109 - r98110;
        double r98112 = exp(r98111);
        double r98113 = r98098 * r98112;
        double r98114 = r98113 / r98099;
        return r98114;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r98115 = x;
        double r98116 = -1.0216426453804218e-94;
        bool r98117 = r98115 <= r98116;
        double r98118 = 1.190356007915927e-222;
        bool r98119 = r98115 <= r98118;
        double r98120 = !r98119;
        bool r98121 = r98117 || r98120;
        double r98122 = 1.0;
        double r98123 = a;
        double r98124 = r98122 / r98123;
        double r98125 = 1.0;
        double r98126 = pow(r98124, r98125);
        double r98127 = z;
        double r98128 = r98122 / r98127;
        double r98129 = log(r98128);
        double r98130 = y;
        double r98131 = log(r98124);
        double r98132 = t;
        double r98133 = b;
        double r98134 = fma(r98131, r98132, r98133);
        double r98135 = fma(r98129, r98130, r98134);
        double r98136 = 3.0;
        double r98137 = pow(r98135, r98136);
        double r98138 = cbrt(r98137);
        double r98139 = exp(r98138);
        double r98140 = r98126 / r98139;
        double r98141 = r98115 * r98140;
        double r98142 = r98141 / r98130;
        double r98143 = pow(r98123, r98125);
        double r98144 = r98122 / r98143;
        double r98145 = pow(r98144, r98125);
        double r98146 = exp(r98135);
        double r98147 = r98145 / r98146;
        double r98148 = r98115 / r98130;
        double r98149 = r98147 * r98148;
        double r98150 = r98121 ? r98142 : r98149;
        return r98150;
}

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0216426453804218e-94 or 1.190356007915927e-222 < x

    1. Initial program 1.5

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

    2. Taylor expanded around inf 1.6

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

    3. Simplified0.8

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

    5. Applied add-cbrt-cube0.8

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

    6. Simplified0.8

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

    if -1.0216426453804218e-94 < x < 1.190356007915927e-222

    1. Initial program 3.3

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

    2. Taylor expanded around inf 3.3

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

    3. Simplified2.5

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

    5. Applied add-cbrt-cube2.5

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

    6. Simplified2.5

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

    7. Taylor expanded around inf 0.1

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

    8. Simplified0.1

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02164264538042179 \cdot 10^{-94} \lor \neg \left(x \le 1.1903560079159271 \cdot 10^{-222}\right):\\ \;\;\;\;\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\sqrt[3]{{\left(\mathsf{fma}\left(\log \left(\frac{1}{z}\right), y, \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)\right)}^{3}}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{{a}^{1}}\right)}^{1}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{z}\right), y, \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \frac{x}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))