Average Error: 2.0 → 0.1
Time: 19.1s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le 3.2005961165294398 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\\ \end{array}\]
\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 3.2005961165294398 \cdot 10^{-5}:\\
\;\;\;\;\frac{x \cdot \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}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{\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}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r111277 = x;
        double r111278 = y;
        double r111279 = z;
        double r111280 = log(r111279);
        double r111281 = r111278 * r111280;
        double r111282 = t;
        double r111283 = 1.0;
        double r111284 = r111282 - r111283;
        double r111285 = a;
        double r111286 = log(r111285);
        double r111287 = r111284 * r111286;
        double r111288 = r111281 + r111287;
        double r111289 = b;
        double r111290 = r111288 - r111289;
        double r111291 = exp(r111290);
        double r111292 = r111277 * r111291;
        double r111293 = r111292 / r111278;
        return r111293;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r111294 = a;
        double r111295 = 3.20059611652944e-05;
        bool r111296 = r111294 <= r111295;
        double r111297 = x;
        double r111298 = 1.0;
        double r111299 = r111298 / r111294;
        double r111300 = 1.0;
        double r111301 = pow(r111299, r111300);
        double r111302 = y;
        double r111303 = z;
        double r111304 = r111298 / r111303;
        double r111305 = log(r111304);
        double r111306 = log(r111299);
        double r111307 = t;
        double r111308 = b;
        double r111309 = fma(r111306, r111307, r111308);
        double r111310 = fma(r111302, r111305, r111309);
        double r111311 = exp(r111310);
        double r111312 = r111301 / r111311;
        double r111313 = r111297 * r111312;
        double r111314 = r111313 / r111302;
        double r111315 = r111312 / r111302;
        double r111316 = r111297 * r111315;
        double r111317 = r111296 ? r111314 : r111316;
        return r111317;
}

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 a < 3.20059611652944e-05

    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

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

    if 3.20059611652944e-05 < a

    1. Initial program 3.1

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

    2. Taylor expanded around inf 3.1

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

    3. Simplified2.3

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

    5. Applied *-un-lft-identity2.3

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

    6. Applied times-frac0.1

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 3.2005961165294398 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))