Average Error: 1.9 → 0.4
Time: 23.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log a \le -72.125010945129787:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{1}{{a}^{1}}}{y}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}\\ \end{array}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\begin{array}{l}
\mathbf{if}\;\log a \le -72.125010945129787:\\
\;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r118196 = x;
        double r118197 = y;
        double r118198 = z;
        double r118199 = log(r118198);
        double r118200 = r118197 * r118199;
        double r118201 = t;
        double r118202 = 1.0;
        double r118203 = r118201 - r118202;
        double r118204 = a;
        double r118205 = log(r118204);
        double r118206 = r118203 * r118205;
        double r118207 = r118200 + r118206;
        double r118208 = b;
        double r118209 = r118207 - r118208;
        double r118210 = exp(r118209);
        double r118211 = r118196 * r118210;
        double r118212 = r118211 / r118197;
        return r118212;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r118213 = a;
        double r118214 = log(r118213);
        double r118215 = -72.12501094512979;
        bool r118216 = r118214 <= r118215;
        double r118217 = x;
        double r118218 = y;
        double r118219 = z;
        double r118220 = log(r118219);
        double r118221 = r118218 * r118220;
        double r118222 = t;
        double r118223 = 1.0;
        double r118224 = r118222 - r118223;
        double r118225 = r118224 * r118214;
        double r118226 = r118221 + r118225;
        double r118227 = b;
        double r118228 = r118226 - r118227;
        double r118229 = exp(r118228);
        double r118230 = r118217 * r118229;
        double r118231 = r118230 / r118218;
        double r118232 = 1.0;
        double r118233 = pow(r118213, r118223);
        double r118234 = r118232 / r118233;
        double r118235 = r118234 / r118218;
        double r118236 = -r118220;
        double r118237 = -r118214;
        double r118238 = fma(r118237, r118222, r118227);
        double r118239 = fma(r118218, r118236, r118238);
        double r118240 = exp(r118239);
        double r118241 = r118235 / r118240;
        double r118242 = r118217 * r118241;
        double r118243 = r118216 ? r118231 : r118242;
        return r118243;
}

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 (log a) < -72.12501094512979

    1. Initial program 0.7

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

    if -72.12501094512979 < (log a)

    1. Initial program 2.8

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

    2. Taylor expanded around inf 2.8

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

    3. Simplified4.1

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

    5. Applied *-un-lft-identity4.1

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

    6. Applied *-un-lft-identity4.1

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

    7. Applied div-inv4.1

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

    8. Applied times-frac2.2

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

    9. Applied times-frac0.2

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

    10. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \le -72.125010945129787:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{1}{{a}^{1}}}{y}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))