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

↓

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

\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 r533116 = x;
        double r533117 = y;
        double r533118 = z;
        double r533119 = log(r533118);
        double r533120 = r533117 * r533119;
        double r533121 = t;
        double r533122 = 1.0;
        double r533123 = r533121 - r533122;
        double r533124 = a;
        double r533125 = log(r533124);
        double r533126 = r533123 * r533125;
        double r533127 = r533120 + r533126;
        double r533128 = b;
        double r533129 = r533127 - r533128;
        double r533130 = exp(r533129);
        double r533131 = r533116 * r533130;
        double r533132 = r533131 / r533117;
        return r533132;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r533133 = a;
        double r533134 = log(r533133);
        double r533135 = -72.12501094512979;
        bool r533136 = r533134 <= r533135;
        double r533137 = x;
        double r533138 = y;
        double r533139 = z;
        double r533140 = log(r533139);
        double r533141 = r533138 * r533140;
        double r533142 = t;
        double r533143 = 1.0;
        double r533144 = r533142 - r533143;
        double r533145 = r533144 * r533134;
        double r533146 = r533141 + r533145;
        double r533147 = b;
        double r533148 = r533146 - r533147;
        double r533149 = exp(r533148);
        double r533150 = r533137 * r533149;
        double r533151 = r533150 / r533138;
        double r533152 = 1.0;
        double r533153 = pow(r533133, r533143);
        double r533154 = r533152 / r533153;
        double r533155 = r533154 / r533138;
        double r533156 = -r533140;
        double r533157 = -r533134;
        double r533158 = fma(r533157, r533142, r533147);
        double r533159 = fma(r533138, r533156, r533158);
        double r533160 = exp(r533159);
        double r533161 = r533155 / r533160;
        double r533162 = r533137 * r533161;
        double r533163 = r533136 ? r533151 : r533162;
        return r533163;
}

Original	1.9
Target	11.7
Herbie	0.4

Derivation

Split input into 2 regimes
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
  \[\leadsto \color{blue}{x} \cdot \frac{\frac{\frac{1}{{a}^{1}}}{y}}{e^{\mathsf{fma}\left(y, -\log z, \mathsf{fma}\left(-\log a, t, b\right)\right)}}\]
Recombined 2 regimes into one program.
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, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if (log a) < -72.12501094512979`

`if -72.12501094512979 < (log a)`

Reproduce