Average Error: 2.1 → 0.6
Time: 35.9s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.350566990880654625576653098125575830144 \cdot 10^{82} \lor \neg \left(x \le 3077508455594739684605952\right):\\ \;\;\;\;\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot e^{\left(y \cdot \log z + \log a \cdot t\right) - b}\right) \cdot {\left(\frac{1}{{a}^{1}}\right)}^{1}\\ \end{array}\]
\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 -7.350566990880654625576653098125575830144 \cdot 10^{82} \lor \neg \left(x \le 3077508455594739684605952\right):\\
\;\;\;\;\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r261780 = x;
        double r261781 = y;
        double r261782 = z;
        double r261783 = log(r261782);
        double r261784 = r261781 * r261783;
        double r261785 = t;
        double r261786 = 1.0;
        double r261787 = r261785 - r261786;
        double r261788 = a;
        double r261789 = log(r261788);
        double r261790 = r261787 * r261789;
        double r261791 = r261784 + r261790;
        double r261792 = b;
        double r261793 = r261791 - r261792;
        double r261794 = exp(r261793);
        double r261795 = r261780 * r261794;
        double r261796 = r261795 / r261781;
        return r261796;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r261797 = x;
        double r261798 = -7.350566990880655e+82;
        bool r261799 = r261797 <= r261798;
        double r261800 = 3.0775084555947397e+24;
        bool r261801 = r261797 <= r261800;
        double r261802 = !r261801;
        bool r261803 = r261799 || r261802;
        double r261804 = y;
        double r261805 = z;
        double r261806 = log(r261805);
        double r261807 = r261804 * r261806;
        double r261808 = t;
        double r261809 = 1.0;
        double r261810 = r261808 - r261809;
        double r261811 = a;
        double r261812 = log(r261811);
        double r261813 = r261810 * r261812;
        double r261814 = r261807 + r261813;
        double r261815 = b;
        double r261816 = r261814 - r261815;
        double r261817 = exp(r261816);
        double r261818 = r261797 * r261817;
        double r261819 = 1.0;
        double r261820 = r261819 / r261804;
        double r261821 = r261818 * r261820;
        double r261822 = r261797 / r261804;
        double r261823 = r261812 * r261808;
        double r261824 = r261807 + r261823;
        double r261825 = r261824 - r261815;
        double r261826 = exp(r261825);
        double r261827 = r261822 * r261826;
        double r261828 = pow(r261811, r261809);
        double r261829 = r261819 / r261828;
        double r261830 = pow(r261829, r261809);
        double r261831 = r261827 * r261830;
        double r261832 = r261803 ? r261821 : r261831;
        return r261832;
}

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

Target

Original2.1
Target10.9
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471478852839936735108494759:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.228837407310493290424346923828125:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.350566990880655e+82 or 3.0775084555947397e+24 < x

    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. Using strategy rm

    3. Applied div-inv0.7

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

    if -7.350566990880655e+82 < x < 3.0775084555947397e+24

    1. Initial program 2.9

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

    3. Applied div-inv2.9

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

    5. Applied associate--l+2.9

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

    6. Applied exp-sum14.0

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

    7. Simplified14.0

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

    8. Simplified18.4

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

    10. Applied pow-sub18.4

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

    11. Taylor expanded around inf 16.5

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

    12. Simplified0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.350566990880654625576653098125575830144 \cdot 10^{82} \lor \neg \left(x \le 3077508455594739684605952\right):\\ \;\;\;\;\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot e^{\left(y \cdot \log z + \log a \cdot t\right) - b}\right) \cdot {\left(\frac{1}{{a}^{1}}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))