Average Error: 3.9 → 2.3
Time: 7.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.1562973914214696 \cdot 10^{-111} \lor \neg \left(t \le 2.42372495958711142 \cdot 10^{-244}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]
\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -8.1562973914214696 \cdot 10^{-111} \lor \neg \left(t \le 2.42372495958711142 \cdot 10^{-244}\right):\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62793 = x;
        double r62794 = y;
        double r62795 = 2.0;
        double r62796 = z;
        double r62797 = t;
        double r62798 = a;
        double r62799 = r62797 + r62798;
        double r62800 = sqrt(r62799);
        double r62801 = r62796 * r62800;
        double r62802 = r62801 / r62797;
        double r62803 = b;
        double r62804 = c;
        double r62805 = r62803 - r62804;
        double r62806 = 5.0;
        double r62807 = 6.0;
        double r62808 = r62806 / r62807;
        double r62809 = r62798 + r62808;
        double r62810 = 3.0;
        double r62811 = r62797 * r62810;
        double r62812 = r62795 / r62811;
        double r62813 = r62809 - r62812;
        double r62814 = r62805 * r62813;
        double r62815 = r62802 - r62814;
        double r62816 = r62795 * r62815;
        double r62817 = exp(r62816);
        double r62818 = r62794 * r62817;
        double r62819 = r62793 + r62818;
        double r62820 = r62793 / r62819;
        return r62820;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r62821 = t;
        double r62822 = -8.15629739142147e-111;
        bool r62823 = r62821 <= r62822;
        double r62824 = 2.4237249595871114e-244;
        bool r62825 = r62821 <= r62824;
        double r62826 = !r62825;
        bool r62827 = r62823 || r62826;
        double r62828 = x;
        double r62829 = y;
        double r62830 = 2.0;
        double r62831 = z;
        double r62832 = a;
        double r62833 = r62821 + r62832;
        double r62834 = sqrt(r62833);
        double r62835 = r62834 / r62821;
        double r62836 = b;
        double r62837 = c;
        double r62838 = r62836 - r62837;
        double r62839 = 5.0;
        double r62840 = 6.0;
        double r62841 = r62839 / r62840;
        double r62842 = r62832 + r62841;
        double r62843 = 3.0;
        double r62844 = r62821 * r62843;
        double r62845 = r62830 / r62844;
        double r62846 = r62842 - r62845;
        double r62847 = r62838 * r62846;
        double r62848 = -r62847;
        double r62849 = fma(r62831, r62835, r62848);
        double r62850 = -r62838;
        double r62851 = r62850 + r62838;
        double r62852 = r62846 * r62851;
        double r62853 = r62849 + r62852;
        double r62854 = r62830 * r62853;
        double r62855 = exp(r62854);
        double r62856 = r62829 * r62855;
        double r62857 = r62828 + r62856;
        double r62858 = r62828 / r62857;
        double r62859 = r62831 * r62834;
        double r62860 = r62832 - r62841;
        double r62861 = r62860 * r62844;
        double r62862 = r62859 * r62861;
        double r62863 = r62832 * r62832;
        double r62864 = r62841 * r62841;
        double r62865 = r62863 - r62864;
        double r62866 = r62865 * r62844;
        double r62867 = r62860 * r62830;
        double r62868 = r62866 - r62867;
        double r62869 = r62838 * r62868;
        double r62870 = r62821 * r62869;
        double r62871 = r62862 - r62870;
        double r62872 = r62821 * r62861;
        double r62873 = r62871 / r62872;
        double r62874 = r62830 * r62873;
        double r62875 = exp(r62874);
        double r62876 = r62829 * r62875;
        double r62877 = r62828 + r62876;
        double r62878 = r62828 / r62877;
        double r62879 = r62827 ? r62858 : r62878;
        return r62879;
}

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

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if t < -8.15629739142147e-111 or 2.4237249595871114e-244 < t

    1. Initial program 2.9

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity2.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    4. Applied times-frac1.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

    5. Applied prod-diff17.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right) + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right), b - c, \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)\right)}}}\]

    6. Simplified17.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)} + \mathsf{fma}\left(-\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right), b - c, \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(b - c\right)\right)\right)}}\]

    7. Simplified0.9

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \color{blue}{\left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)}\right)}}\]

    if -8.15629739142147e-111 < t < 2.4237249595871114e-244

    1. Initial program 8.6

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
    2. Using strategy rm

    3. Applied flip-+11.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{2}{t \cdot 3}\right)\right)}}\]

    4. Applied frac-sub11.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

    5. Applied associate-*r/11.8

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)}{\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)}}\right)}}\]

    6. Applied frac-sub8.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.1562973914214696 \cdot 10^{-111} \lor \neg \left(t \le 2.42372495958711142 \cdot 10^{-244}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot \left(t \cdot 3\right) - \left(a - \frac{5}{6}\right) \cdot 2\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(t \cdot 3\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))