Average Error: 3.7 → 3.3
Time: 38.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\ \end{array}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\
\;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r4621792 = x;
        double r4621793 = y;
        double r4621794 = 2.0;
        double r4621795 = z;
        double r4621796 = t;
        double r4621797 = a;
        double r4621798 = r4621796 + r4621797;
        double r4621799 = sqrt(r4621798);
        double r4621800 = r4621795 * r4621799;
        double r4621801 = r4621800 / r4621796;
        double r4621802 = b;
        double r4621803 = c;
        double r4621804 = r4621802 - r4621803;
        double r4621805 = 5.0;
        double r4621806 = 6.0;
        double r4621807 = r4621805 / r4621806;
        double r4621808 = r4621797 + r4621807;
        double r4621809 = 3.0;
        double r4621810 = r4621796 * r4621809;
        double r4621811 = r4621794 / r4621810;
        double r4621812 = r4621808 - r4621811;
        double r4621813 = r4621804 * r4621812;
        double r4621814 = r4621801 - r4621813;
        double r4621815 = r4621794 * r4621814;
        double r4621816 = exp(r4621815);
        double r4621817 = r4621793 * r4621816;
        double r4621818 = r4621792 + r4621817;
        double r4621819 = r4621792 / r4621818;
        return r4621819;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r4621820 = b;
        double r4621821 = -2.978001093890994e+55;
        bool r4621822 = r4621820 <= r4621821;
        double r4621823 = x;
        double r4621824 = t;
        double r4621825 = a;
        double r4621826 = r4621824 + r4621825;
        double r4621827 = sqrt(r4621826);
        double r4621828 = z;
        double r4621829 = r4621827 * r4621828;
        double r4621830 = /* ERROR: no posit support in C */;
        double r4621831 = /* ERROR: no posit support in C */;
        double r4621832 = r4621831 / r4621824;
        double r4621833 = 5.0;
        double r4621834 = 6.0;
        double r4621835 = r4621833 / r4621834;
        double r4621836 = r4621835 + r4621825;
        double r4621837 = 2.0;
        double r4621838 = 3.0;
        double r4621839 = r4621838 * r4621824;
        double r4621840 = r4621837 / r4621839;
        double r4621841 = r4621836 - r4621840;
        double r4621842 = c;
        double r4621843 = r4621820 - r4621842;
        double r4621844 = r4621841 * r4621843;
        double r4621845 = r4621832 - r4621844;
        double r4621846 = r4621845 * r4621837;
        double r4621847 = exp(r4621846);
        double r4621848 = y;
        double r4621849 = r4621847 * r4621848;
        double r4621850 = r4621823 + r4621849;
        double r4621851 = r4621823 / r4621850;
        double r4621852 = r4621828 / r4621824;
        double r4621853 = r4621827 * r4621852;
        double r4621854 = r4621837 / r4621824;
        double r4621855 = r4621854 / r4621838;
        double r4621856 = r4621825 - r4621855;
        double r4621857 = r4621835 + r4621856;
        double r4621858 = r4621857 * r4621843;
        double r4621859 = r4621853 - r4621858;
        double r4621860 = exp(r4621859);
        double r4621861 = sqrt(r4621860);
        double r4621862 = log(r4621861);
        double r4621863 = r4621862 + r4621862;
        double r4621864 = r4621837 * r4621863;
        double r4621865 = exp(r4621864);
        double r4621866 = r4621848 * r4621865;
        double r4621867 = r4621823 + r4621866;
        double r4621868 = r4621823 / r4621867;
        double r4621869 = r4621822 ? r4621851 : r4621868;
        return r4621869;
}

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 b < -2.978001093890994e+55

    1. Initial program 5.4

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
    2. Using strategy rm

    3. Applied insert-posit165.7

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{\color{blue}{\left(\left(z \cdot \sqrt{t + a}\right)\right)}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

    if -2.978001093890994e+55 < b

    1. Initial program 3.2

      \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
    2. Using strategy rm

    3. Applied add-log-exp8.1

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\log \left(e^{\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}\right)}\right)}}\]

    4. Applied add-log-exp15.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \left(\color{blue}{\log \left(e^{\frac{z \cdot \sqrt{t + a}}{t}}\right)} - \log \left(e^{\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}\right)\right)}}\]

    5. Applied diff-log15.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\log \left(\frac{e^{\frac{z \cdot \sqrt{t + a}}{t}}}{e^{\left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)}}\right)}}}\]

    6. Simplified2.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \log \color{blue}{\left(e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}\right)}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt2.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \log \color{blue}{\left(\sqrt{e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}} \cdot \sqrt{e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)}}}\]

    9. Applied log-prod2.6

      \[\leadsto \frac{x}{x + y \cdot e^{2.0 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\frac{z}{t} \cdot \sqrt{t + a} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.978001093890994 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{x + e^{\left(\frac{\left(\left(\sqrt{t + a} \cdot z\right)\right)}{t} - \left(\left(\frac{5.0}{6.0} + a\right) - \frac{2.0}{3.0 \cdot t}\right) \cdot \left(b - c\right)\right) \cdot 2.0} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2.0 \cdot \left(\log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right) + \log \left(\sqrt{e^{\sqrt{t + a} \cdot \frac{z}{t} - \left(\frac{5.0}{6.0} + \left(a - \frac{\frac{2.0}{t}}{3.0}\right)\right) \cdot \left(b - c\right)}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))