Average Error: 3.9 → 2.3
Time: 45.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{\log \left(e^{\mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{a + t}}{t}\right)}\right) \cdot 2.0}, x\right)}\]
\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)}}
↓
\frac{x}{\mathsf{fma}\left(y, e^{\log \left(e^{\mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{a + t}}{t}\right)}\right) \cdot 2.0}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2331785 = x;
        double r2331786 = y;
        double r2331787 = 2.0;
        double r2331788 = z;
        double r2331789 = t;
        double r2331790 = a;
        double r2331791 = r2331789 + r2331790;
        double r2331792 = sqrt(r2331791);
        double r2331793 = r2331788 * r2331792;
        double r2331794 = r2331793 / r2331789;
        double r2331795 = b;
        double r2331796 = c;
        double r2331797 = r2331795 - r2331796;
        double r2331798 = 5.0;
        double r2331799 = 6.0;
        double r2331800 = r2331798 / r2331799;
        double r2331801 = r2331790 + r2331800;
        double r2331802 = 3.0;
        double r2331803 = r2331789 * r2331802;
        double r2331804 = r2331787 / r2331803;
        double r2331805 = r2331801 - r2331804;
        double r2331806 = r2331797 * r2331805;
        double r2331807 = r2331794 - r2331806;
        double r2331808 = r2331787 * r2331807;
        double r2331809 = exp(r2331808);
        double r2331810 = r2331786 * r2331809;
        double r2331811 = r2331785 + r2331810;
        double r2331812 = r2331785 / r2331811;
        return r2331812;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2331813 = x;
        double r2331814 = y;
        double r2331815 = c;
        double r2331816 = b;
        double r2331817 = r2331815 - r2331816;
        double r2331818 = 5.0;
        double r2331819 = 6.0;
        double r2331820 = r2331818 / r2331819;
        double r2331821 = 2.0;
        double r2331822 = t;
        double r2331823 = r2331821 / r2331822;
        double r2331824 = 3.0;
        double r2331825 = r2331823 / r2331824;
        double r2331826 = a;
        double r2331827 = r2331825 - r2331826;
        double r2331828 = r2331820 - r2331827;
        double r2331829 = z;
        double r2331830 = r2331826 + r2331822;
        double r2331831 = sqrt(r2331830);
        double r2331832 = r2331831 / r2331822;
        double r2331833 = r2331829 * r2331832;
        double r2331834 = fma(r2331817, r2331828, r2331833);
        double r2331835 = exp(r2331834);
        double r2331836 = log(r2331835);
        double r2331837 = r2331836 * r2331821;
        double r2331838 = exp(r2331837);
        double r2331839 = fma(r2331814, r2331838, r2331813);
        double r2331840 = r2331813 / r2331839;
        return r2331840;
}

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. Initial program 3.9

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

  2. Simplified2.3

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

  4. Applied add-log-exp2.3

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

  5. Final simplification2.3

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

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(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)))))))))))