Average Error: 3.9 → 2.0
Time: 7.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\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)}}
↓
\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r74830 = x;
        double r74831 = y;
        double r74832 = 2.0;
        double r74833 = z;
        double r74834 = t;
        double r74835 = a;
        double r74836 = r74834 + r74835;
        double r74837 = sqrt(r74836);
        double r74838 = r74833 * r74837;
        double r74839 = r74838 / r74834;
        double r74840 = b;
        double r74841 = c;
        double r74842 = r74840 - r74841;
        double r74843 = 5.0;
        double r74844 = 6.0;
        double r74845 = r74843 / r74844;
        double r74846 = r74835 + r74845;
        double r74847 = 3.0;
        double r74848 = r74834 * r74847;
        double r74849 = r74832 / r74848;
        double r74850 = r74846 - r74849;
        double r74851 = r74842 * r74850;
        double r74852 = r74839 - r74851;
        double r74853 = r74832 * r74852;
        double r74854 = exp(r74853);
        double r74855 = r74831 * r74854;
        double r74856 = r74830 + r74855;
        double r74857 = r74830 / r74856;
        return r74857;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r74858 = x;
        double r74859 = y;
        double r74860 = 2.0;
        double r74861 = z;
        double r74862 = 1.0;
        double r74863 = r74861 / r74862;
        double r74864 = t;
        double r74865 = a;
        double r74866 = r74864 + r74865;
        double r74867 = sqrt(r74866);
        double r74868 = r74867 / r74864;
        double r74869 = b;
        double r74870 = c;
        double r74871 = r74869 - r74870;
        double r74872 = 5.0;
        double r74873 = 6.0;
        double r74874 = r74872 / r74873;
        double r74875 = r74865 + r74874;
        double r74876 = 3.0;
        double r74877 = r74864 * r74876;
        double r74878 = r74860 / r74877;
        double r74879 = r74875 - r74878;
        double r74880 = r74871 * r74879;
        double r74881 = -r74880;
        double r74882 = fma(r74863, r74868, r74881);
        double r74883 = r74860 * r74882;
        double r74884 = exp(r74883);
        double r74885 = r74859 * r74884;
        double r74886 = r74858 + r74885;
        double r74887 = r74858 / r74886;
        return r74887;
}

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

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

  3. Applied *-un-lft-identity3.9

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

  4. Applied times-frac3.2

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

  5. Applied fma-neg2.0

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

  6. Final simplification2.0

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

Reproduce

herbie shell --seed 2020018 +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)))))))))))