Average Error: 3.8 → 1.4
Time: 27.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r92796 = x;
        double r92797 = y;
        double r92798 = 2.0;
        double r92799 = z;
        double r92800 = t;
        double r92801 = a;
        double r92802 = r92800 + r92801;
        double r92803 = sqrt(r92802);
        double r92804 = r92799 * r92803;
        double r92805 = r92804 / r92800;
        double r92806 = b;
        double r92807 = c;
        double r92808 = r92806 - r92807;
        double r92809 = 5.0;
        double r92810 = 6.0;
        double r92811 = r92809 / r92810;
        double r92812 = r92801 + r92811;
        double r92813 = 3.0;
        double r92814 = r92800 * r92813;
        double r92815 = r92798 / r92814;
        double r92816 = r92812 - r92815;
        double r92817 = r92808 * r92816;
        double r92818 = r92805 - r92817;
        double r92819 = r92798 * r92818;
        double r92820 = exp(r92819);
        double r92821 = r92797 * r92820;
        double r92822 = r92796 + r92821;
        double r92823 = r92796 / r92822;
        return r92823;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r92824 = x;
        double r92825 = y;
        double r92826 = 2.0;
        double r92827 = exp(r92826);
        double r92828 = t;
        double r92829 = r92826 / r92828;
        double r92830 = 3.0;
        double r92831 = r92829 / r92830;
        double r92832 = a;
        double r92833 = 5.0;
        double r92834 = 6.0;
        double r92835 = r92833 / r92834;
        double r92836 = r92832 + r92835;
        double r92837 = r92831 - r92836;
        double r92838 = b;
        double r92839 = c;
        double r92840 = r92838 - r92839;
        double r92841 = z;
        double r92842 = cbrt(r92828);
        double r92843 = r92842 * r92842;
        double r92844 = r92841 / r92843;
        double r92845 = r92828 + r92832;
        double r92846 = sqrt(r92845);
        double r92847 = r92846 / r92842;
        double r92848 = r92844 * r92847;
        double r92849 = fma(r92837, r92840, r92848);
        double r92850 = pow(r92827, r92849);
        double r92851 = fma(r92825, r92850, r92824);
        double r92852 = r92824 / r92851;
        return r92852;
}

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 3.8

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

  2. Simplified2.5

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

  4. Applied add-cube-cbrt2.5

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2019304 +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)))))))))))