Average Error: 3.8 → 1.4
Time: 15.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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{2}{t \cdot 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 r98787 = x;
        double r98788 = y;
        double r98789 = 2.0;
        double r98790 = z;
        double r98791 = t;
        double r98792 = a;
        double r98793 = r98791 + r98792;
        double r98794 = sqrt(r98793);
        double r98795 = r98790 * r98794;
        double r98796 = r98795 / r98791;
        double r98797 = b;
        double r98798 = c;
        double r98799 = r98797 - r98798;
        double r98800 = 5.0;
        double r98801 = 6.0;
        double r98802 = r98800 / r98801;
        double r98803 = r98792 + r98802;
        double r98804 = 3.0;
        double r98805 = r98791 * r98804;
        double r98806 = r98789 / r98805;
        double r98807 = r98803 - r98806;
        double r98808 = r98799 * r98807;
        double r98809 = r98796 - r98808;
        double r98810 = r98789 * r98809;
        double r98811 = exp(r98810);
        double r98812 = r98788 * r98811;
        double r98813 = r98787 + r98812;
        double r98814 = r98787 / r98813;
        return r98814;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r98815 = x;
        double r98816 = y;
        double r98817 = 2.0;
        double r98818 = exp(r98817);
        double r98819 = t;
        double r98820 = 3.0;
        double r98821 = r98819 * r98820;
        double r98822 = r98817 / r98821;
        double r98823 = a;
        double r98824 = 5.0;
        double r98825 = 6.0;
        double r98826 = r98824 / r98825;
        double r98827 = r98823 + r98826;
        double r98828 = r98822 - r98827;
        double r98829 = b;
        double r98830 = c;
        double r98831 = r98829 - r98830;
        double r98832 = z;
        double r98833 = cbrt(r98819);
        double r98834 = r98833 * r98833;
        double r98835 = r98832 / r98834;
        double r98836 = r98819 + r98823;
        double r98837 = sqrt(r98836);
        double r98838 = r98837 / r98833;
        double r98839 = r98835 * r98838;
        double r98840 = fma(r98828, r98831, r98839);
        double r98841 = pow(r98818, r98840);
        double r98842 = fma(r98816, r98841, r98815);
        double r98843 = r98815 / r98842;
        return r98843;
}

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. Simplified2.6

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

  4. Applied add-cube-cbrt2.6

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2020045 +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)))))))))))