Average Error: 3.8 → 2.0
Time: 23.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\sqrt[3]{\sqrt{t + a}} \cdot \sqrt[3]{\sqrt{t + a}}\right) \cdot \frac{z \cdot \sqrt[3]{\sqrt{t + a}}}{t}\right)}, x\right)}\]
\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, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \left(\sqrt[3]{\sqrt{t + a}} \cdot \sqrt[3]{\sqrt{t + a}}\right) \cdot \frac{z \cdot \sqrt[3]{\sqrt{t + a}}}{t}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r67778 = x;
        double r67779 = y;
        double r67780 = 2.0;
        double r67781 = z;
        double r67782 = t;
        double r67783 = a;
        double r67784 = r67782 + r67783;
        double r67785 = sqrt(r67784);
        double r67786 = r67781 * r67785;
        double r67787 = r67786 / r67782;
        double r67788 = b;
        double r67789 = c;
        double r67790 = r67788 - r67789;
        double r67791 = 5.0;
        double r67792 = 6.0;
        double r67793 = r67791 / r67792;
        double r67794 = r67783 + r67793;
        double r67795 = 3.0;
        double r67796 = r67782 * r67795;
        double r67797 = r67780 / r67796;
        double r67798 = r67794 - r67797;
        double r67799 = r67790 * r67798;
        double r67800 = r67787 - r67799;
        double r67801 = r67780 * r67800;
        double r67802 = exp(r67801);
        double r67803 = r67779 * r67802;
        double r67804 = r67778 + r67803;
        double r67805 = r67778 / r67804;
        return r67805;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r67806 = x;
        double r67807 = y;
        double r67808 = 2.0;
        double r67809 = c;
        double r67810 = b;
        double r67811 = r67809 - r67810;
        double r67812 = 5.0;
        double r67813 = 6.0;
        double r67814 = r67812 / r67813;
        double r67815 = a;
        double r67816 = t;
        double r67817 = r67808 / r67816;
        double r67818 = 3.0;
        double r67819 = r67817 / r67818;
        double r67820 = r67815 - r67819;
        double r67821 = r67814 + r67820;
        double r67822 = r67816 + r67815;
        double r67823 = sqrt(r67822);
        double r67824 = cbrt(r67823);
        double r67825 = r67824 * r67824;
        double r67826 = z;
        double r67827 = r67826 * r67824;
        double r67828 = r67827 / r67816;
        double r67829 = r67825 * r67828;
        double r67830 = fma(r67811, r67821, r67829);
        double r67831 = r67808 * r67830;
        double r67832 = exp(r67831);
        double r67833 = fma(r67807, r67832, r67806);
        double r67834 = r67806 / r67833;
        return r67834;
}

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. Simplified1.9

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

  4. Applied add-cube-cbrt1.9

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

  5. Applied associate-*l*1.9

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

  6. Simplified2.0

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

  7. Final simplification2.0

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

Reproduce

herbie shell --seed 2019194 +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)))))))))))