Average Error: 3.7 → 1.4
Time: 36.1s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}\right)}, x\right)}\]
\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^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2310763 = x;
        double r2310764 = y;
        double r2310765 = 2.0;
        double r2310766 = z;
        double r2310767 = t;
        double r2310768 = a;
        double r2310769 = r2310767 + r2310768;
        double r2310770 = sqrt(r2310769);
        double r2310771 = r2310766 * r2310770;
        double r2310772 = r2310771 / r2310767;
        double r2310773 = b;
        double r2310774 = c;
        double r2310775 = r2310773 - r2310774;
        double r2310776 = 5.0;
        double r2310777 = 6.0;
        double r2310778 = r2310776 / r2310777;
        double r2310779 = r2310768 + r2310778;
        double r2310780 = 3.0;
        double r2310781 = r2310767 * r2310780;
        double r2310782 = r2310765 / r2310781;
        double r2310783 = r2310779 - r2310782;
        double r2310784 = r2310775 * r2310783;
        double r2310785 = r2310772 - r2310784;
        double r2310786 = r2310765 * r2310785;
        double r2310787 = exp(r2310786);
        double r2310788 = r2310764 * r2310787;
        double r2310789 = r2310763 + r2310788;
        double r2310790 = r2310763 / r2310789;
        return r2310790;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2310791 = x;
        double r2310792 = y;
        double r2310793 = 2.0;
        double r2310794 = c;
        double r2310795 = b;
        double r2310796 = r2310794 - r2310795;
        double r2310797 = 5.0;
        double r2310798 = 6.0;
        double r2310799 = r2310797 / r2310798;
        double r2310800 = t;
        double r2310801 = r2310793 / r2310800;
        double r2310802 = 3.0;
        double r2310803 = r2310801 / r2310802;
        double r2310804 = a;
        double r2310805 = r2310803 - r2310804;
        double r2310806 = r2310799 - r2310805;
        double r2310807 = z;
        double r2310808 = cbrt(r2310800);
        double r2310809 = r2310808 * r2310808;
        double r2310810 = r2310807 / r2310809;
        double r2310811 = r2310804 + r2310800;
        double r2310812 = sqrt(r2310811);
        double r2310813 = r2310812 / r2310808;
        double r2310814 = r2310810 * r2310813;
        double r2310815 = fma(r2310796, r2310806, r2310814);
        double r2310816 = r2310793 * r2310815;
        double r2310817 = exp(r2310816);
        double r2310818 = fma(r2310792, r2310817, r2310791);
        double r2310819 = r2310791 / r2310818;
        return r2310819;
}

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 3.7

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

  2. Simplified2.0

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

  4. Applied add-cube-cbrt2.0

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

  5. Applied *-un-lft-identity2.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{a + \color{blue}{1 \cdot t}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  6. Applied *-un-lft-identity2.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{\color{blue}{1 \cdot a} + 1 \cdot t}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  7. Applied distribute-lft-out2.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\sqrt{\color{blue}{1 \cdot \left(a + t\right)}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  8. Applied sqrt-prod2.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{a + t}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\right)}, x\right)}\]

  9. Applied times-frac2.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), z \cdot \color{blue}{\left(\frac{\sqrt{1}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}\right)}\right)}, x\right)}\]

  10. Applied associate-*r*1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \color{blue}{\left(z \cdot \frac{\sqrt{1}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt{a + t}}{\sqrt[3]{t}}}\right)}, x\right)}\]

  11. Simplified1.4

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

  12. Final simplification1.4

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

Reproduce

herbie shell --seed 2019151 +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)))))))))))