Average Error: 3.8 → 1.8
Time: 16.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\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}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{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 r101742 = x;
        double r101743 = y;
        double r101744 = 2.0;
        double r101745 = z;
        double r101746 = t;
        double r101747 = a;
        double r101748 = r101746 + r101747;
        double r101749 = sqrt(r101748);
        double r101750 = r101745 * r101749;
        double r101751 = r101750 / r101746;
        double r101752 = b;
        double r101753 = c;
        double r101754 = r101752 - r101753;
        double r101755 = 5.0;
        double r101756 = 6.0;
        double r101757 = r101755 / r101756;
        double r101758 = r101747 + r101757;
        double r101759 = 3.0;
        double r101760 = r101746 * r101759;
        double r101761 = r101744 / r101760;
        double r101762 = r101758 - r101761;
        double r101763 = r101754 * r101762;
        double r101764 = r101751 - r101763;
        double r101765 = r101744 * r101764;
        double r101766 = exp(r101765);
        double r101767 = r101743 * r101766;
        double r101768 = r101742 + r101767;
        double r101769 = r101742 / r101768;
        return r101769;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r101770 = x;
        double r101771 = y;
        double r101772 = 2.0;
        double r101773 = z;
        double r101774 = t;
        double r101775 = cbrt(r101774);
        double r101776 = r101775 * r101775;
        double r101777 = r101773 / r101776;
        double r101778 = a;
        double r101779 = r101774 + r101778;
        double r101780 = sqrt(r101779);
        double r101781 = r101780 / r101775;
        double r101782 = b;
        double r101783 = c;
        double r101784 = r101782 - r101783;
        double r101785 = 5.0;
        double r101786 = 6.0;
        double r101787 = r101785 / r101786;
        double r101788 = r101778 + r101787;
        double r101789 = 3.0;
        double r101790 = r101774 * r101789;
        double r101791 = r101772 / r101790;
        double r101792 = r101788 - r101791;
        double r101793 = r101784 * r101792;
        double r101794 = -r101793;
        double r101795 = fma(r101777, r101781, r101794);
        double r101796 = r101772 * r101795;
        double r101797 = exp(r101796);
        double r101798 = r101771 * r101797;
        double r101799 = r101770 + r101798;
        double r101800 = r101770 / r101799;
        return r101800;
}

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. Using strategy rm

  3. Applied add-cube-cbrt3.8

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

  4. Applied times-frac2.6

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

  5. Applied fma-neg1.8

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

  6. Final simplification1.8

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

Reproduce

herbie shell --seed 2019354 +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)))))))))))