Average Error: 3.8 → 1.4
Time: 34.6s
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 r81683 = x;
        double r81684 = y;
        double r81685 = 2.0;
        double r81686 = z;
        double r81687 = t;
        double r81688 = a;
        double r81689 = r81687 + r81688;
        double r81690 = sqrt(r81689);
        double r81691 = r81686 * r81690;
        double r81692 = r81691 / r81687;
        double r81693 = b;
        double r81694 = c;
        double r81695 = r81693 - r81694;
        double r81696 = 5.0;
        double r81697 = 6.0;
        double r81698 = r81696 / r81697;
        double r81699 = r81688 + r81698;
        double r81700 = 3.0;
        double r81701 = r81687 * r81700;
        double r81702 = r81685 / r81701;
        double r81703 = r81699 - r81702;
        double r81704 = r81695 * r81703;
        double r81705 = r81692 - r81704;
        double r81706 = r81685 * r81705;
        double r81707 = exp(r81706);
        double r81708 = r81684 * r81707;
        double r81709 = r81683 + r81708;
        double r81710 = r81683 / r81709;
        return r81710;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r81711 = x;
        double r81712 = y;
        double r81713 = 2.0;
        double r81714 = exp(r81713);
        double r81715 = t;
        double r81716 = r81713 / r81715;
        double r81717 = 3.0;
        double r81718 = r81716 / r81717;
        double r81719 = a;
        double r81720 = 5.0;
        double r81721 = 6.0;
        double r81722 = r81720 / r81721;
        double r81723 = r81719 + r81722;
        double r81724 = r81718 - r81723;
        double r81725 = b;
        double r81726 = c;
        double r81727 = r81725 - r81726;
        double r81728 = z;
        double r81729 = cbrt(r81715);
        double r81730 = r81729 * r81729;
        double r81731 = r81728 / r81730;
        double r81732 = r81715 + r81719;
        double r81733 = sqrt(r81732);
        double r81734 = r81733 / r81729;
        double r81735 = r81731 * r81734;
        double r81736 = fma(r81724, r81727, r81735);
        double r81737 = pow(r81714, r81736);
        double r81738 = fma(r81712, r81737, r81711);
        double r81739 = r81711 / r81738;
        return r81739;
}

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

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

  4. Applied add-cube-cbrt2.6

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

  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 2019305 +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)))))))))))