Average Error: 4.1 → 1.4
Time: 23.6s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}\]
\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{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t}} \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r60671 = x;
        double r60672 = y;
        double r60673 = 2.0;
        double r60674 = z;
        double r60675 = t;
        double r60676 = a;
        double r60677 = r60675 + r60676;
        double r60678 = sqrt(r60677);
        double r60679 = r60674 * r60678;
        double r60680 = r60679 / r60675;
        double r60681 = b;
        double r60682 = c;
        double r60683 = r60681 - r60682;
        double r60684 = 5.0;
        double r60685 = 6.0;
        double r60686 = r60684 / r60685;
        double r60687 = r60676 + r60686;
        double r60688 = 3.0;
        double r60689 = r60675 * r60688;
        double r60690 = r60673 / r60689;
        double r60691 = r60687 - r60690;
        double r60692 = r60683 * r60691;
        double r60693 = r60680 - r60692;
        double r60694 = r60673 * r60693;
        double r60695 = exp(r60694);
        double r60696 = r60672 * r60695;
        double r60697 = r60671 + r60696;
        double r60698 = r60671 / r60697;
        return r60698;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r60699 = x;
        double r60700 = y;
        double r60701 = 2.0;
        double r60702 = exp(r60701);
        double r60703 = t;
        double r60704 = r60701 / r60703;
        double r60705 = 3.0;
        double r60706 = r60704 / r60705;
        double r60707 = a;
        double r60708 = 5.0;
        double r60709 = 6.0;
        double r60710 = r60708 / r60709;
        double r60711 = r60707 + r60710;
        double r60712 = r60706 - r60711;
        double r60713 = b;
        double r60714 = c;
        double r60715 = r60713 - r60714;
        double r60716 = z;
        double r60717 = cbrt(r60716);
        double r60718 = r60717 * r60717;
        double r60719 = cbrt(r60703);
        double r60720 = r60718 / r60719;
        double r60721 = r60717 / r60719;
        double r60722 = r60703 + r60707;
        double r60723 = sqrt(r60722);
        double r60724 = r60723 / r60719;
        double r60725 = r60721 * r60724;
        double r60726 = r60720 * r60725;
        double r60727 = fma(r60712, r60715, r60726);
        double r60728 = pow(r60702, r60727);
        double r60729 = fma(r60700, r60728, r60699);
        double r60730 = r60699 / r60729;
        return r60730;
}

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 4.1

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

  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-frac1.5

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

  7. Applied add-cube-cbrt1.5

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

  8. Applied times-frac1.5

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

  9. Applied associate-*l*1.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2019325 +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)))))))))))