Average Error: 4.1 → 2.3
Time: 7.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)}}\]
\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}{1}, \frac{\sqrt{t + a}}{t}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r76582 = x;
        double r76583 = y;
        double r76584 = 2.0;
        double r76585 = z;
        double r76586 = t;
        double r76587 = a;
        double r76588 = r76586 + r76587;
        double r76589 = sqrt(r76588);
        double r76590 = r76585 * r76589;
        double r76591 = r76590 / r76586;
        double r76592 = b;
        double r76593 = c;
        double r76594 = r76592 - r76593;
        double r76595 = 5.0;
        double r76596 = 6.0;
        double r76597 = r76595 / r76596;
        double r76598 = r76587 + r76597;
        double r76599 = 3.0;
        double r76600 = r76586 * r76599;
        double r76601 = r76584 / r76600;
        double r76602 = r76598 - r76601;
        double r76603 = r76594 * r76602;
        double r76604 = r76591 - r76603;
        double r76605 = r76584 * r76604;
        double r76606 = exp(r76605);
        double r76607 = r76583 * r76606;
        double r76608 = r76582 + r76607;
        double r76609 = r76582 / r76608;
        return r76609;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r76610 = x;
        double r76611 = y;
        double r76612 = 2.0;
        double r76613 = z;
        double r76614 = 1.0;
        double r76615 = r76613 / r76614;
        double r76616 = t;
        double r76617 = a;
        double r76618 = r76616 + r76617;
        double r76619 = sqrt(r76618);
        double r76620 = r76619 / r76616;
        double r76621 = 5.0;
        double r76622 = 6.0;
        double r76623 = r76621 / r76622;
        double r76624 = r76617 + r76623;
        double r76625 = 3.0;
        double r76626 = r76616 * r76625;
        double r76627 = r76612 / r76626;
        double r76628 = -r76627;
        double r76629 = fma(r76614, r76624, r76628);
        double r76630 = -r76629;
        double r76631 = b;
        double r76632 = c;
        double r76633 = r76631 - r76632;
        double r76634 = r76630 * r76633;
        double r76635 = fma(r76615, r76620, r76634);
        double r76636 = r76612 * r76635;
        double r76637 = exp(r76636);
        double r76638 = r76611 * r76637;
        double r76639 = r76610 + r76638;
        double r76640 = r76610 / r76639;
        return r76640;
}

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. Using strategy rm

  3. Applied add-cbrt-cube4.1

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

  4. Applied add-cbrt-cube7.1

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

  5. Applied cbrt-unprod7.1

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

  6. Applied add-cbrt-cube7.1

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

  7. Applied cbrt-undiv7.3

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

  8. Simplified7.3

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

  10. Applied *-un-lft-identity7.3

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}\]

  11. Applied times-frac6.6

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}\]

  12. Applied fma-neg5.4

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}}\]

  13. Simplified2.3

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, \color{blue}{\left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)}\right)}}\]

  14. Final simplification2.3

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{1}, \frac{\sqrt{t + a}}{t}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)}}\]

Reproduce

herbie shell --seed 2019362 +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)))))))))))