Average Error: 3.9 → 1.7
Time: 8.5s
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 r84603 = x;
        double r84604 = y;
        double r84605 = 2.0;
        double r84606 = z;
        double r84607 = t;
        double r84608 = a;
        double r84609 = r84607 + r84608;
        double r84610 = sqrt(r84609);
        double r84611 = r84606 * r84610;
        double r84612 = r84611 / r84607;
        double r84613 = b;
        double r84614 = c;
        double r84615 = r84613 - r84614;
        double r84616 = 5.0;
        double r84617 = 6.0;
        double r84618 = r84616 / r84617;
        double r84619 = r84608 + r84618;
        double r84620 = 3.0;
        double r84621 = r84607 * r84620;
        double r84622 = r84605 / r84621;
        double r84623 = r84619 - r84622;
        double r84624 = r84615 * r84623;
        double r84625 = r84612 - r84624;
        double r84626 = r84605 * r84625;
        double r84627 = exp(r84626);
        double r84628 = r84604 * r84627;
        double r84629 = r84603 + r84628;
        double r84630 = r84603 / r84629;
        return r84630;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r84631 = x;
        double r84632 = y;
        double r84633 = 2.0;
        double r84634 = z;
        double r84635 = t;
        double r84636 = cbrt(r84635);
        double r84637 = r84636 * r84636;
        double r84638 = r84634 / r84637;
        double r84639 = a;
        double r84640 = r84635 + r84639;
        double r84641 = sqrt(r84640);
        double r84642 = r84641 / r84636;
        double r84643 = b;
        double r84644 = c;
        double r84645 = r84643 - r84644;
        double r84646 = 5.0;
        double r84647 = 6.0;
        double r84648 = r84646 / r84647;
        double r84649 = r84639 + r84648;
        double r84650 = 3.0;
        double r84651 = r84635 * r84650;
        double r84652 = r84633 / r84651;
        double r84653 = r84649 - r84652;
        double r84654 = r84645 * r84653;
        double r84655 = -r84654;
        double r84656 = fma(r84638, r84642, r84655);
        double r84657 = r84633 * r84656;
        double r84658 = exp(r84657);
        double r84659 = r84632 * r84658;
        double r84660 = r84631 + r84659;
        double r84661 = r84631 / r84660;
        return r84661;
}

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 3.9

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

  3. Applied add-cube-cbrt3.9

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

  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-neg1.7

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

  6. Final simplification1.7

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

Reproduce

herbie shell --seed 2019352 +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)))))))))))