Average Error: 4.0 → 1.9
Time: 15.1s
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 r114573 = x;
        double r114574 = y;
        double r114575 = 2.0;
        double r114576 = z;
        double r114577 = t;
        double r114578 = a;
        double r114579 = r114577 + r114578;
        double r114580 = sqrt(r114579);
        double r114581 = r114576 * r114580;
        double r114582 = r114581 / r114577;
        double r114583 = b;
        double r114584 = c;
        double r114585 = r114583 - r114584;
        double r114586 = 5.0;
        double r114587 = 6.0;
        double r114588 = r114586 / r114587;
        double r114589 = r114578 + r114588;
        double r114590 = 3.0;
        double r114591 = r114577 * r114590;
        double r114592 = r114575 / r114591;
        double r114593 = r114589 - r114592;
        double r114594 = r114585 * r114593;
        double r114595 = r114582 - r114594;
        double r114596 = r114575 * r114595;
        double r114597 = exp(r114596);
        double r114598 = r114574 * r114597;
        double r114599 = r114573 + r114598;
        double r114600 = r114573 / r114599;
        return r114600;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r114601 = x;
        double r114602 = y;
        double r114603 = 2.0;
        double r114604 = z;
        double r114605 = t;
        double r114606 = cbrt(r114605);
        double r114607 = r114606 * r114606;
        double r114608 = r114604 / r114607;
        double r114609 = a;
        double r114610 = r114605 + r114609;
        double r114611 = sqrt(r114610);
        double r114612 = r114611 / r114606;
        double r114613 = b;
        double r114614 = c;
        double r114615 = r114613 - r114614;
        double r114616 = 5.0;
        double r114617 = 6.0;
        double r114618 = r114616 / r114617;
        double r114619 = r114609 + r114618;
        double r114620 = 3.0;
        double r114621 = r114605 * r114620;
        double r114622 = r114603 / r114621;
        double r114623 = r114619 - r114622;
        double r114624 = r114615 * r114623;
        double r114625 = -r114624;
        double r114626 = fma(r114608, r114612, r114625);
        double r114627 = r114603 * r114626;
        double r114628 = exp(r114627);
        double r114629 = r114602 * r114628;
        double r114630 = r114601 + r114629;
        double r114631 = r114601 / r114630;
        return r114631;
}

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 4.0

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

  3. Applied add-cube-cbrt4.0

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

  4. Applied times-frac2.8

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

  5. Applied fma-neg1.9

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

  6. Final simplification1.9

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

Reproduce

herbie shell --seed 2020057 +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)))))))))))