Average Error: 4.0 → 1.4
Time: 12.5s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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}{3}}{t} - \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 r88572 = x;
        double r88573 = y;
        double r88574 = 2.0;
        double r88575 = z;
        double r88576 = t;
        double r88577 = a;
        double r88578 = r88576 + r88577;
        double r88579 = sqrt(r88578);
        double r88580 = r88575 * r88579;
        double r88581 = r88580 / r88576;
        double r88582 = b;
        double r88583 = c;
        double r88584 = r88582 - r88583;
        double r88585 = 5.0;
        double r88586 = 6.0;
        double r88587 = r88585 / r88586;
        double r88588 = r88577 + r88587;
        double r88589 = 3.0;
        double r88590 = r88576 * r88589;
        double r88591 = r88574 / r88590;
        double r88592 = r88588 - r88591;
        double r88593 = r88584 * r88592;
        double r88594 = r88581 - r88593;
        double r88595 = r88574 * r88594;
        double r88596 = exp(r88595);
        double r88597 = r88573 * r88596;
        double r88598 = r88572 + r88597;
        double r88599 = r88572 / r88598;
        return r88599;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r88600 = x;
        double r88601 = y;
        double r88602 = 2.0;
        double r88603 = exp(r88602);
        double r88604 = 3.0;
        double r88605 = r88602 / r88604;
        double r88606 = t;
        double r88607 = r88605 / r88606;
        double r88608 = a;
        double r88609 = 5.0;
        double r88610 = 6.0;
        double r88611 = r88609 / r88610;
        double r88612 = r88608 + r88611;
        double r88613 = r88607 - r88612;
        double r88614 = b;
        double r88615 = c;
        double r88616 = r88614 - r88615;
        double r88617 = z;
        double r88618 = cbrt(r88606);
        double r88619 = r88618 * r88618;
        double r88620 = r88617 / r88619;
        double r88621 = r88606 + r88608;
        double r88622 = sqrt(r88621);
        double r88623 = r88622 / r88618;
        double r88624 = r88620 * r88623;
        double r88625 = fma(r88613, r88616, r88624);
        double r88626 = pow(r88603, r88625);
        double r88627 = fma(r88601, r88626, r88600);
        double r88628 = r88600 / r88627;
        return r88628;
}

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. Simplified2.6

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

  4. Applied add-cube-cbrt2.6

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2020047 +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)))))))))))