Average Error: 3.7 → 1.3
Time: 29.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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}{t}}{3} - \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 r76570 = x;
        double r76571 = y;
        double r76572 = 2.0;
        double r76573 = z;
        double r76574 = t;
        double r76575 = a;
        double r76576 = r76574 + r76575;
        double r76577 = sqrt(r76576);
        double r76578 = r76573 * r76577;
        double r76579 = r76578 / r76574;
        double r76580 = b;
        double r76581 = c;
        double r76582 = r76580 - r76581;
        double r76583 = 5.0;
        double r76584 = 6.0;
        double r76585 = r76583 / r76584;
        double r76586 = r76575 + r76585;
        double r76587 = 3.0;
        double r76588 = r76574 * r76587;
        double r76589 = r76572 / r76588;
        double r76590 = r76586 - r76589;
        double r76591 = r76582 * r76590;
        double r76592 = r76579 - r76591;
        double r76593 = r76572 * r76592;
        double r76594 = exp(r76593);
        double r76595 = r76571 * r76594;
        double r76596 = r76570 + r76595;
        double r76597 = r76570 / r76596;
        return r76597;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r76598 = x;
        double r76599 = y;
        double r76600 = 2.0;
        double r76601 = exp(r76600);
        double r76602 = t;
        double r76603 = r76600 / r76602;
        double r76604 = 3.0;
        double r76605 = r76603 / r76604;
        double r76606 = a;
        double r76607 = 5.0;
        double r76608 = 6.0;
        double r76609 = r76607 / r76608;
        double r76610 = r76606 + r76609;
        double r76611 = r76605 - r76610;
        double r76612 = b;
        double r76613 = c;
        double r76614 = r76612 - r76613;
        double r76615 = z;
        double r76616 = cbrt(r76602);
        double r76617 = r76616 * r76616;
        double r76618 = r76615 / r76617;
        double r76619 = r76602 + r76606;
        double r76620 = sqrt(r76619);
        double r76621 = r76620 / r76616;
        double r76622 = r76618 * r76621;
        double r76623 = fma(r76611, r76614, r76622);
        double r76624 = pow(r76601, r76623);
        double r76625 = fma(r76599, r76624, r76598);
        double r76626 = r76598 / r76625;
        return r76626;
}

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 3.7

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

  2. Simplified2.4

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

  4. Applied add-cube-cbrt2.4

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

  5. Applied times-frac1.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2019303 +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)))))))))))