Average Error: 4.0 → 1.6
Time: 30.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\frac{\frac{\sqrt{a + t}}{\sqrt[3]{t}}}{\sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}\]
\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, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} - \left(\frac{\frac{2}{t}}{3} - a\right), \frac{\frac{\frac{\sqrt{a + t}}{\sqrt[3]{t}}}{\sqrt[3]{t}}}{\frac{\sqrt[3]{t}}{z}}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3121691 = x;
        double r3121692 = y;
        double r3121693 = 2.0;
        double r3121694 = z;
        double r3121695 = t;
        double r3121696 = a;
        double r3121697 = r3121695 + r3121696;
        double r3121698 = sqrt(r3121697);
        double r3121699 = r3121694 * r3121698;
        double r3121700 = r3121699 / r3121695;
        double r3121701 = b;
        double r3121702 = c;
        double r3121703 = r3121701 - r3121702;
        double r3121704 = 5.0;
        double r3121705 = 6.0;
        double r3121706 = r3121704 / r3121705;
        double r3121707 = r3121696 + r3121706;
        double r3121708 = 3.0;
        double r3121709 = r3121695 * r3121708;
        double r3121710 = r3121693 / r3121709;
        double r3121711 = r3121707 - r3121710;
        double r3121712 = r3121703 * r3121711;
        double r3121713 = r3121700 - r3121712;
        double r3121714 = r3121693 * r3121713;
        double r3121715 = exp(r3121714);
        double r3121716 = r3121692 * r3121715;
        double r3121717 = r3121691 + r3121716;
        double r3121718 = r3121691 / r3121717;
        return r3121718;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3121719 = x;
        double r3121720 = y;
        double r3121721 = 2.0;
        double r3121722 = c;
        double r3121723 = b;
        double r3121724 = r3121722 - r3121723;
        double r3121725 = 5.0;
        double r3121726 = 6.0;
        double r3121727 = r3121725 / r3121726;
        double r3121728 = t;
        double r3121729 = r3121721 / r3121728;
        double r3121730 = 3.0;
        double r3121731 = r3121729 / r3121730;
        double r3121732 = a;
        double r3121733 = r3121731 - r3121732;
        double r3121734 = r3121727 - r3121733;
        double r3121735 = r3121732 + r3121728;
        double r3121736 = sqrt(r3121735);
        double r3121737 = cbrt(r3121728);
        double r3121738 = r3121736 / r3121737;
        double r3121739 = r3121738 / r3121737;
        double r3121740 = z;
        double r3121741 = r3121737 / r3121740;
        double r3121742 = r3121739 / r3121741;
        double r3121743 = fma(r3121724, r3121734, r3121742);
        double r3121744 = r3121721 * r3121743;
        double r3121745 = exp(r3121744);
        double r3121746 = fma(r3121720, r3121745, r3121719);
        double r3121747 = r3121719 / r3121746;
        return r3121747;
}

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. Simplified1.7

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

  4. Applied *-un-lft-identity1.7

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

  5. Applied add-cube-cbrt1.7

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

  6. Applied times-frac1.7

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

  7. Applied associate-/r*1.6

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

  8. Simplified1.6

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

  9. Final simplification1.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))