Average Error: 4.0 → 1.7
Time: 14.2s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
\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 r116747 = x;
        double r116748 = y;
        double r116749 = 2.0;
        double r116750 = z;
        double r116751 = t;
        double r116752 = a;
        double r116753 = r116751 + r116752;
        double r116754 = sqrt(r116753);
        double r116755 = r116750 * r116754;
        double r116756 = r116755 / r116751;
        double r116757 = b;
        double r116758 = c;
        double r116759 = r116757 - r116758;
        double r116760 = 5.0;
        double r116761 = 6.0;
        double r116762 = r116760 / r116761;
        double r116763 = r116752 + r116762;
        double r116764 = 3.0;
        double r116765 = r116751 * r116764;
        double r116766 = r116749 / r116765;
        double r116767 = r116763 - r116766;
        double r116768 = r116759 * r116767;
        double r116769 = r116756 - r116768;
        double r116770 = r116749 * r116769;
        double r116771 = exp(r116770);
        double r116772 = r116748 * r116771;
        double r116773 = r116747 + r116772;
        double r116774 = r116747 / r116773;
        return r116774;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r116775 = x;
        double r116776 = y;
        double r116777 = 2.0;
        double r116778 = z;
        double r116779 = t;
        double r116780 = cbrt(r116779);
        double r116781 = r116780 * r116780;
        double r116782 = r116778 / r116781;
        double r116783 = a;
        double r116784 = r116779 + r116783;
        double r116785 = sqrt(r116784);
        double r116786 = r116785 / r116780;
        double r116787 = b;
        double r116788 = c;
        double r116789 = r116787 - r116788;
        double r116790 = 5.0;
        double r116791 = 6.0;
        double r116792 = r116790 / r116791;
        double r116793 = r116783 + r116792;
        double r116794 = 3.0;
        double r116795 = r116779 * r116794;
        double r116796 = r116777 / r116795;
        double r116797 = r116793 - r116796;
        double r116798 = r116789 * r116797;
        double r116799 = -r116798;
        double r116800 = fma(r116782, r116786, r116799);
        double r116801 = r116777 * r116800;
        double r116802 = exp(r116801);
        double r116803 = r116776 * r116802;
        double r116804 = r116775 + r116803;
        double r116805 = r116775 / r116804;
        return r116805;
}

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-frac2.7

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

  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 2020065 +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)))))))))))