Average Error: 4.0 → 1.5
Time: 19.3s
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 r125743 = x;
        double r125744 = y;
        double r125745 = 2.0;
        double r125746 = z;
        double r125747 = t;
        double r125748 = a;
        double r125749 = r125747 + r125748;
        double r125750 = sqrt(r125749);
        double r125751 = r125746 * r125750;
        double r125752 = r125751 / r125747;
        double r125753 = b;
        double r125754 = c;
        double r125755 = r125753 - r125754;
        double r125756 = 5.0;
        double r125757 = 6.0;
        double r125758 = r125756 / r125757;
        double r125759 = r125748 + r125758;
        double r125760 = 3.0;
        double r125761 = r125747 * r125760;
        double r125762 = r125745 / r125761;
        double r125763 = r125759 - r125762;
        double r125764 = r125755 * r125763;
        double r125765 = r125752 - r125764;
        double r125766 = r125745 * r125765;
        double r125767 = exp(r125766);
        double r125768 = r125744 * r125767;
        double r125769 = r125743 + r125768;
        double r125770 = r125743 / r125769;
        return r125770;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r125771 = x;
        double r125772 = y;
        double r125773 = 2.0;
        double r125774 = exp(r125773);
        double r125775 = 3.0;
        double r125776 = r125773 / r125775;
        double r125777 = t;
        double r125778 = r125776 / r125777;
        double r125779 = a;
        double r125780 = 5.0;
        double r125781 = 6.0;
        double r125782 = r125780 / r125781;
        double r125783 = r125779 + r125782;
        double r125784 = r125778 - r125783;
        double r125785 = b;
        double r125786 = c;
        double r125787 = r125785 - r125786;
        double r125788 = z;
        double r125789 = cbrt(r125777);
        double r125790 = r125789 * r125789;
        double r125791 = r125788 / r125790;
        double r125792 = r125777 + r125779;
        double r125793 = sqrt(r125792);
        double r125794 = r125793 / r125789;
        double r125795 = r125791 * r125794;
        double r125796 = fma(r125784, r125787, r125795);
        double r125797 = pow(r125774, r125796);
        double r125798 = fma(r125772, r125797, r125771);
        double r125799 = r125771 / r125798;
        return r125799;
}

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. Simplified2.7

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

  4. Applied add-cube-cbrt2.7

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

  5. Applied times-frac1.5

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

  6. Final simplification1.5

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

Reproduce

herbie shell --seed 2020043 +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)))))))))))