Average Error: 4.0 → 1.4
Time: 2.9m
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, \left(e^{\mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\frac{5.0}{6.0} + \left(a - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot \left(c - b\right)\right)\right) \cdot 2.0}\right), x\right)}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, \left(e^{\mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\frac{5.0}{6.0} + \left(a - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot \left(c - b\right)\right)\right) \cdot 2.0}\right), x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r15794709 = x;
        double r15794710 = y;
        double r15794711 = 2.0;
        double r15794712 = z;
        double r15794713 = t;
        double r15794714 = a;
        double r15794715 = r15794713 + r15794714;
        double r15794716 = sqrt(r15794715);
        double r15794717 = r15794712 * r15794716;
        double r15794718 = r15794717 / r15794713;
        double r15794719 = b;
        double r15794720 = c;
        double r15794721 = r15794719 - r15794720;
        double r15794722 = 5.0;
        double r15794723 = 6.0;
        double r15794724 = r15794722 / r15794723;
        double r15794725 = r15794714 + r15794724;
        double r15794726 = 3.0;
        double r15794727 = r15794713 * r15794726;
        double r15794728 = r15794711 / r15794727;
        double r15794729 = r15794725 - r15794728;
        double r15794730 = r15794721 * r15794729;
        double r15794731 = r15794718 - r15794730;
        double r15794732 = r15794711 * r15794731;
        double r15794733 = exp(r15794732);
        double r15794734 = r15794710 * r15794733;
        double r15794735 = r15794709 + r15794734;
        double r15794736 = r15794709 / r15794735;
        return r15794736;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r15794737 = x;
        double r15794738 = y;
        double r15794739 = 1.0;
        double r15794740 = t;
        double r15794741 = cbrt(r15794740);
        double r15794742 = r15794741 * r15794741;
        double r15794743 = z;
        double r15794744 = cbrt(r15794743);
        double r15794745 = r15794744 * r15794744;
        double r15794746 = r15794742 / r15794745;
        double r15794747 = r15794739 / r15794746;
        double r15794748 = a;
        double r15794749 = r15794748 + r15794740;
        double r15794750 = sqrt(r15794749);
        double r15794751 = r15794741 / r15794744;
        double r15794752 = r15794750 / r15794751;
        double r15794753 = 5.0;
        double r15794754 = 6.0;
        double r15794755 = r15794753 / r15794754;
        double r15794756 = 2.0;
        double r15794757 = 3.0;
        double r15794758 = r15794740 * r15794757;
        double r15794759 = r15794756 / r15794758;
        double r15794760 = r15794748 - r15794759;
        double r15794761 = r15794755 + r15794760;
        double r15794762 = c;
        double r15794763 = b;
        double r15794764 = r15794762 - r15794763;
        double r15794765 = r15794761 * r15794764;
        double r15794766 = fma(r15794747, r15794752, r15794765);
        double r15794767 = r15794766 * r15794756;
        double r15794768 = exp(r15794767);
        double r15794769 = fma(r15794738, r15794768, r15794737);
        double r15794770 = r15794737 / r15794769;
        return r15794770;
}

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

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Simplified3.0

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{t}{z}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt3.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  5. Applied add-cube-cbrt3.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  6. Applied times-frac3.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  7. Applied *-un-lft-identity3.0

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\color{blue}{1 \cdot \sqrt{a + t}}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  8. Applied times-frac2.6

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\color{blue}{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  9. Applied fma-neg1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}}\right), x\right)}\]

  10. Simplified1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \color{blue}{\left(\left(c - b\right) \cdot \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right)\right)}\right)}\right), x\right)}\]

  11. Final simplification1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{\mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(\frac{5.0}{6.0} + \left(a - \frac{2.0}{t \cdot 3.0}\right)\right) \cdot \left(c - b\right)\right)\right) \cdot 2.0}\right), x\right)}\]

Reproduce

herbie shell --seed 2019128 +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)))))))))))