Average Error: 29.2 → 29.4
Time: 7.4s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r84722 = x;
        double r84723 = y;
        double r84724 = r84722 * r84723;
        double r84725 = z;
        double r84726 = r84724 + r84725;
        double r84727 = r84726 * r84723;
        double r84728 = 27464.7644705;
        double r84729 = r84727 + r84728;
        double r84730 = r84729 * r84723;
        double r84731 = 230661.510616;
        double r84732 = r84730 + r84731;
        double r84733 = r84732 * r84723;
        double r84734 = t;
        double r84735 = r84733 + r84734;
        double r84736 = a;
        double r84737 = r84723 + r84736;
        double r84738 = r84737 * r84723;
        double r84739 = b;
        double r84740 = r84738 + r84739;
        double r84741 = r84740 * r84723;
        double r84742 = c;
        double r84743 = r84741 + r84742;
        double r84744 = r84743 * r84723;
        double r84745 = i;
        double r84746 = r84744 + r84745;
        double r84747 = r84735 / r84746;
        return r84747;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r84748 = x;
        double r84749 = y;
        double r84750 = z;
        double r84751 = fma(r84748, r84749, r84750);
        double r84752 = cbrt(r84749);
        double r84753 = r84751 * r84752;
        double r84754 = r84753 * r84752;
        double r84755 = r84754 * r84752;
        double r84756 = 27464.7644705;
        double r84757 = r84755 + r84756;
        double r84758 = r84757 * r84749;
        double r84759 = 230661.510616;
        double r84760 = r84758 + r84759;
        double r84761 = r84760 * r84749;
        double r84762 = t;
        double r84763 = r84761 + r84762;
        double r84764 = 1.0;
        double r84765 = a;
        double r84766 = r84749 + r84765;
        double r84767 = b;
        double r84768 = fma(r84766, r84749, r84767);
        double r84769 = c;
        double r84770 = fma(r84768, r84749, r84769);
        double r84771 = i;
        double r84772 = fma(r84770, r84749, r84771);
        double r84773 = r84772 * r84764;
        double r84774 = r84764 / r84773;
        double r84775 = r84763 * r84774;
        return r84775;
}

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

Bits error versus i

Derivation

  1. Initial program 29.2

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
  2. Using strategy rm

  3. Applied div-inv29.3

    \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}\]

  4. Simplified29.3

    \[\leadsto \left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt29.4

    \[\leadsto \left(\left(\left(\left(x \cdot y + z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  7. Applied associate-*r*29.4

    \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot y + z\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \sqrt[3]{y}} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  8. Simplified29.4

    \[\leadsto \left(\left(\left(\color{blue}{\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

  9. Final simplification29.4

    \[\leadsto \left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))