Average Error: 28.5 → 28.6
Time: 8.2s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\left(\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right) \cdot 1}\]
\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(x \cdot y + z\right) \cdot 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 r100704 = x;
        double r100705 = y;
        double r100706 = r100704 * r100705;
        double r100707 = z;
        double r100708 = r100706 + r100707;
        double r100709 = r100708 * r100705;
        double r100710 = 27464.7644705;
        double r100711 = r100709 + r100710;
        double r100712 = r100711 * r100705;
        double r100713 = 230661.510616;
        double r100714 = r100712 + r100713;
        double r100715 = r100714 * r100705;
        double r100716 = t;
        double r100717 = r100715 + r100716;
        double r100718 = a;
        double r100719 = r100705 + r100718;
        double r100720 = r100719 * r100705;
        double r100721 = b;
        double r100722 = r100720 + r100721;
        double r100723 = r100722 * r100705;
        double r100724 = c;
        double r100725 = r100723 + r100724;
        double r100726 = r100725 * r100705;
        double r100727 = i;
        double r100728 = r100726 + r100727;
        double r100729 = r100717 / r100728;
        return r100729;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r100730 = x;
        double r100731 = y;
        double r100732 = r100730 * r100731;
        double r100733 = z;
        double r100734 = r100732 + r100733;
        double r100735 = r100734 * r100731;
        double r100736 = 27464.7644705;
        double r100737 = r100735 + r100736;
        double r100738 = r100737 * r100731;
        double r100739 = 230661.510616;
        double r100740 = r100738 + r100739;
        double r100741 = r100740 * r100731;
        double r100742 = t;
        double r100743 = r100741 + r100742;
        double r100744 = 1.0;
        double r100745 = a;
        double r100746 = r100731 + r100745;
        double r100747 = b;
        double r100748 = fma(r100746, r100731, r100747);
        double r100749 = c;
        double r100750 = fma(r100748, r100731, r100749);
        double r100751 = i;
        double r100752 = fma(r100750, r100731, r100751);
        double r100753 = r100752 * r100744;
        double r100754 = r100744 / r100753;
        double r100755 = r100743 * r100754;
        return r100755;
}

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 28.5

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

  3. Applied div-inv28.6

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

  4. Simplified28.6

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

  5. Final simplification28.6

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

Reproduce

herbie shell --seed 2020039 +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)))