Average Error: 28.7 → 28.8
Time: 19.9s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.764470499998\right) \cdot y + 230661.510616000014\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]
\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}
↓
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.764470499998\right), y, 230661.510616000014\right), 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)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r65742 = x;
        double r65743 = y;
        double r65744 = r65742 * r65743;
        double r65745 = z;
        double r65746 = r65744 + r65745;
        double r65747 = r65746 * r65743;
        double r65748 = 27464.7644705;
        double r65749 = r65747 + r65748;
        double r65750 = r65749 * r65743;
        double r65751 = 230661.510616;
        double r65752 = r65750 + r65751;
        double r65753 = r65752 * r65743;
        double r65754 = t;
        double r65755 = r65753 + r65754;
        double r65756 = a;
        double r65757 = r65743 + r65756;
        double r65758 = r65757 * r65743;
        double r65759 = b;
        double r65760 = r65758 + r65759;
        double r65761 = r65760 * r65743;
        double r65762 = c;
        double r65763 = r65761 + r65762;
        double r65764 = r65763 * r65743;
        double r65765 = i;
        double r65766 = r65764 + r65765;
        double r65767 = r65755 / r65766;
        return r65767;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r65768 = x;
        double r65769 = y;
        double r65770 = z;
        double r65771 = fma(r65768, r65769, r65770);
        double r65772 = 27464.7644705;
        double r65773 = fma(r65771, r65769, r65772);
        double r65774 = 230661.510616;
        double r65775 = fma(r65773, r65769, r65774);
        double r65776 = t;
        double r65777 = fma(r65775, r65769, r65776);
        double r65778 = 1.0;
        double r65779 = a;
        double r65780 = r65769 + r65779;
        double r65781 = b;
        double r65782 = fma(r65780, r65769, r65781);
        double r65783 = c;
        double r65784 = fma(r65782, r65769, r65783);
        double r65785 = i;
        double r65786 = fma(r65784, r65769, r65785);
        double r65787 = r65778 / r65786;
        double r65788 = r65777 * r65787;
        return r65788;
}

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 28.7

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

  2. Simplified28.7

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

  4. Applied div-inv28.8

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

  5. Final simplification28.8

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

Reproduce

herbie shell --seed 2020047 +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)))