Average Error: 28.7 → 28.8
Time: 16.5s
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 r98776 = x;
        double r98777 = y;
        double r98778 = r98776 * r98777;
        double r98779 = z;
        double r98780 = r98778 + r98779;
        double r98781 = r98780 * r98777;
        double r98782 = 27464.7644705;
        double r98783 = r98781 + r98782;
        double r98784 = r98783 * r98777;
        double r98785 = 230661.510616;
        double r98786 = r98784 + r98785;
        double r98787 = r98786 * r98777;
        double r98788 = t;
        double r98789 = r98787 + r98788;
        double r98790 = a;
        double r98791 = r98777 + r98790;
        double r98792 = r98791 * r98777;
        double r98793 = b;
        double r98794 = r98792 + r98793;
        double r98795 = r98794 * r98777;
        double r98796 = c;
        double r98797 = r98795 + r98796;
        double r98798 = r98797 * r98777;
        double r98799 = i;
        double r98800 = r98798 + r98799;
        double r98801 = r98789 / r98800;
        return r98801;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r98802 = x;
        double r98803 = y;
        double r98804 = z;
        double r98805 = fma(r98802, r98803, r98804);
        double r98806 = 27464.7644705;
        double r98807 = fma(r98805, r98803, r98806);
        double r98808 = 230661.510616;
        double r98809 = fma(r98807, r98803, r98808);
        double r98810 = t;
        double r98811 = fma(r98809, r98803, r98810);
        double r98812 = 1.0;
        double r98813 = a;
        double r98814 = r98803 + r98813;
        double r98815 = b;
        double r98816 = fma(r98814, r98803, r98815);
        double r98817 = c;
        double r98818 = fma(r98816, r98803, r98817);
        double r98819 = i;
        double r98820 = fma(r98818, r98803, r98819);
        double r98821 = r98812 / r98820;
        double r98822 = r98811 * r98821;
        return r98822;
}

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)))