Average Error: 28.2 → 28.3
Time: 36.8s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)}}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)}}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r1952800 = x;
        double r1952801 = y;
        double r1952802 = r1952800 * r1952801;
        double r1952803 = z;
        double r1952804 = r1952802 + r1952803;
        double r1952805 = r1952804 * r1952801;
        double r1952806 = 27464.7644705;
        double r1952807 = r1952805 + r1952806;
        double r1952808 = r1952807 * r1952801;
        double r1952809 = 230661.510616;
        double r1952810 = r1952808 + r1952809;
        double r1952811 = r1952810 * r1952801;
        double r1952812 = t;
        double r1952813 = r1952811 + r1952812;
        double r1952814 = a;
        double r1952815 = r1952801 + r1952814;
        double r1952816 = r1952815 * r1952801;
        double r1952817 = b;
        double r1952818 = r1952816 + r1952817;
        double r1952819 = r1952818 * r1952801;
        double r1952820 = c;
        double r1952821 = r1952819 + r1952820;
        double r1952822 = r1952821 * r1952801;
        double r1952823 = i;
        double r1952824 = r1952822 + r1952823;
        double r1952825 = r1952813 / r1952824;
        return r1952825;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r1952826 = 1.0;
        double r1952827 = y;
        double r1952828 = a;
        double r1952829 = r1952827 + r1952828;
        double r1952830 = b;
        double r1952831 = fma(r1952829, r1952827, r1952830);
        double r1952832 = c;
        double r1952833 = fma(r1952827, r1952831, r1952832);
        double r1952834 = i;
        double r1952835 = fma(r1952833, r1952827, r1952834);
        double r1952836 = r1952826 / r1952835;
        double r1952837 = x;
        double r1952838 = z;
        double r1952839 = fma(r1952827, r1952837, r1952838);
        double r1952840 = 27464.7644705;
        double r1952841 = fma(r1952827, r1952839, r1952840);
        double r1952842 = 230661.510616;
        double r1952843 = fma(r1952827, r1952841, r1952842);
        double r1952844 = t;
        double r1952845 = fma(r1952827, r1952843, r1952844);
        double r1952846 = r1952826 / r1952845;
        double r1952847 = r1952836 / r1952846;
        return r1952847;
}

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 28.2

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  2. Simplified28.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)}}\]
  3. Using strategy rm

  4. Applied clear-num28.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}}\]
  5. Using strategy rm

  6. Applied div-inv28.5

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}}\]

  7. Applied associate-/r*28.3

    \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)}}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}}\]

  8. Final simplification28.3

    \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y + a, y, b\right), c\right), y, i\right)}}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))