Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.8 → 28.8

Time: 34.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r1065054 = x;
        double r1065055 = y;
        double r1065056 = r1065054 * r1065055;
        double r1065057 = z;
        double r1065058 = r1065056 + r1065057;
        double r1065059 = r1065058 * r1065055;
        double r1065060 = 27464.7644705;
        double r1065061 = r1065059 + r1065060;
        double r1065062 = r1065061 * r1065055;
        double r1065063 = 230661.510616;
        double r1065064 = r1065062 + r1065063;
        double r1065065 = r1065064 * r1065055;
        double r1065066 = t;
        double r1065067 = r1065065 + r1065066;
        double r1065068 = a;
        double r1065069 = r1065055 + r1065068;
        double r1065070 = r1065069 * r1065055;
        double r1065071 = b;
        double r1065072 = r1065070 + r1065071;
        double r1065073 = r1065072 * r1065055;
        double r1065074 = c;
        double r1065075 = r1065073 + r1065074;
        double r1065076 = r1065075 * r1065055;
        double r1065077 = i;
        double r1065078 = r1065076 + r1065077;
        double r1065079 = r1065067 / r1065078;
        return r1065079;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r1065080 = 1.0;
        double r1065081 = y;
        double r1065082 = a;
        double r1065083 = r1065081 + r1065082;
        double r1065084 = b;
        double r1065085 = fma(r1065083, r1065081, r1065084);
        double r1065086 = c;
        double r1065087 = fma(r1065081, r1065085, r1065086);
        double r1065088 = i;
        double r1065089 = fma(r1065087, r1065081, r1065088);
        double r1065090 = r1065080 / r1065089;
        double r1065091 = x;
        double r1065092 = z;
        double r1065093 = fma(r1065091, r1065081, r1065092);
        double r1065094 = 27464.7644705;
        double r1065095 = fma(r1065093, r1065081, r1065094);
        double r1065096 = 230661.510616;
        double r1065097 = fma(r1065095, r1065081, r1065096);
        double r1065098 = t;
        double r1065099 = fma(r1065081, r1065097, r1065098);
        double r1065100 = r1065090 * r1065099;
        return r1065100;
}

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 28.8

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

3. Applied *-un-lft-identity 28.8

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

4. Applied associate-/l* 29.0

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

5. Simplified 29.0

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

7. Applied div-inv 29.0

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

9. Applied *-un-lft-identity 29.0

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

10. Applied times-frac 28.9

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

11. Simplified 28.8

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

12. Final simplification 28.8

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

Reproduce

herbie shell --seed 2019152 +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)))