Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.6 → 28.6

Time: 33.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2525169 = x;
        double r2525170 = y;
        double r2525171 = r2525169 * r2525170;
        double r2525172 = z;
        double r2525173 = r2525171 + r2525172;
        double r2525174 = r2525173 * r2525170;
        double r2525175 = 27464.7644705;
        double r2525176 = r2525174 + r2525175;
        double r2525177 = r2525176 * r2525170;
        double r2525178 = 230661.510616;
        double r2525179 = r2525177 + r2525178;
        double r2525180 = r2525179 * r2525170;
        double r2525181 = t;
        double r2525182 = r2525180 + r2525181;
        double r2525183 = a;
        double r2525184 = r2525170 + r2525183;
        double r2525185 = r2525184 * r2525170;
        double r2525186 = b;
        double r2525187 = r2525185 + r2525186;
        double r2525188 = r2525187 * r2525170;
        double r2525189 = c;
        double r2525190 = r2525188 + r2525189;
        double r2525191 = r2525190 * r2525170;
        double r2525192 = i;
        double r2525193 = r2525191 + r2525192;
        double r2525194 = r2525182 / r2525193;
        return r2525194;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2525195 = y;
        double r2525196 = x;
        double r2525197 = z;
        double r2525198 = fma(r2525195, r2525196, r2525197);
        double r2525199 = 27464.7644705;
        double r2525200 = fma(r2525198, r2525195, r2525199);
        double r2525201 = 230661.510616;
        double r2525202 = fma(r2525195, r2525200, r2525201);
        double r2525203 = t;
        double r2525204 = fma(r2525195, r2525202, r2525203);
        double r2525205 = a;
        double r2525206 = r2525205 + r2525195;
        double r2525207 = b;
        double r2525208 = fma(r2525206, r2525195, r2525207);
        double r2525209 = c;
        double r2525210 = fma(r2525208, r2525195, r2525209);
        double r2525211 = i;
        double r2525212 = fma(r2525210, r2525195, r2525211);
        double r2525213 = r2525204 / r2525212;
        return r2525213;
}

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 28.6

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

2. Simplified 28.6

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

4. Applied *-un-lft-identity 28.6

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

5. Applied associate-/l* 28.8

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

7. Applied div-inv 28.8

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

8. Simplified 28.6

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

9. Final simplification 28.6

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

Reproduce

herbie shell --seed 2019129 +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)))