Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.1 → 29.1

Time: 31.3s

Precision: 64

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

↓

\[\frac{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right) + 230661.5106160000141244381666183471679688, t\right)}{i + y \cdot \left(c + \left(\left(y + a\right) \cdot y + b\right) \cdot y\right)}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2540158 = x;
        double r2540159 = y;
        double r2540160 = r2540158 * r2540159;
        double r2540161 = z;
        double r2540162 = r2540160 + r2540161;
        double r2540163 = r2540162 * r2540159;
        double r2540164 = 27464.7644705;
        double r2540165 = r2540163 + r2540164;
        double r2540166 = r2540165 * r2540159;
        double r2540167 = 230661.510616;
        double r2540168 = r2540166 + r2540167;
        double r2540169 = r2540168 * r2540159;
        double r2540170 = t;
        double r2540171 = r2540169 + r2540170;
        double r2540172 = a;
        double r2540173 = r2540159 + r2540172;
        double r2540174 = r2540173 * r2540159;
        double r2540175 = b;
        double r2540176 = r2540174 + r2540175;
        double r2540177 = r2540176 * r2540159;
        double r2540178 = c;
        double r2540179 = r2540177 + r2540178;
        double r2540180 = r2540179 * r2540159;
        double r2540181 = i;
        double r2540182 = r2540180 + r2540181;
        double r2540183 = r2540171 / r2540182;
        return r2540183;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r2540184 = y;
        double r2540185 = x;
        double r2540186 = z;
        double r2540187 = fma(r2540184, r2540185, r2540186);
        double r2540188 = 27464.7644705;
        double r2540189 = fma(r2540184, r2540187, r2540188);
        double r2540190 = r2540184 * r2540189;
        double r2540191 = 230661.510616;
        double r2540192 = r2540190 + r2540191;
        double r2540193 = t;
        double r2540194 = fma(r2540184, r2540192, r2540193);
        double r2540195 = i;
        double r2540196 = c;
        double r2540197 = a;
        double r2540198 = r2540184 + r2540197;
        double r2540199 = r2540198 * r2540184;
        double r2540200 = b;
        double r2540201 = r2540199 + r2540200;
        double r2540202 = r2540201 * r2540184;
        double r2540203 = r2540196 + r2540202;
        double r2540204 = r2540184 * r2540203;
        double r2540205 = r2540195 + r2540204;
        double r2540206 = r2540194 / r2540205;
        return r2540206;
}

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 29.1

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
2. Using strategy rm

3. Applied add-cube-cbrt 29.1

   \[\leadsto \frac{\left(\color{blue}{\left(\sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y} \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y}\right) \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y}} + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
4. Using strategy rm

5. Applied *-un-lft-identity 29.1

   \[\leadsto \frac{\left(\left(\sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y} \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y}\right) \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y} + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\color{blue}{1 \cdot \left(\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i\right)}}\]

6. Applied associate-/r* 29.1

   \[\leadsto \color{blue}{\frac{\frac{\left(\left(\sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y} \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y}\right) \cdot \sqrt[3]{\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y} + 230661.5106160000141244381666183471679688\right) \cdot y + t}{1}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}}\]

7. Simplified 29.1

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 230661.5106160000141244381666183471679688 + y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

8. Final simplification 29.1

   \[\leadsto \frac{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right) + 230661.5106160000141244381666183471679688, t\right)}{i + y \cdot \left(c + \left(\left(y + a\right) \cdot y + b\right) \cdot y\right)}\]

Reproduce

herbie shell --seed 2019179 +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)))