Average Error: 28.7 → 29.0
Time: 8.5s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}\]
\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{1}{\frac{\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, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r93213 = x;
        double r93214 = y;
        double r93215 = r93213 * r93214;
        double r93216 = z;
        double r93217 = r93215 + r93216;
        double r93218 = r93217 * r93214;
        double r93219 = 27464.7644705;
        double r93220 = r93218 + r93219;
        double r93221 = r93220 * r93214;
        double r93222 = 230661.510616;
        double r93223 = r93221 + r93222;
        double r93224 = r93223 * r93214;
        double r93225 = t;
        double r93226 = r93224 + r93225;
        double r93227 = a;
        double r93228 = r93214 + r93227;
        double r93229 = r93228 * r93214;
        double r93230 = b;
        double r93231 = r93229 + r93230;
        double r93232 = r93231 * r93214;
        double r93233 = c;
        double r93234 = r93232 + r93233;
        double r93235 = r93234 * r93214;
        double r93236 = i;
        double r93237 = r93235 + r93236;
        double r93238 = r93226 / r93237;
        return r93238;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r93239 = 1.0;
        double r93240 = y;
        double r93241 = a;
        double r93242 = r93240 + r93241;
        double r93243 = r93242 * r93240;
        double r93244 = b;
        double r93245 = r93243 + r93244;
        double r93246 = r93245 * r93240;
        double r93247 = c;
        double r93248 = r93246 + r93247;
        double r93249 = r93248 * r93240;
        double r93250 = i;
        double r93251 = r93249 + r93250;
        double r93252 = x;
        double r93253 = z;
        double r93254 = fma(r93252, r93240, r93253);
        double r93255 = 27464.7644705;
        double r93256 = fma(r93254, r93240, r93255);
        double r93257 = 230661.510616;
        double r93258 = fma(r93256, r93240, r93257);
        double r93259 = t;
        double r93260 = fma(r93258, r93240, r93259);
        double r93261 = r93251 / r93260;
        double r93262 = r93239 / r93261;
        return r93262;
}

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

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

  3. Applied *-un-lft-identity28.7

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

  4. Applied associate-/r*28.7

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

  5. Simplified28.7

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

  7. Applied clear-num29.0

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

  8. Final simplification29.0

    \[\leadsto \frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}}\]

Reproduce

herbie shell --seed 2019353 +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)))