Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.8 → 28.9

Time: 31.6s

Precision: 64

Math

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

↓

\[\frac{\mathsf{fma}\left(\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right)}, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r78382 = x;
        double r78383 = y;
        double r78384 = r78382 * r78383;
        double r78385 = z;
        double r78386 = r78384 + r78385;
        double r78387 = r78386 * r78383;
        double r78388 = 27464.7644705;
        double r78389 = r78387 + r78388;
        double r78390 = r78389 * r78383;
        double r78391 = 230661.510616;
        double r78392 = r78390 + r78391;
        double r78393 = r78392 * r78383;
        double r78394 = t;
        double r78395 = r78393 + r78394;
        double r78396 = a;
        double r78397 = r78383 + r78396;
        double r78398 = r78397 * r78383;
        double r78399 = b;
        double r78400 = r78398 + r78399;
        double r78401 = r78400 * r78383;
        double r78402 = c;
        double r78403 = r78401 + r78402;
        double r78404 = r78403 * r78383;
        double r78405 = i;
        double r78406 = r78404 + r78405;
        double r78407 = r78395 / r78406;
        return r78407;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r78408 = x;
        double r78409 = y;
        double r78410 = z;
        double r78411 = fma(r78408, r78409, r78410);
        double r78412 = 27464.7644705;
        double r78413 = fma(r78411, r78409, r78412);
        double r78414 = 230661.510616;
        double r78415 = fma(r78413, r78409, r78414);
        double r78416 = cbrt(r78415);
        double r78417 = r78416 * r78416;
        double r78418 = r78417 * r78416;
        double r78419 = t;
        double r78420 = fma(r78418, r78409, r78419);
        double r78421 = a;
        double r78422 = r78409 + r78421;
        double r78423 = b;
        double r78424 = fma(r78422, r78409, r78423);
        double r78425 = c;
        double r78426 = fma(r78424, r78409, r78425);
        double r78427 = i;
        double r78428 = fma(r78426, r78409, r78427);
        double r78429 = r78420 / r78428;
        return r78429;
}

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

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

2. Simplified 28.8

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

4. Applied add-cube-cbrt 28.9

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

5. Final simplification 28.9

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

Reproduce

herbie shell --seed 2019303 +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) 27464.764470499998) y) 230661.510616000014) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))