Average Error: 29.2 → 29.2
Time: 22.5s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, y + a, b \cdot y\right) + c, y, i\right)}\]
\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{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, y + a, b \cdot y\right) + c, y, i\right)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r61385 = x;
        double r61386 = y;
        double r61387 = r61385 * r61386;
        double r61388 = z;
        double r61389 = r61387 + r61388;
        double r61390 = r61389 * r61386;
        double r61391 = 27464.7644705;
        double r61392 = r61390 + r61391;
        double r61393 = r61392 * r61386;
        double r61394 = 230661.510616;
        double r61395 = r61393 + r61394;
        double r61396 = r61395 * r61386;
        double r61397 = t;
        double r61398 = r61396 + r61397;
        double r61399 = a;
        double r61400 = r61386 + r61399;
        double r61401 = r61400 * r61386;
        double r61402 = b;
        double r61403 = r61401 + r61402;
        double r61404 = r61403 * r61386;
        double r61405 = c;
        double r61406 = r61404 + r61405;
        double r61407 = r61406 * r61386;
        double r61408 = i;
        double r61409 = r61407 + r61408;
        double r61410 = r61398 / r61409;
        return r61410;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r61411 = y;
        double r61412 = x;
        double r61413 = z;
        double r61414 = fma(r61411, r61412, r61413);
        double r61415 = 27464.7644705;
        double r61416 = fma(r61411, r61414, r61415);
        double r61417 = 230661.510616;
        double r61418 = fma(r61411, r61416, r61417);
        double r61419 = t;
        double r61420 = fma(r61418, r61411, r61419);
        double r61421 = r61411 * r61411;
        double r61422 = a;
        double r61423 = r61411 + r61422;
        double r61424 = b;
        double r61425 = r61424 * r61411;
        double r61426 = fma(r61421, r61423, r61425);
        double r61427 = c;
        double r61428 = r61426 + r61427;
        double r61429 = i;
        double r61430 = fma(r61428, r61411, r61429);
        double r61431 = r61420 / r61430;
        return r61431;
}

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 29.2

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

  2. Simplified29.2

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

  4. Applied fma-udef29.2

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

  5. Taylor expanded around inf 29.3

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644704999984242022037506103515625\right), 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\color{blue}{\left(a \cdot {y}^{2} + \left({y}^{3} + y \cdot b\right)\right)} + c, y, i\right)}\]

  6. Simplified29.2

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

  7. Final simplification29.2

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

Reproduce

herbie shell --seed 2019174 +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)))