Average Error: 25.1 → 0.6
Time: 28.6s
Precision: 64
\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.9717766738087274 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811\\ \mathbf{elif}\;x \le 2.0084271042117127 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{x - 2.0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811\\ \end{array}\]
\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.9717766738087274 \cdot 10^{+52}:\\
\;\;\;\;\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811\\

\mathbf{elif}\;x \le 2.0084271042117127 \cdot 10^{+34}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{x - 2.0}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811\\

\end{array}
double f(double x, double y, double z) {
        double r7078258 = x;
        double r7078259 = 2.0;
        double r7078260 = r7078258 - r7078259;
        double r7078261 = 4.16438922228;
        double r7078262 = r7078258 * r7078261;
        double r7078263 = 78.6994924154;
        double r7078264 = r7078262 + r7078263;
        double r7078265 = r7078264 * r7078258;
        double r7078266 = 137.519416416;
        double r7078267 = r7078265 + r7078266;
        double r7078268 = r7078267 * r7078258;
        double r7078269 = y;
        double r7078270 = r7078268 + r7078269;
        double r7078271 = r7078270 * r7078258;
        double r7078272 = z;
        double r7078273 = r7078271 + r7078272;
        double r7078274 = r7078260 * r7078273;
        double r7078275 = 43.3400022514;
        double r7078276 = r7078258 + r7078275;
        double r7078277 = r7078276 * r7078258;
        double r7078278 = 263.505074721;
        double r7078279 = r7078277 + r7078278;
        double r7078280 = r7078279 * r7078258;
        double r7078281 = 313.399215894;
        double r7078282 = r7078280 + r7078281;
        double r7078283 = r7078282 * r7078258;
        double r7078284 = 47.066876606;
        double r7078285 = r7078283 + r7078284;
        double r7078286 = r7078274 / r7078285;
        return r7078286;
}
↓
double f(double x, double y, double z) {
        double r7078287 = x;
        double r7078288 = -3.9717766738087274e+52;
        bool r7078289 = r7078287 <= r7078288;
        double r7078290 = 4.16438922228;
        double r7078291 = y;
        double r7078292 = r7078291 / r7078287;
        double r7078293 = r7078292 / r7078287;
        double r7078294 = fma(r7078290, r7078287, r7078293);
        double r7078295 = 110.1139242984811;
        double r7078296 = r7078294 - r7078295;
        double r7078297 = 2.0084271042117127e+34;
        bool r7078298 = r7078287 <= r7078297;
        double r7078299 = 78.6994924154;
        double r7078300 = fma(r7078287, r7078290, r7078299);
        double r7078301 = 137.519416416;
        double r7078302 = fma(r7078287, r7078300, r7078301);
        double r7078303 = fma(r7078287, r7078302, r7078291);
        double r7078304 = z;
        double r7078305 = fma(r7078287, r7078303, r7078304);
        double r7078306 = 43.3400022514;
        double r7078307 = r7078287 + r7078306;
        double r7078308 = 263.505074721;
        double r7078309 = fma(r7078307, r7078287, r7078308);
        double r7078310 = 313.399215894;
        double r7078311 = fma(r7078309, r7078287, r7078310);
        double r7078312 = 47.066876606;
        double r7078313 = fma(r7078311, r7078287, r7078312);
        double r7078314 = 2.0;
        double r7078315 = r7078287 - r7078314;
        double r7078316 = r7078313 / r7078315;
        double r7078317 = r7078305 / r7078316;
        double r7078318 = r7078298 ? r7078317 : r7078296;
        double r7078319 = r7078289 ? r7078296 : r7078318;
        return r7078319;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original25.1
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.9717766738087274e+52 or 2.0084271042117127e+34 < x

    1. Initial program 58.8

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    2. Simplified58.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \left(x - 2.0\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*54.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{x - 2.0}}}\]

    5. Taylor expanded around inf 0.9

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922228 \cdot x\right) - 110.1139242984811}\]

    6. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811}\]

    if -3.9717766738087274e+52 < x < 2.0084271042117127e+34

    1. Initial program 0.8

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    2. Simplified0.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \left(x - 2.0\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*0.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{x - 2.0}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.9717766738087274 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811\\ \mathbf{elif}\;x \le 2.0084271042117127 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{x - 2.0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4.16438922228, x, \frac{\frac{y}{x}}{x}\right) - 110.1139242984811\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))