Average Error: 27.1 → 0.6
Time: 1.7m
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.328424586542639866198193472263284120987 \cdot 10^{62} \lor \neg \left(x \le 2.123566528240809367630637301063751523391 \cdot 10^{49}\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.328424586542639866198193472263284120987 \cdot 10^{62} \lor \neg \left(x \le 2.123566528240809367630637301063751523391 \cdot 10^{49}\right):\\
\;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}\\

\end{array}
double f(double x, double y, double z) {
        double r409240 = x;
        double r409241 = 2.0;
        double r409242 = r409240 - r409241;
        double r409243 = 4.16438922228;
        double r409244 = r409240 * r409243;
        double r409245 = 78.6994924154;
        double r409246 = r409244 + r409245;
        double r409247 = r409246 * r409240;
        double r409248 = 137.519416416;
        double r409249 = r409247 + r409248;
        double r409250 = r409249 * r409240;
        double r409251 = y;
        double r409252 = r409250 + r409251;
        double r409253 = r409252 * r409240;
        double r409254 = z;
        double r409255 = r409253 + r409254;
        double r409256 = r409242 * r409255;
        double r409257 = 43.3400022514;
        double r409258 = r409240 + r409257;
        double r409259 = r409258 * r409240;
        double r409260 = 263.505074721;
        double r409261 = r409259 + r409260;
        double r409262 = r409261 * r409240;
        double r409263 = 313.399215894;
        double r409264 = r409262 + r409263;
        double r409265 = r409264 * r409240;
        double r409266 = 47.066876606;
        double r409267 = r409265 + r409266;
        double r409268 = r409256 / r409267;
        return r409268;
}
↓
double f(double x, double y, double z) {
        double r409269 = x;
        double r409270 = -3.32842458654264e+62;
        bool r409271 = r409269 <= r409270;
        double r409272 = 2.1235665282408094e+49;
        bool r409273 = r409269 <= r409272;
        double r409274 = !r409273;
        bool r409275 = r409271 || r409274;
        double r409276 = 4.16438922228;
        double r409277 = y;
        double r409278 = 2.0;
        double r409279 = pow(r409269, r409278);
        double r409280 = r409277 / r409279;
        double r409281 = 110.1139242984811;
        double r409282 = r409280 - r409281;
        double r409283 = fma(r409276, r409269, r409282);
        double r409284 = 78.6994924154;
        double r409285 = fma(r409269, r409276, r409284);
        double r409286 = 137.519416416;
        double r409287 = fma(r409269, r409285, r409286);
        double r409288 = fma(r409269, r409287, r409277);
        double r409289 = z;
        double r409290 = fma(r409269, r409288, r409289);
        double r409291 = 43.3400022514;
        double r409292 = r409291 + r409269;
        double r409293 = 263.505074721;
        double r409294 = fma(r409292, r409269, r409293);
        double r409295 = 313.399215894;
        double r409296 = fma(r409294, r409269, r409295);
        double r409297 = 47.066876606;
        double r409298 = fma(r409296, r409269, r409297);
        double r409299 = 2.0;
        double r409300 = r409269 - r409299;
        double r409301 = r409298 / r409300;
        double r409302 = r409290 / r409301;
        double r409303 = r409275 ? r409283 : r409302;
        return r409303;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original27.1
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870004842699683658678411714981 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \mathbf{elif}\;x \lt 9.429991714554672672712552870340896976735 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.5050747210000281484099105000495910645 \cdot x + \left(43.3400022514000013984514225739985704422 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.32842458654264e+62 or 2.1235665282408094e+49 < x

    1. Initial program 62.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Simplified59.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}}\]

    3. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]

    4. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)}\]

    if -3.32842458654264e+62 < x < 2.1235665282408094e+49

    1. Initial program 1.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

    2. Simplified0.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.328424586542639866198193472263284120987 \cdot 10^{62} \lor \neg \left(x \le 2.123566528240809367630637301063751523391 \cdot 10^{49}\right):\\ \;\;\;\;\mathsf{fma}\left(4.16438922227999963610045597306452691555, x, \frac{y}{{x}^{2}} - 110.1139242984810948655649553984403610229\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922227999963610045597306452691555, 78.69949241540000173245061887428164482117\right), 137.5194164160000127594685181975364685059\right), y\right), z\right)}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514000013984514225739985704422 + x, x, 263.5050747210000281484099105000495910645\right), x, 313.3992158940000081202015280723571777344\right), x, 47.06687660600000100430406746454536914825\right)}{x - 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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) 1.0) (/ (+ (* (+ (* (+ (* (+ (* 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)))