Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 26.9 → 1.2

Time: 39.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -63074748513986640340139900928:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810806547102401964366436005\right)\\ \mathbf{elif}\;x \le 620615997107907.625:\\ \;\;\;\;\frac{\left(z + \left(\left(\frac{x \cdot \left(78.69949241540000173245061887428164482117 \cdot \left(78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117\right) + \left(\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right)\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right)\right)}{\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) + \left(78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117 - \left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot 78.69949241540000173245061887428164482117\right)} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x\right) \cdot \left(x - 2\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810806547102401964366436005\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r18820253 = x;
        double r18820254 = 2.0;
        double r18820255 = r18820253 - r18820254;
        double r18820256 = 4.16438922228;
        double r18820257 = r18820253 * r18820256;
        double r18820258 = 78.6994924154;
        double r18820259 = r18820257 + r18820258;
        double r18820260 = r18820259 * r18820253;
        double r18820261 = 137.519416416;
        double r18820262 = r18820260 + r18820261;
        double r18820263 = r18820262 * r18820253;
        double r18820264 = y;
        double r18820265 = r18820263 + r18820264;
        double r18820266 = r18820265 * r18820253;
        double r18820267 = z;
        double r18820268 = r18820266 + r18820267;
        double r18820269 = r18820255 * r18820268;
        double r18820270 = 43.3400022514;
        double r18820271 = r18820253 + r18820270;
        double r18820272 = r18820271 * r18820253;
        double r18820273 = 263.505074721;
        double r18820274 = r18820272 + r18820273;
        double r18820275 = r18820274 * r18820253;
        double r18820276 = 313.399215894;
        double r18820277 = r18820275 + r18820276;
        double r18820278 = r18820277 * r18820253;
        double r18820279 = 47.066876606;
        double r18820280 = r18820278 + r18820279;
        double r18820281 = r18820269 / r18820280;
        return r18820281;
}

↓

double f(double x, double y, double z) {
        double r18820282 = x;
        double r18820283 = -6.307474851398664e+28;
        bool r18820284 = r18820282 <= r18820283;
        double r18820285 = y;
        double r18820286 = r18820282 * r18820282;
        double r18820287 = r18820285 / r18820286;
        double r18820288 = 4.16438922228;
        double r18820289 = r18820282 * r18820288;
        double r18820290 = 110.11392429848108;
        double r18820291 = r18820289 - r18820290;
        double r18820292 = r18820287 + r18820291;
        double r18820293 = 620615997107907.6;
        bool r18820294 = r18820282 <= r18820293;
        double r18820295 = z;
        double r18820296 = 78.6994924154;
        double r18820297 = r18820296 * r18820296;
        double r18820298 = r18820296 * r18820297;
        double r18820299 = r18820289 * r18820289;
        double r18820300 = r18820299 * r18820289;
        double r18820301 = r18820298 + r18820300;
        double r18820302 = r18820282 * r18820301;
        double r18820303 = r18820289 * r18820296;
        double r18820304 = r18820297 - r18820303;
        double r18820305 = r18820299 + r18820304;
        double r18820306 = r18820302 / r18820305;
        double r18820307 = 137.519416416;
        double r18820308 = r18820306 + r18820307;
        double r18820309 = r18820308 * r18820282;
        double r18820310 = r18820309 + r18820285;
        double r18820311 = r18820310 * r18820282;
        double r18820312 = r18820295 + r18820311;
        double r18820313 = 2.0;
        double r18820314 = r18820282 - r18820313;
        double r18820315 = r18820312 * r18820314;
        double r18820316 = 47.066876606;
        double r18820317 = 313.399215894;
        double r18820318 = 263.505074721;
        double r18820319 = 43.3400022514;
        double r18820320 = r18820319 + r18820282;
        double r18820321 = r18820320 * r18820282;
        double r18820322 = r18820318 + r18820321;
        double r18820323 = r18820282 * r18820322;
        double r18820324 = r18820317 + r18820323;
        double r18820325 = r18820282 * r18820324;
        double r18820326 = r18820316 + r18820325;
        double r18820327 = r18820315 / r18820326;
        double r18820328 = r18820294 ? r18820327 : r18820292;
        double r18820329 = r18820284 ? r18820292 : r18820328;
        return r18820329;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.9
Target	0.6
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if x < -6.307474851398664e+28 or 620615997107907.6 < x

   1. Initial program 56.8

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

   3. Applied associate-/l* 53.1

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}}\]
   4. Using strategy rm

   5. Applied flip3-- 53.1

      \[\leadsto \frac{\color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}}}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}}\]

   6. Applied associate-/l/ 53.1

      \[\leadsto \color{blue}{\frac{{x}^{3} - {2}^{3}}{\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z} \cdot \left(x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)\right)}}\]

   7. Simplified 53.1

      \[\leadsto \frac{{x}^{3} - {2}^{3}}{\color{blue}{\frac{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + \left(x \cdot \left(43.3400022514000013984514225739985704422 + x\right) + 263.5050747210000281484099105000495910645\right) \cdot x\right)}{\frac{z + \left(x \cdot \left(\left(4.16438922227999963610045597306452691555 \cdot x + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) + y\right) \cdot x}{x \cdot x + 2 \cdot \left(x + 2\right)}}}}\]

   8. Taylor expanded around inf 2.1

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810806547102401964366436005}\]

   9. Simplified 2.1

      \[\leadsto \color{blue}{\frac{y}{x \cdot x} + \left(4.16438922227999963610045597306452691555 \cdot x - 110.1139242984810806547102401964366436005\right)}\]

   if -6.307474851398664e+28 < x < 620615997107907.6

   1. Initial program 0.5

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

   3. Applied flip3-+ 0.5

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\frac{{\left(x \cdot 4.16438922227999963610045597306452691555\right)}^{3} + {78.69949241540000173245061887428164482117}^{3}}{\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) + \left(78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117 - \left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot 78.69949241540000173245061887428164482117\right)}} \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

   4. Applied associate-*l/ 0.5

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\frac{\left({\left(x \cdot 4.16438922227999963610045597306452691555\right)}^{3} + {78.69949241540000173245061887428164482117}^{3}\right) \cdot x}{\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) + \left(78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117 - \left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot 78.69949241540000173245061887428164482117\right)}} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\]

   5. Simplified 0.5

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -63074748513986640340139900928:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810806547102401964366436005\right)\\ \mathbf{elif}\;x \le 620615997107907.625:\\ \;\;\;\;\frac{\left(z + \left(\left(\frac{x \cdot \left(78.69949241540000173245061887428164482117 \cdot \left(78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117\right) + \left(\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right)\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right)\right)}{\left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot \left(x \cdot 4.16438922227999963610045597306452691555\right) + \left(78.69949241540000173245061887428164482117 \cdot 78.69949241540000173245061887428164482117 - \left(x \cdot 4.16438922227999963610045597306452691555\right) \cdot 78.69949241540000173245061887428164482117\right)} + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x\right) \cdot \left(x - 2\right)}{47.06687660600000100430406746454536914825 + x \cdot \left(313.3992158940000081202015280723571777344 + x \cdot \left(263.5050747210000281484099105000495910645 + \left(43.3400022514000013984514225739985704422 + x\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot x} + \left(x \cdot 4.16438922227999963610045597306452691555 - 110.1139242984810806547102401964366436005\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))