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

Average Error: 26.7 → 0.8

Time: 23.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -394748133220946787888436044907157651456:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \mathbf{elif}\;x \le 998303336278827940708352:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) - 263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot x}{\left(x + 43.3400022514000013984514225739985704422\right) \cdot x - 263.5050747210000281484099105000495910645} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r290255 = x;
        double r290256 = 2.0;
        double r290257 = r290255 - r290256;
        double r290258 = 4.16438922228;
        double r290259 = r290255 * r290258;
        double r290260 = 78.6994924154;
        double r290261 = r290259 + r290260;
        double r290262 = r290261 * r290255;
        double r290263 = 137.519416416;
        double r290264 = r290262 + r290263;
        double r290265 = r290264 * r290255;
        double r290266 = y;
        double r290267 = r290265 + r290266;
        double r290268 = r290267 * r290255;
        double r290269 = z;
        double r290270 = r290268 + r290269;
        double r290271 = r290257 * r290270;
        double r290272 = 43.3400022514;
        double r290273 = r290255 + r290272;
        double r290274 = r290273 * r290255;
        double r290275 = 263.505074721;
        double r290276 = r290274 + r290275;
        double r290277 = r290276 * r290255;
        double r290278 = 313.399215894;
        double r290279 = r290277 + r290278;
        double r290280 = r290279 * r290255;
        double r290281 = 47.066876606;
        double r290282 = r290280 + r290281;
        double r290283 = r290271 / r290282;
        return r290283;
}

↓

double f(double x, double y, double z) {
        double r290284 = x;
        double r290285 = -3.947481332209468e+38;
        bool r290286 = r290284 <= r290285;
        double r290287 = 2.0;
        double r290288 = r290284 - r290287;
        double r290289 = y;
        double r290290 = 3.0;
        double r290291 = pow(r290284, r290290);
        double r290292 = r290289 / r290291;
        double r290293 = 4.16438922228;
        double r290294 = r290292 + r290293;
        double r290295 = 101.7851458539211;
        double r290296 = r290295 / r290284;
        double r290297 = r290294 - r290296;
        double r290298 = r290288 * r290297;
        double r290299 = 9.98303336278828e+23;
        bool r290300 = r290284 <= r290299;
        double r290301 = r290284 * r290293;
        double r290302 = 78.6994924154;
        double r290303 = r290301 + r290302;
        double r290304 = r290303 * r290284;
        double r290305 = 137.519416416;
        double r290306 = r290304 + r290305;
        double r290307 = r290306 * r290284;
        double r290308 = r290307 + r290289;
        double r290309 = r290308 * r290284;
        double r290310 = z;
        double r290311 = r290309 + r290310;
        double r290312 = 43.3400022514;
        double r290313 = r290284 + r290312;
        double r290314 = r290313 * r290284;
        double r290315 = r290314 * r290314;
        double r290316 = 263.505074721;
        double r290317 = r290316 * r290316;
        double r290318 = r290315 - r290317;
        double r290319 = r290318 * r290284;
        double r290320 = r290314 - r290316;
        double r290321 = r290319 / r290320;
        double r290322 = 313.399215894;
        double r290323 = r290321 + r290322;
        double r290324 = r290323 * r290284;
        double r290325 = 47.066876606;
        double r290326 = r290324 + r290325;
        double r290327 = r290311 / r290326;
        double r290328 = r290288 * r290327;
        double r290329 = 2.0;
        double r290330 = pow(r290284, r290329);
        double r290331 = r290289 / r290330;
        double r290332 = r290293 * r290284;
        double r290333 = r290331 + r290332;
        double r290334 = 110.1139242984811;
        double r290335 = r290333 - r290334;
        double r290336 = r290300 ? r290328 : r290335;
        double r290337 = r290286 ? r290298 : r290336;
        return r290337;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.7
Target	0.6
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if x < -3.947481332209468e+38

   1. Initial program 60.4

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

   3. Applied *-un-lft-identity 60.4

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}}\]

   4. Applied times-frac 56.5

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

   5. Simplified 56.5

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

   6. Taylor expanded around inf 0.7

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - 101.785145853921093817007204052060842514 \cdot \frac{1}{x}\right)}\]

   7. Simplified 0.7

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)}\]

   if -3.947481332209468e+38 < x < 9.98303336278828e+23

   1. Initial program 0.7

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

   3. Applied *-un-lft-identity 0.7

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825\right)}}\]

   4. Applied times-frac 0.3

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x + 263.5050747210000281484099105000495910645\right) \cdot x + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}}\]

   5. Simplified 0.3

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

   7. Applied flip-+ 0.3

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

   8. Applied associate-*l/ 0.3

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

   if 9.98303336278828e+23 < x

   1. Initial program 56.5

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

   2. Taylor expanded around inf 2.2

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -394748133220946787888436044907157651456:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999963610045597306452691555\right) - \frac{101.785145853921093817007204052060842514}{x}\right)\\ \mathbf{elif}\;x \le 998303336278827940708352:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999963610045597306452691555 + 78.69949241540000173245061887428164482117\right) \cdot x + 137.5194164160000127594685181975364685059\right) \cdot x + y\right) \cdot x + z}{\left(\frac{\left(\left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) \cdot \left(\left(x + 43.3400022514000013984514225739985704422\right) \cdot x\right) - 263.5050747210000281484099105000495910645 \cdot 263.5050747210000281484099105000495910645\right) \cdot x}{\left(x + 43.3400022514000013984514225739985704422\right) \cdot x - 263.5050747210000281484099105000495910645} + 313.3992158940000081202015280723571777344\right) \cdot x + 47.06687660600000100430406746454536914825}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999963610045597306452691555 \cdot x\right) - 110.1139242984810948655649553984403610229\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))