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

Average Error: 26.8 → 0.6

Time: 10.9s

Precision: 64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.2270580557324376 \cdot 10^{54} \lor \neg \left(x \le 1.17686368158435667 \cdot 10^{46}\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r364239 = x;
        double r364240 = 2.0;
        double r364241 = r364239 - r364240;
        double r364242 = 4.16438922228;
        double r364243 = r364239 * r364242;
        double r364244 = 78.6994924154;
        double r364245 = r364243 + r364244;
        double r364246 = r364245 * r364239;
        double r364247 = 137.519416416;
        double r364248 = r364246 + r364247;
        double r364249 = r364248 * r364239;
        double r364250 = y;
        double r364251 = r364249 + r364250;
        double r364252 = r364251 * r364239;
        double r364253 = z;
        double r364254 = r364252 + r364253;
        double r364255 = r364241 * r364254;
        double r364256 = 43.3400022514;
        double r364257 = r364239 + r364256;
        double r364258 = r364257 * r364239;
        double r364259 = 263.505074721;
        double r364260 = r364258 + r364259;
        double r364261 = r364260 * r364239;
        double r364262 = 313.399215894;
        double r364263 = r364261 + r364262;
        double r364264 = r364263 * r364239;
        double r364265 = 47.066876606;
        double r364266 = r364264 + r364265;
        double r364267 = r364255 / r364266;
        return r364267;
}

↓

double f(double x, double y, double z) {
        double r364268 = x;
        double r364269 = -1.2270580557324376e+54;
        bool r364270 = r364268 <= r364269;
        double r364271 = 1.1768636815843567e+46;
        bool r364272 = r364268 <= r364271;
        double r364273 = !r364272;
        bool r364274 = r364270 || r364273;
        double r364275 = 2.0;
        double r364276 = r364268 - r364275;
        double r364277 = y;
        double r364278 = 3.0;
        double r364279 = pow(r364268, r364278);
        double r364280 = r364277 / r364279;
        double r364281 = 4.16438922228;
        double r364282 = r364280 + r364281;
        double r364283 = 101.7851458539211;
        double r364284 = 1.0;
        double r364285 = r364284 / r364268;
        double r364286 = r364283 * r364285;
        double r364287 = r364282 - r364286;
        double r364288 = r364276 * r364287;
        double r364289 = r364268 * r364281;
        double r364290 = 78.6994924154;
        double r364291 = r364289 + r364290;
        double r364292 = r364291 * r364268;
        double r364293 = 137.519416416;
        double r364294 = r364292 + r364293;
        double r364295 = r364294 * r364268;
        double r364296 = r364295 + r364277;
        double r364297 = r364296 * r364268;
        double r364298 = z;
        double r364299 = r364297 + r364298;
        double r364300 = 43.3400022514;
        double r364301 = r364268 + r364300;
        double r364302 = r364301 * r364268;
        double r364303 = 263.505074721;
        double r364304 = r364302 + r364303;
        double r364305 = r364304 * r364268;
        double r364306 = 313.399215894;
        double r364307 = r364305 + r364306;
        double r364308 = r364307 * r364268;
        double r364309 = 47.066876606;
        double r364310 = r364308 + r364309;
        double r364311 = r364299 / r364310;
        double r364312 = r364276 * r364311;
        double r364313 = r364274 ? r364288 : r364312;
        return r364313;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.8
Target	0.6
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.2270580557324376e+54 or 1.1768636815843567e+46 < x

   1. Initial program 61.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 61.8

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 58.2

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 58.2

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   6. Taylor expanded around inf 0.5

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)}\]

   if -1.2270580557324376e+54 < x < 1.1768636815843567e+46

   1. Initial program 1.3

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.3

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 0.6

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 0.6

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2270580557324376 \cdot 10^{54} \lor \neg \left(x \le 1.17686368158435667 \cdot 10^{46}\right):\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{y}{{x}^{3}} + 4.16438922227999964\right) - 101.785145853921094 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(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)))