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

Average Error: 26.5 → 0.8

Time: 32.3s

Precision: binary64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4777063698332427 \cdot 10^{+22} \lor \neg \left(x \leq 2.339839825086441 \cdot 10^{+45}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + \left(x \cdot 4.16438922228 + 3655.120465407641 \cdot \frac{1}{x}\right)\right) - \left(130977.50649958356 \cdot \frac{1}{{x}^{2}} + 110.11392429848108\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \end{array}\]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- 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)))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4777063698332427e+22) (not (<= x 2.339839825086441e+45)))
   (-
    (+
     (/ y (pow x 2.0))
     (+ (* x 4.16438922228) (* 3655.120465407641 (/ 1.0 x))))
    (+ (* 130977.50649958356 (/ 1.0 (pow x 2.0))) 110.11392429848108))
   (*
    (- x 2.0)
    (/
     (+
      z
      (*
       x
       (+
        y
        (* x (+ 137.519416416 (* x (+ (* x 4.16438922228) 78.6994924154)))))))
     (+
      47.066876606
      (*
       x
       (+
        313.399215894
        (* x (+ 263.505074721 (* x (+ x 43.3400022514)))))))))))

C

double code(double x, double y, double z) {
	return ((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);
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4777063698332427e+22) || !(x <= 2.339839825086441e+45)) {
		tmp = ((y / pow(x, 2.0)) + ((x * 4.16438922228) + (3655.120465407641 * (1.0 / x)))) - ((130977.50649958356 * (1.0 / pow(x, 2.0))) + 110.11392429848108);
	} else {
		tmp = (x - 2.0) * ((z + (x * (y + (x * (137.519416416 + (x * ((x * 4.16438922228) + 78.6994924154))))))) / (47.066876606 + (x * (313.399215894 + (x * (263.505074721 + (x * (x + 43.3400022514))))))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.5
Target	0.5
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -1.47770636983324272e22 or 2.33983982508644097e45 < x

   1. Initial program 58.7

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

   2. Taylor expanded around inf 1.2

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(4.16438922228 \cdot x + 3655.120465407641 \cdot \frac{1}{x}\right)\right) - \left(130977.50649958356 \cdot \frac{1}{{x}^{2}} + 110.11392429848108\right)}\]

   if -1.47770636983324272e22 < x < 2.33983982508644097e45

   1. Initial program 0.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity_binary64_12013 0.8

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606\right)}}\]

   4. Applied times-frac_binary64_12019 0.4

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

   6. Simplified 0.4

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4777063698332427 \cdot 10^{+22} \lor \neg \left(x \leq 2.339839825086441 \cdot 10^{+45}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + \left(x \cdot 4.16438922228 + 3655.120465407641 \cdot \frac{1}{x}\right)\right) - \left(130977.50649958356 \cdot \frac{1}{{x}^{2}} + 110.11392429848108\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{z + x \cdot \left(y + x \cdot \left(137.519416416 + x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right)\right)\right)}{47.066876606 + x \cdot \left(313.399215894 + x \cdot \left(263.505074721 + x \cdot \left(x + 43.3400022514\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(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 2.0) 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)))