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

Average Error: 26.6 → 0.8

Time: 1.2min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6281278315350795 \cdot 10^{+39} \lor \neg \left(x \leq 6.795307804012348 \cdot 10^{+46}\right):\\ \;\;\;\;\left(\left(x \cdot 4.16438922228 + \frac{3655.120465407641}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.11392429848108 + \frac{130977.50649958356}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} + \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} + \frac{x \cdot \left(x \cdot y\right)}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)}\right)\right) + \frac{z}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} \cdot \left(x - 2\right)\right) - \left(19.87956841479999 \cdot \frac{{x}^{3}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} + 275.038832832 \cdot \frac{x \cdot x}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)}\right)\right) + -2 \cdot \frac{x \cdot y}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)}\\ \end{array}\]

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 -1.6281278315350795e+39) (not (<= x 6.795307804012348e+46)))
   (-
    (+ (+ (* x 4.16438922228) (/ 3655.120465407641 x)) (/ y (* x x)))
    (+ 110.11392429848108 (/ 130977.50649958356 (* x x))))
   (+
    (-
     (+
      (+
       (*
        4.16438922228
        (/
         (pow x 5.0)
         (+
          (pow x 4.0)
          (+
           (* x 313.399215894)
           (+
            47.066876606
            (* x (* x (+ 263.505074721 (* x 43.3400022514)))))))))
       (+
        (*
         70.37071397084
         (/
          (pow x 4.0)
          (+
           (pow x 4.0)
           (+
            (* x 313.399215894)
            (+
             47.066876606
             (* x (* x (+ 263.505074721 (* x 43.3400022514)))))))))
        (/
         (* x (* x y))
         (+
          (pow x 4.0)
          (+
           (* x 313.399215894)
           (+
            47.066876606
            (* x (* x (+ 263.505074721 (* x 43.3400022514))))))))))
      (*
       (/
        z
        (+
         (pow x 4.0)
         (+
          (* x 313.399215894)
          (+ 47.066876606 (* x (* x (+ 263.505074721 (* x 43.3400022514))))))))
       (- x 2.0)))
     (+
      (*
       19.87956841479999
       (/
        (pow x 3.0)
        (+
         (pow x 4.0)
         (+
          (* x 313.399215894)
          (+
           47.066876606
           (* x (* x (+ 263.505074721 (* x 43.3400022514)))))))))
      (*
       275.038832832
       (/
        (* x x)
        (+
         (pow x 4.0)
         (+
          (* x 313.399215894)
          (+
           47.066876606
           (* x (* x (+ 263.505074721 (* x 43.3400022514)))))))))))
    (*
     -2.0
     (/
      (* x y)
      (+
       (pow x 4.0)
       (+
        (* x 313.399215894)
        (+
         47.066876606
         (* x (* x (+ 263.505074721 (* 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 <= -1.6281278315350795e+39) || !(x <= 6.795307804012348e+46)) {
		tmp = (((x * 4.16438922228) + (3655.120465407641 / x)) + (y / (x * x))) - (110.11392429848108 + (130977.50649958356 / (x * x)));
	} else {
		tmp = ((((4.16438922228 * (pow(x, 5.0) / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514))))))))) + ((70.37071397084 * (pow(x, 4.0) / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514))))))))) + ((x * (x * y)) / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514)))))))))) + ((z / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514)))))))) * (x - 2.0))) - ((19.87956841479999 * (pow(x, 3.0) / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514))))))))) + (275.038832832 * ((x * x) / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514))))))))))) + (-2.0 * ((x * y) / (pow(x, 4.0) + ((x * 313.399215894) + (47.066876606 + (x * (x * (263.505074721 + (x * 43.3400022514)))))))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.6
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 < -1.62812783153507955e39 or 6.79530780401234759e46 < x

   1. Initial program 60.5

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

   2. Taylor expanded around inf 0.9

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

   3. Simplified 0.9

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

   if -1.62812783153507955e39 < x < 6.79530780401234759e46

   1. Initial program 1.0

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

   2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\left(\frac{x \cdot z}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)} + \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)} + \frac{{x}^{2} \cdot y}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{z}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)} + \left(2 \cdot \frac{x \cdot y}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)} + \left(19.87956841479999 \cdot \frac{{x}^{3}}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)} + 275.038832832 \cdot \frac{{x}^{2}}{{x}^{4} + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 47.066876606\right)\right)\right)}\right)\right)\right)}\]

   3. Simplified 0.7

      \[\leadsto \color{blue}{\left(\left(\left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)} + \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)} + \frac{x \cdot \left(x \cdot y\right)}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)}\right)\right) + \frac{z}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)} \cdot \left(x - 2\right)\right) - \left(19.87956841479999 \cdot \frac{{x}^{3}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)} + 275.038832832 \cdot \frac{x \cdot x}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)}\right)\right) + -2 \cdot \frac{x \cdot y}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + 43.3400022514 \cdot x\right)\right)\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6281278315350795 \cdot 10^{+39} \lor \neg \left(x \leq 6.795307804012348 \cdot 10^{+46}\right):\\ \;\;\;\;\left(\left(x \cdot 4.16438922228 + \frac{3655.120465407641}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.11392429848108 + \frac{130977.50649958356}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} + \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} + \frac{x \cdot \left(x \cdot y\right)}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)}\right)\right) + \frac{z}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} \cdot \left(x - 2\right)\right) - \left(19.87956841479999 \cdot \frac{{x}^{3}}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)} + 275.038832832 \cdot \frac{x \cdot x}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)}\right)\right) + -2 \cdot \frac{x \cdot y}{{x}^{4} + \left(x \cdot 313.399215894 + \left(47.066876606 + x \cdot \left(x \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))