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

Average Error: 26.7 → 0.7

Time: 5.0min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.530021484323837 \cdot 10^{+58} \lor \neg \left(x \leq 2.684414262750609 \cdot 10^{+23}\right):\\ \;\;\;\;\left(x \cdot 4.16438922228 + \frac{3655.120465407641}{x}\right) + \left(\frac{y}{x \cdot x} - \left(110.11392429848108 + \frac{130977.50649958356}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} + \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} + y \cdot \frac{x}{\frac{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)}{x}}\right)\right) + \frac{z}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} \cdot \left(x - 2\right)\right) - \left(\frac{19.87956841479999}{\frac{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)}{{x}^{3}}} + \left(\frac{2 \cdot \left(x \cdot y\right)}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} + \frac{\left(x \cdot x\right) \cdot 275.038832832}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)}\right)\right)\\ \end{array}\]

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 -2.530021484323837e+58) (not (<= x 2.684414262750609e+23)))
   (+
    (+ (* 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)
         (+
          (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
          (+ (* x 313.399215894) (* x (* x 263.505074721)))))))
      (+
       (*
        70.37071397084
        (/
         (pow x 4.0)
         (+
          (pow x 4.0)
          (+
           (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
           (+ (* x 313.399215894) (* x (* x 263.505074721)))))))
       (*
        y
        (/
         x
         (/
          (+
           (pow x 4.0)
           (+
            (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
            (+ (* x 313.399215894) (* x (* x 263.505074721)))))
          x)))))
     (*
      (/
       z
       (+
        (pow x 4.0)
        (+
         (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
         (+ (* x 313.399215894) (* x (* x 263.505074721))))))
      (- x 2.0)))
    (+
     (/
      19.87956841479999
      (/
       (+
        (pow x 4.0)
        (+
         (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
         (+ (* x 313.399215894) (* x (* x 263.505074721)))))
       (pow x 3.0)))
     (+
      (/
       (* 2.0 (* x y))
       (+
        (pow x 4.0)
        (+
         (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
         (+ (* x 313.399215894) (* x (* x 263.505074721))))))
      (/
       (* (* x x) 275.038832832)
       (+
        (pow x 4.0)
        (+
         (+ (* (pow x 3.0) 43.3400022514) 47.066876606)
         (+ (* x 313.399215894) (* x (* x 263.505074721)))))))))))

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 <= -2.530021484323837e+58) || !(x <= 2.684414262750609e+23)) {
		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) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721))))))) + ((70.37071397084 * (pow(x, 4.0) / (pow(x, 4.0) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721))))))) + (y * (x / ((pow(x, 4.0) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721))))) / x))))) + ((z / (pow(x, 4.0) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721)))))) * (x - 2.0))) - ((19.87956841479999 / ((pow(x, 4.0) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721))))) / pow(x, 3.0))) + (((2.0 * (x * y)) / (pow(x, 4.0) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721)))))) + (((x * x) * 275.038832832) / (pow(x, 4.0) + (((pow(x, 3.0) * 43.3400022514) + 47.066876606) + ((x * 313.399215894) + (x * (x * 263.505074721))))))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.7
Target	0.5
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if x < -2.53002148432383677e58 or 2.6844142627506089e23 < x

   1. Initial program 60.0

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

   2. Simplified 56.1

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

   3. Taylor expanded around inf 1.1

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

   4. Simplified 1.1

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

   if -2.53002148432383677e58 < x < 2.6844142627506089e23

   1. Initial program 1.1

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

   2. Simplified 0.5

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

   3. Taylor expanded around 0 1.0

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

   4. Simplified 0.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.530021484323837 \cdot 10^{+58} \lor \neg \left(x \leq 2.684414262750609 \cdot 10^{+23}\right):\\ \;\;\;\;\left(x \cdot 4.16438922228 + \frac{3655.120465407641}{x}\right) + \left(\frac{y}{x \cdot x} - \left(110.11392429848108 + \frac{130977.50649958356}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} + \left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} + y \cdot \frac{x}{\frac{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)}{x}}\right)\right) + \frac{z}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} \cdot \left(x - 2\right)\right) - \left(\frac{19.87956841479999}{\frac{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)}{{x}^{3}}} + \left(\frac{2 \cdot \left(x \cdot y\right)}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)} + \frac{\left(x \cdot x\right) \cdot 275.038832832}{{x}^{4} + \left(\left({x}^{3} \cdot 43.3400022514 + 47.066876606\right) + \left(x \cdot 313.399215894 + x \cdot \left(x \cdot 263.505074721\right)\right)\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))