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

Average Error: 26.2 → 0.7

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -3.361760792985159 \cdot 10^{+41}:\\ \;\;\;\;\left(x \cdot 4.16438922228 + \frac{y}{x \cdot x}\right) - 110.11392429848108\\ \mathbf{elif}\;x \leq 1.1146806748453102 \cdot 10^{+41}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{x \cdot \left(y + x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right)\right) + z}{x \cdot \left(\left(x \cdot \left(x \cdot \left(x + 43.3400022514\right)\right) + x \cdot 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922228 + \frac{y}{{x}^{3}}\right) - \frac{101.78514585392108}{x}\right)\\ \end{array}\]

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 (<= x -3.361760792985159e+41)
   (- (+ (* x 4.16438922228) (/ y (* x x))) 110.11392429848108)
   (if (<= x 1.1146806748453102e+41)
     (*
      (- x 2.0)
      (/
       (+
        (*
         x
         (+
          y
          (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))))
        z)
       (+
        (*
         x
         (+
          (+ (* x (* x (+ x 43.3400022514))) (* x 263.505074721))
          313.399215894))
        47.066876606)))
     (*
      (- x 2.0)
      (- (+ 4.16438922228 (/ y (pow x 3.0))) (/ 101.78514585392108 x))))))

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 <= -3.361760792985159e+41) {
		tmp = ((x * 4.16438922228) + (y / (x * x))) - 110.11392429848108;
	} else if (x <= 1.1146806748453102e+41) {
		tmp = (x - 2.0) * (((x * (y + (x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)))) + z) / ((x * (((x * (x * (x + 43.3400022514))) + (x * 263.505074721)) + 313.399215894)) + 47.066876606));
	} else {
		tmp = (x - 2.0) * ((4.16438922228 + (y / pow(x, 3.0))) - (101.78514585392108 / x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.2
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 3 regimes

2. if x < -3.36176079298515914e41

   1. Initial program 60.6

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

   2. Taylor expanded around inf 1.0

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

   3. Simplified 1.0

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

   if -3.36176079298515914e41 < x < 1.11468067484531015e41

   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. Using strategy rm

   3. Applied *-un-lft-identity_binary64_12695 1.1

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

   4. Applied times-frac_binary64_12701 0.5

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

   5. Simplified 0.5

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

   6. Simplified 0.5

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

   8. Applied distribute-rgt-in_binary64_12645 0.5

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

   9. Simplified 0.5

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

   10. Simplified 0.5

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

   if 1.11468067484531015e41 < x

   1. Initial program 60.1

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

   3. Applied *-un-lft-identity_binary64_12695 60.1

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

   4. Applied times-frac_binary64_12701 56.2

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

   5. Simplified 56.2

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

   6. Simplified 56.2

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

   7. Taylor expanded around inf 0.9

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922228\right) - 101.78514585392108 \cdot \frac{1}{x}\right)}\]

   8. Simplified 0.9

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(4.16438922228 + \frac{y}{{x}^{3}}\right) - \frac{101.78514585392108}{x}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.361760792985159 \cdot 10^{+41}:\\ \;\;\;\;\left(x \cdot 4.16438922228 + \frac{y}{x \cdot x}\right) - 110.11392429848108\\ \mathbf{elif}\;x \leq 1.1146806748453102 \cdot 10^{+41}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{x \cdot \left(y + x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right)\right) + z}{x \cdot \left(\left(x \cdot \left(x \cdot \left(x + 43.3400022514\right)\right) + x \cdot 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(4.16438922228 + \frac{y}{{x}^{3}}\right) - \frac{101.78514585392108}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(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)))