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

Average Error: 26.7 → 0.7

Time: 26.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.369172497454679 \cdot 10^{+37}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{3451.550173699799}{x \cdot x} + \left(4.16438922228 + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{101.78514585392108}{x} + \frac{124074.40615218396}{{x}^{3}}\right)\right)\\ \mathbf{elif}\;x \leq 3.88076255453383 \cdot 10^{+73}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{x \cdot y}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + \left(\frac{z}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + \left(137.519416416 \cdot \frac{{x}^{2}}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{4}}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + 78.6994924154 \cdot \frac{{x}^{3}}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array}\]

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 -2.369172497454679e+37)
   (*
    (- x 2.0)
    (-
     (+ (/ 3451.550173699799 (* x x)) (+ 4.16438922228 (/ y (pow x 3.0))))
     (+ (/ 101.78514585392108 x) (/ 124074.40615218396 (pow x 3.0)))))
   (if (<= x 3.88076255453383e+73)
     (*
      (- x 2.0)
      (+
       (/
        (* x y)
        (+
         47.066876606
         (+
          (* x 313.399215894)
          (+
           (* 263.505074721 (pow x 2.0))
           (+ (* (pow x 3.0) 43.3400022514) (pow x 4.0))))))
       (+
        (/
         z
         (+
          47.066876606
          (+
           (* x 313.399215894)
           (+
            (* 263.505074721 (pow x 2.0))
            (+ (* (pow x 3.0) 43.3400022514) (pow x 4.0))))))
        (+
         (*
          137.519416416
          (/
           (pow x 2.0)
           (+
            47.066876606
            (+
             (* x 313.399215894)
             (+
              (* 263.505074721 (pow x 2.0))
              (+ (* (pow x 3.0) 43.3400022514) (pow x 4.0)))))))
         (+
          (*
           4.16438922228
           (/
            (pow x 4.0)
            (+
             47.066876606
             (+
              (* x 313.399215894)
              (+
               (* 263.505074721 (pow x 2.0))
               (+ (* (pow x 3.0) 43.3400022514) (pow x 4.0)))))))
          (*
           78.6994924154
           (/
            (pow x 3.0)
            (+
             47.066876606
             (+
              (* x 313.399215894)
              (+
               (* 263.505074721 (pow x 2.0))
               (+ (* (pow x 3.0) 43.3400022514) (pow x 4.0))))))))))))
     (* x 4.16438922228))))

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 <= -2.369172497454679e+37) {
		tmp = (x - 2.0) * (((3451.550173699799 / (x * x)) + (4.16438922228 + (y / pow(x, 3.0)))) - ((101.78514585392108 / x) + (124074.40615218396 / pow(x, 3.0))));
	} else if (x <= 3.88076255453383e+73) {
		tmp = (x - 2.0) * (((x * y) / (47.066876606 + ((x * 313.399215894) + ((263.505074721 * pow(x, 2.0)) + ((pow(x, 3.0) * 43.3400022514) + pow(x, 4.0)))))) + ((z / (47.066876606 + ((x * 313.399215894) + ((263.505074721 * pow(x, 2.0)) + ((pow(x, 3.0) * 43.3400022514) + pow(x, 4.0)))))) + ((137.519416416 * (pow(x, 2.0) / (47.066876606 + ((x * 313.399215894) + ((263.505074721 * pow(x, 2.0)) + ((pow(x, 3.0) * 43.3400022514) + pow(x, 4.0))))))) + ((4.16438922228 * (pow(x, 4.0) / (47.066876606 + ((x * 313.399215894) + ((263.505074721 * pow(x, 2.0)) + ((pow(x, 3.0) * 43.3400022514) + pow(x, 4.0))))))) + (78.6994924154 * (pow(x, 3.0) / (47.066876606 + ((x * 313.399215894) + ((263.505074721 * pow(x, 2.0)) + ((pow(x, 3.0) * 43.3400022514) + pow(x, 4.0)))))))))));
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.7
Target	0.6
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 < -2.3691724974546789e37

   1. Initial program 58.9

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

   3. Applied *-un-lft-identity_binary64_10649 58.9

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

   4. Applied times-frac_binary64_10655 54.8

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

   5. Simplified 54.8

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

   6. Simplified 54.8

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

   7. Taylor expanded around inf 0.9

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(3451.550173699799 \cdot \frac{1}{{x}^{2}} + \left(\frac{y}{{x}^{3}} + 4.16438922228\right)\right) - \left(101.78514585392108 \cdot \frac{1}{x} + 124074.40615218396 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

   8. Simplified 0.9

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\left(\frac{3451.550173699799}{x \cdot x} + \left(4.16438922228 + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{101.78514585392108}{x} + \frac{124074.40615218396}{{x}^{3}}\right)\right)}\]

   if -2.3691724974546789e37 < x < 3.88076255453383013e73

   1. Initial program 2.6

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

   3. Applied *-un-lft-identity_binary64_10649 2.6

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

   4. Applied times-frac_binary64_10655 0.8

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

   5. Simplified 0.8

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

   6. Simplified 0.8

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

   7. Taylor expanded around 0 0.7

      \[\leadsto \left(x - 2\right) \cdot \color{blue}{\left(\frac{x \cdot y}{47.066876606 + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + {x}^{4}\right)\right)\right)} + \left(\frac{z}{47.066876606 + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + {x}^{4}\right)\right)\right)} + \left(137.519416416 \cdot \frac{{x}^{2}}{47.066876606 + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + {x}^{4}\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{4}}{47.066876606 + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + {x}^{4}\right)\right)\right)} + 78.6994924154 \cdot \frac{{x}^{3}}{47.066876606 + \left(313.399215894 \cdot x + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + {x}^{4}\right)\right)\right)}\right)\right)\right)\right)}\]

   if 3.88076255453383013e73 < x

   1. Initial program 64.0

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

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{4.16438922228 \cdot x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.369172497454679 \cdot 10^{+37}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\left(\frac{3451.550173699799}{x \cdot x} + \left(4.16438922228 + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{101.78514585392108}{x} + \frac{124074.40615218396}{{x}^{3}}\right)\right)\\ \mathbf{elif}\;x \leq 3.88076255453383 \cdot 10^{+73}:\\ \;\;\;\;\left(x - 2\right) \cdot \left(\frac{x \cdot y}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + \left(\frac{z}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + \left(137.519416416 \cdot \frac{{x}^{2}}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{4}}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)} + 78.6994924154 \cdot \frac{{x}^{3}}{47.066876606 + \left(x \cdot 313.399215894 + \left(263.505074721 \cdot {x}^{2} + \left({x}^{3} \cdot 43.3400022514 + {x}^{4}\right)\right)\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array}\]

Reproduce

herbie shell --seed 2021093 
(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)))