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

Average Error: 26.8 → 1.3

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -3.019861234299114 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.264914738129263 \cdot 10^{+25}:\\ \;\;\;\;\begin{array}{l} t_1 := {x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)\\ \mathsf{fma}\left(70.37071397084, \frac{{x}^{4}}{t_1}, \frac{x \cdot \left(x \cdot y\right)}{t_1} + \mathsf{fma}\left(4.16438922228, \frac{{x}^{5}}{t_1}, \frac{x \cdot z}{t_1}\right)\right) - \mathsf{fma}\left(2, \frac{x \cdot y}{t_1}, \mathsf{fma}\left(2, \frac{z}{t_1}, \mathsf{fma}\left(275.038832832, \frac{x \cdot x}{t_1}, 19.8795684148 \cdot \frac{{x}^{3}}{t_1}\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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
 (let* ((t_0
         (-
          (+ (fma x 4.16438922228 (/ 3655.1204654076414 x)) (/ y (* x x)))
          (+ 110.1139242984811 (/ 130977.50649958357 (* x x))))))
   (if (<= x -3.019861234299114e+31)
     t_0
     (if (<= x 1.264914738129263e+25)
       (let* ((t_1
               (+
                (pow x 4.0)
                (+
                 47.066876606
                 (fma
                  (* x x)
                  263.505074721
                  (fma x 313.399215894 (* (pow x 3.0) 43.3400022514)))))))
         (-
          (fma
           70.37071397084
           (/ (pow x 4.0) t_1)
           (+
            (/ (* x (* x y)) t_1)
            (fma 4.16438922228 (/ (pow x 5.0) t_1) (/ (* x z) t_1))))
          (fma
           2.0
           (/ (* x y) t_1)
           (fma
            2.0
            (/ z t_1)
            (fma
             275.038832832
             (/ (* x x) t_1)
             (* 19.8795684148 (/ (pow x 3.0) t_1)))))))
       t_0))))

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 t_0 = (fma(x, 4.16438922228, (3655.1204654076414 / x)) + (y / (x * x))) - (110.1139242984811 + (130977.50649958357 / (x * x)));
	double tmp;
	if (x <= -3.019861234299114e+31) {
		tmp = t_0;
	} else if (x <= 1.264914738129263e+25) {
		double t_1 = pow(x, 4.0) + (47.066876606 + fma((x * x), 263.505074721, fma(x, 313.399215894, (pow(x, 3.0) * 43.3400022514))));
		tmp = fma(70.37071397084, (pow(x, 4.0) / t_1), (((x * (x * y)) / t_1) + fma(4.16438922228, (pow(x, 5.0) / t_1), ((x * z) / t_1)))) - fma(2.0, ((x * y) / t_1), fma(2.0, (z / t_1), fma(275.038832832, ((x * x) / t_1), (19.8795684148 * (pow(x, 3.0) / t_1)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.8
Target	0.8
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if x < -3.0198612342991141e31 or 1.26491473812926299e25 < x

   1. Initial program 58.0

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

   2. Taylor expanded in x around inf 2.3

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(3655.1204654076414 \cdot \frac{1}{x} + 4.16438922228 \cdot x\right)\right) - \left(110.1139242984811 + 130977.50649958357 \cdot \frac{1}{{x}^{2}}\right)} \]

   3. Simplified 2.3

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)} \]

   if -3.0198612342991141e31 < x < 1.26491473812926299e25

   1. Initial program 0.5

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

   2. Taylor expanded in y around 0 0.5

      \[\leadsto \color{blue}{\left(70.37071397084 \cdot \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(\frac{y \cdot {x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(4.16438922228 \cdot \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right) - \left(2 \cdot \frac{y \cdot x}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(2 \cdot \frac{z}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + \left(275.038832832 \cdot \frac{{x}^{2}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)} + 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \left(263.505074721 \cdot {x}^{2} + \left(43.3400022514 \cdot {x}^{3} + 313.399215894 \cdot x\right)\right)\right)}\right)\right)\right)} \]

   3. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(70.37071397084, \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \frac{x \cdot \left(x \cdot y\right)}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)} + \mathsf{fma}\left(4.16438922228, \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \frac{z \cdot x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}\right)\right) - \mathsf{fma}\left(2, \frac{x \cdot y}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \mathsf{fma}\left(2, \frac{z}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \mathsf{fma}\left(275.038832832, \frac{x \cdot x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.019861234299114 \cdot 10^{+31}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.264914738129263 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(70.37071397084, \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \frac{x \cdot \left(x \cdot y\right)}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)} + \mathsf{fma}\left(4.16438922228, \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \frac{x \cdot z}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}\right)\right) - \mathsf{fma}\left(2, \frac{x \cdot y}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \mathsf{fma}\left(2, \frac{z}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, \mathsf{fma}\left(275.038832832, \frac{x \cdot x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}, 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x \cdot x, 263.505074721, \mathsf{fma}\left(x, 313.399215894, {x}^{3} \cdot 43.3400022514\right)\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022076 
(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)))