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

Average Error: 26.6 → 1.0

Time: 39.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.737801379487723 \cdot 10^{+38} \lor \neg \left(x \leq 6.654331205362247 \cdot 10^{+36}\right):\\ \;\;\;\;\left(\mathsf{fma}\left(x, 4.16438922228, \frac{3655.1204654076414}{x}\right) + \frac{y}{x \cdot x}\right) - \left(110.1139242984811 + \frac{130977.50649958357}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := {x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)\\ t_1 := \frac{z}{t_0}\\ \mathsf{fma}\left(70.37071397084, \frac{{x}^{4}}{t_0}, \mathsf{fma}\left(4.16438922228, \frac{{x}^{5}}{t_0}, \mathsf{fma}\left(t_1, x, \frac{x \cdot \left(x \cdot y\right)}{t_0}\right)\right)\right) - \mathsf{fma}\left(2, \mathsf{fma}\left(\frac{y}{t_0}, x, t_1\right), \mathsf{fma}\left(275.038832832, \frac{x \cdot x}{t_0}, 19.8795684148 \cdot \frac{{x}^{3}}{t_0}\right)\right) \end{array}\\ \end{array} \]

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 -4.737801379487723e+38) (not (<= x 6.654331205362247e+36)))
   (-
    (+ (fma x 4.16438922228 (/ 3655.1204654076414 x)) (/ y (* x x)))
    (+ 110.1139242984811 (/ 130977.50649958357 (* x x))))
   (let* ((t_0
           (+
            (pow x 4.0)
            (+
             47.066876606
             (fma
              x
              313.399215894
              (* (* x x) (+ 263.505074721 (* x 43.3400022514)))))))
          (t_1 (/ z t_0)))
     (-
      (fma
       70.37071397084
       (/ (pow x 4.0) t_0)
       (fma
        4.16438922228
        (/ (pow x 5.0) t_0)
        (fma t_1 x (/ (* x (* x y)) t_0))))
      (fma
       2.0
       (fma (/ y t_0) x t_1)
       (fma
        275.038832832
        (/ (* x x) t_0)
        (* 19.8795684148 (/ (pow x 3.0) 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 tmp;
	if ((x <= -4.737801379487723e+38) || !(x <= 6.654331205362247e+36)) {
		tmp = (fma(x, 4.16438922228, (3655.1204654076414 / x)) + (y / (x * x))) - (110.1139242984811 + (130977.50649958357 / (x * x)));
	} else {
		double t_0 = pow(x, 4.0) + (47.066876606 + fma(x, 313.399215894, ((x * x) * (263.505074721 + (x * 43.3400022514)))));
		double t_1 = z / t_0;
		tmp = fma(70.37071397084, (pow(x, 4.0) / t_0), fma(4.16438922228, (pow(x, 5.0) / t_0), fma(t_1, x, ((x * (x * y)) / t_0)))) - fma(2.0, fma((y / t_0), x, t_1), fma(275.038832832, ((x * x) / t_0), (19.8795684148 * (pow(x, 3.0) / t_0))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	26.6
Target	0.8
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if x < -4.73780137948772316e38 or 6.65433120536224705e36 < x

   1. Initial program 59.7

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

   2. Simplified 59.7

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

   3. Taylor expanded in x around inf 1.6

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

   4. Simplified 1.6

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

   if -4.73780137948772316e38 < x < 6.65433120536224705e36

   1. Initial program 0.8

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

   2. Simplified 0.8

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

   3. Taylor expanded in y around 0 0.8

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

   4. Simplified 0.6

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.737801379487723 \cdot 10^{+38} \lor \neg \left(x \leq 6.654331205362247 \cdot 10^{+36}\right):\\ \;\;\;\;\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{else}:\\ \;\;\;\;\mathsf{fma}\left(70.37071397084, \frac{{x}^{4}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}, \mathsf{fma}\left(4.16438922228, \frac{{x}^{5}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}, \mathsf{fma}\left(\frac{z}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}, x, \frac{x \cdot \left(x \cdot y\right)}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}\right)\right)\right) - \mathsf{fma}\left(2, \mathsf{fma}\left(\frac{y}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}, x, \frac{z}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}\right), \mathsf{fma}\left(275.038832832, \frac{x \cdot x}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}, 19.8795684148 \cdot \frac{{x}^{3}}{{x}^{4} + \left(47.066876606 + \mathsf{fma}\left(x, 313.399215894, \left(x \cdot x\right) \cdot \left(263.505074721 + x \cdot 43.3400022514\right)\right)\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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)))