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

Average Error: 29.9 → 0.6

Time: 12.2s

Precision: binary64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

↓

\[\begin{array}{l} t_1 := \frac{457.9610022158428}{z \cdot z}\\ t_2 := \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right)\\ t_3 := \frac{t}{z \cdot z}\\ \mathbf{if}\;z \leq -5.725184660733055 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(t_3 + \left(t_1 + \frac{a}{{z}^{3}}\right)\right)\right) - t_2, x\right)\\ \mathbf{elif}\;z \leq 4.7714490306862366 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(t_3 + \left(t_1 + \frac{\frac{a}{z \cdot z}}{z}\right)\right)\right) - t_2, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ 457.9610022158428 (* z z)))
        (t_2
         (fma
          15.234687407
          (/ t (pow z 3.0))
          (+ (/ 36.52704169880642 z) (/ 5864.8025282699045 (pow z 3.0)))))
        (t_3 (/ t (* z z))))
   (if (<= z -5.725184660733055e+21)
     (fma y (- (+ 3.13060547623 (+ t_3 (+ t_1 (/ a (pow z 3.0))))) t_2) x)
     (if (<= z 4.7714490306862366e+30)
       (fma
        y
        (*
         (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
         (/
          1.0
          (fma
           z
           (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
           0.607771387771)))
        x)
       (fma
        y
        (- (+ 3.13060547623 (+ t_3 (+ t_1 (/ (/ a (* z z)) z)))) t_2)
        x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 457.9610022158428 / (z * z);
	double t_2 = fma(15.234687407, (t / pow(z, 3.0)), ((36.52704169880642 / z) + (5864.8025282699045 / pow(z, 3.0))));
	double t_3 = t / (z * z);
	double tmp;
	if (z <= -5.725184660733055e+21) {
		tmp = fma(y, ((3.13060547623 + (t_3 + (t_1 + (a / pow(z, 3.0))))) - t_2), x);
	} else if (z <= 4.7714490306862366e+30) {
		tmp = fma(y, (fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b) * (1.0 / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771))), x);
	} else {
		tmp = fma(y, ((3.13060547623 + (t_3 + (t_1 + ((a / (z * z)) / z)))) - t_2), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.9
Target	1.2
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if z < -5725184660733055470000

   1. Initial program 58.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 54.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   3. Taylor expanded in z around inf 1.0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) - \left(15.234687407 \cdot \frac{t}{{z}^{3}} + \left(5864.8025282699045 \cdot \frac{1}{{z}^{3}} + 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]

   4. Simplified 1.0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right)}, x\right) \]

   if -5725184660733055470000 < z < 4.7714490306862366e30

   1. Initial program 0.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   3. Applied div-inv_binary64 0.4

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right) \]

   if 4.7714490306862366e30 < z

   1. Initial program 59.3

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   2. Simplified 56.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   3. Taylor expanded in z around inf 0.7

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{{z}^{2}} + \left(457.9610022158428 \cdot \frac{1}{{z}^{2}} + \frac{a}{{z}^{3}}\right)\right)\right) - \left(15.234687407 \cdot \frac{t}{{z}^{3}} + \left(5864.8025282699045 \cdot \frac{1}{{z}^{3}} + 36.52704169880642 \cdot \frac{1}{z}\right)\right)}, x\right) \]

   4. Simplified 0.7

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right)}, x\right) \]

   5. Applied unpow3_binary64 0.7

      \[\leadsto \mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{\color{blue}{\left(z \cdot z\right) \cdot z}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right) \]

   6. Applied associate-/r*_binary64 0.7

      \[\leadsto \mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \color{blue}{\frac{\frac{a}{z \cdot z}}{z}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.725184660733055 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{a}{{z}^{3}}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\ \mathbf{elif}\;z \leq 4.7714490306862366 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(3.13060547623 + \left(\frac{t}{z \cdot z} + \left(\frac{457.9610022158428}{z \cdot z} + \frac{\frac{a}{z \cdot z}}{z}\right)\right)\right) - \mathsf{fma}\left(15.234687407, \frac{t}{{z}^{3}}, \frac{36.52704169880642}{z} + \frac{5864.8025282699045}{{z}^{3}}\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022076 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))