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

Average Error: 29.4 → 1.5

Time: 10.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1911814461531339 \cdot 10^{+33}:\\ \;\;\;\;x + y \cdot \left(\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}\right)\\ \mathbf{elif}\;z \leq 176105132395.4339:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(z \cdot \left(t + \frac{z \cdot \left(z \cdot \left(z \cdot 9.800690647801265\right) + -124.69639771500474\right)}{z \cdot 3.13060547623 - 11.1667541262}\right) + a\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot 3.13060547623 + \frac{y}{z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)\\ \end{array}\]

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
 (if (<= z -1.1911814461531339e+33)
   (+ x (* y (- (+ 3.13060547623 (/ t (* z z))) (/ 36.527041698806414 z))))
   (if (<= z 176105132395.4339)
     (+
      x
      (*
       y
       (/
        (+
         (*
          z
          (+
           (*
            z
            (+
             t
             (/
              (* z (+ (* z (* z 9.800690647801265)) -124.69639771500474))
              (- (* z 3.13060547623) 11.1667541262))))
           a))
         b)
        (+
         (*
          z
          (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
         0.607771387771))))
     (+
      x
      (+ (* y 3.13060547623) (* (/ y z) (+ (/ t z) -36.527041698806414)))))))

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 tmp;
	if (z <= -1.1911814461531339e+33) {
		tmp = x + (y * ((3.13060547623 + (t / (z * z))) - (36.527041698806414 / z)));
	} else if (z <= 176105132395.4339) {
		tmp = x + (y * (((z * ((z * (t + ((z * ((z * (z * 9.800690647801265)) + -124.69639771500474)) / ((z * 3.13060547623) - 11.1667541262)))) + a)) + b) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)));
	} else {
		tmp = x + ((y * 3.13060547623) + ((y / z) * ((t / z) + -36.527041698806414)));
	}
	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	29.4
Target	1.1
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if z < -1.19118144615313387e33

   1. Initial program 59.0

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

   3. Applied *-un-lft-identity_binary64_13557 59.0

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

   4. Applied times-frac_binary64_13563 56.5

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

   5. Simplified 56.5

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

   6. Simplified 56.5

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

   7. Taylor expanded around inf 1.7

      \[\leadsto x + y \cdot \color{blue}{\left(\left(3.13060547623 + \frac{t}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{1}{z}\right)}\]

   8. Simplified 1.7

      \[\leadsto x + y \cdot \color{blue}{\left(\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}\right)}\]

   if -1.19118144615313387e33 < z < 176105132395.4339

   1. Initial program 0.9

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

   3. Applied *-un-lft-identity_binary64_13557 0.9

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

   4. Applied times-frac_binary64_13563 0.5

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

   5. Simplified 0.5

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

   6. Simplified 0.5

      \[\leadsto x + y \cdot \color{blue}{\frac{z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}}\]
   7. Using strategy rm

   8. Applied flip-+_binary64_13531 0.5

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

   9. Applied associate-*r/_binary64_13501 0.5

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

   10. Simplified 0.5

       \[\leadsto x + y \cdot \frac{z \cdot \left(z \cdot \left(\frac{\color{blue}{z \cdot \left(z \cdot \left(9.800690647801265 \cdot z\right) + -124.69639771500474\right)}}{z \cdot 3.13060547623 - 11.1667541262} + t\right) + a\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\]

   if 176105132395.4339 < z

   1. Initial program 56.6

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

   2. Taylor expanded around inf 10.7

      \[\leadsto x + \color{blue}{\left(\left(3.13060547623 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

   3. Simplified 3.0

      \[\leadsto x + \color{blue}{\left(y \cdot 3.13060547623 + \frac{y}{z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1911814461531339 \cdot 10^{+33}:\\ \;\;\;\;x + y \cdot \left(\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}\right)\\ \mathbf{elif}\;z \leq 176105132395.4339:\\ \;\;\;\;x + y \cdot \frac{z \cdot \left(z \cdot \left(t + \frac{z \cdot \left(z \cdot \left(z \cdot 9.800690647801265\right) + -124.69639771500474\right)}{z \cdot 3.13060547623 - 11.1667541262}\right) + a\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot 3.13060547623 + \frac{y}{z} \cdot \left(\frac{t}{z} + -36.527041698806414\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))))