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

Average Error: 27.5 → 0.5

Time: 13.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.67035503993276071 \cdot 10^{43} \lor \neg \left(x \le 3.7248869126440915 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r440762 = x;
        double r440763 = 2.0;
        double r440764 = r440762 - r440763;
        double r440765 = 4.16438922228;
        double r440766 = r440762 * r440765;
        double r440767 = 78.6994924154;
        double r440768 = r440766 + r440767;
        double r440769 = r440768 * r440762;
        double r440770 = 137.519416416;
        double r440771 = r440769 + r440770;
        double r440772 = r440771 * r440762;
        double r440773 = y;
        double r440774 = r440772 + r440773;
        double r440775 = r440774 * r440762;
        double r440776 = z;
        double r440777 = r440775 + r440776;
        double r440778 = r440764 * r440777;
        double r440779 = 43.3400022514;
        double r440780 = r440762 + r440779;
        double r440781 = r440780 * r440762;
        double r440782 = 263.505074721;
        double r440783 = r440781 + r440782;
        double r440784 = r440783 * r440762;
        double r440785 = 313.399215894;
        double r440786 = r440784 + r440785;
        double r440787 = r440786 * r440762;
        double r440788 = 47.066876606;
        double r440789 = r440787 + r440788;
        double r440790 = r440778 / r440789;
        return r440790;
}

↓

double f(double x, double y, double z) {
        double r440791 = x;
        double r440792 = -3.6703550399327607e+43;
        bool r440793 = r440791 <= r440792;
        double r440794 = 3.7248869126440915e+43;
        bool r440795 = r440791 <= r440794;
        double r440796 = !r440795;
        bool r440797 = r440793 || r440796;
        double r440798 = y;
        double r440799 = 2.0;
        double r440800 = pow(r440791, r440799);
        double r440801 = r440798 / r440800;
        double r440802 = 4.16438922228;
        double r440803 = r440802 * r440791;
        double r440804 = r440801 + r440803;
        double r440805 = 110.1139242984811;
        double r440806 = r440804 - r440805;
        double r440807 = 2.0;
        double r440808 = r440791 - r440807;
        double r440809 = r440791 * r440802;
        double r440810 = 78.6994924154;
        double r440811 = r440809 + r440810;
        double r440812 = r440811 * r440791;
        double r440813 = 3.0;
        double r440814 = pow(r440812, r440813);
        double r440815 = 137.519416416;
        double r440816 = pow(r440815, r440813);
        double r440817 = r440814 + r440816;
        double r440818 = r440817 * r440791;
        double r440819 = r440812 * r440812;
        double r440820 = r440815 * r440815;
        double r440821 = r440812 * r440815;
        double r440822 = r440820 - r440821;
        double r440823 = r440819 + r440822;
        double r440824 = r440818 / r440823;
        double r440825 = r440824 + r440798;
        double r440826 = r440825 * r440791;
        double r440827 = z;
        double r440828 = r440826 + r440827;
        double r440829 = 43.3400022514;
        double r440830 = r440791 + r440829;
        double r440831 = r440830 * r440791;
        double r440832 = 263.505074721;
        double r440833 = r440831 + r440832;
        double r440834 = r440833 * r440791;
        double r440835 = 313.399215894;
        double r440836 = r440834 + r440835;
        double r440837 = r440836 * r440791;
        double r440838 = 47.066876606;
        double r440839 = r440837 + r440838;
        double r440840 = r440828 / r440839;
        double r440841 = r440808 * r440840;
        double r440842 = r440797 ? r440806 : r440841;
        return r440842;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.5
Target	0.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -3.6703550399327607e+43 or 3.7248869126440915e+43 < x

   1. Initial program 61.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   2. Taylor expanded around inf 0.6

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

   if -3.6703550399327607e+43 < x < 3.7248869126440915e+43

   1. Initial program 1.1

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.1

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001\right)}}\]

   4. Applied times-frac 0.4

      \[\leadsto \color{blue}{\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}}\]

   5. Simplified 0.4

      \[\leadsto \color{blue}{\left(x - 2\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
   6. Using strategy rm

   7. Applied flip3-+ 0.5

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\color{blue}{\frac{{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

   8. Applied associate-*l/ 0.5

      \[\leadsto \left(x - 2\right) \cdot \frac{\left(\color{blue}{\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)}} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.67035503993276071 \cdot 10^{43} \lor \neg \left(x \le 3.7248869126440915 \cdot 10^{43}\right):\\ \;\;\;\;\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2\right) \cdot \frac{\left(\frac{\left({\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right)}^{3} + {137.51941641600001}^{3}\right) \cdot x}{\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) + \left(137.51941641600001 \cdot 137.51941641600001 - \left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x\right) \cdot 137.51941641600001\right)} + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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) 1) (/ (+ (* (+ (* (+ (* (+ (* 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))