Table 3 Expressions from the Nguyen (first two columns on the left-hand side) and Feynman (last three columns on the right-hand side) benchmarks

NG-1

\(x^3 + x^2 + x\)

FM-1

I.6.2a

\((2\pi )^{-0.5} e^{-\frac{x^2}{2}}\)

NG-2

\(x^4 + x^3 + x^2 + x\)

FM-2

I.6.2

\((\sqrt{2\pi }\cdot y)^{-1} e^{-\frac{(x/y)^2}{2}}\)

NG-3

\(x^5 + x^4 + x^3 + x^2 + x\)

FM-3

I.12.1, I.12.5

xy

NG-4

\(x^6 + x^5 + x^4 + x^3 + x^2 + x\)

FM-4

I.14.4, II.8.31

\(0.5\, x y^2\)

NG-5

\(\sin x^2 \cdot \cos x - 1\)

FM-5

I.25.13, I.29.4

x/y

NG-6

\(\sin x + \sin (x + x^2)\)

FM-6

I.26.2

\(\arcsin (x\sin {y})\)

NG-7

\(\ln (x+1) + \ln (x^2 + 1)\)

FM-7

I.34.27, III.12.43

\((2\pi )^{-1}xy\)

NG-8

\(\sqrt{x}\)

FM-8

I.39.1

\(1.5\, xy\)

NG-9

\(\sin x + \sin y^2\)

FM-9

II.3.24

\(\frac{x}{4\pi y^2}\)

NG-10

\(2\sin x \cdot \cos y\)

FM-10

II.11.28

\(\frac{1+xy}{1-(0.\overline{3}\, xy)}\)

FM-11

II.27.18

\(xy^2\)

FM-12

II.38.14

\(\frac{x}{2\cdot (1 + y)}\)