For these kind of tasks, I love to use the program Ries by Robert Munafo. It was essentially made for such tasks. Let me show you how it gives outputs (on the examples you ask).
For $1.414$, it gives
Your target value: T = 1.414 mrob.com/ries
x^2 = 2 for x = T + 0.000213562 {37}
x^5 = e^sqrt(3) for x = T - 1.75419e-05 {75}
1/(log_5(x)) = sqrt(7)+2 for x = T + 1.36403e-05 {92}
e^(x^5) = 8^e for x = T - 3.80911e-06 {92}
1/(x-sqrt(phi)) = e^phi+2 for x = T + 1.25986e-06 {102}
1/ln(x)-2 = 1/4"/phi for x = T - 6.46979e-07 {101}
(x^2-1)^2 = cospi(1/8^2) for x = T + 5.46898e-07 {96}
x^2-sqrt(phi) = 1/e^(1/pi) for x = T + 3.53186e-07 {97}
sinpi(1/(e^x)^2) = 1/(sqrt(2)+4) for x = T - 2.92598e-07 {106}
1/sinpi(ln(sqrt(x))) = 8/(1+pi) for x = T + 8.71324e-08 {109}
log_(2/x)(x) = sqrt(sinpi(ln(phi))) for x = T + 7.7742e-09 {110}
sinpi(-x+sqrt(2)) = 2/e^8 for x = T + 2.1526e-10 {111}
(3-1/7) x = 1/5^2+4 ('exact' match) {117}
(for more results, use the option '-l3')
NOTE: 'exact' match may result from floating-point roundoff error.
log_A(B) = logarithm to base A of B = ln(B) / ln(A) cospi(X) = cos(pi * x)
e = base of natural logarithms, 2.71828... sinpi(X) = sin(pi * x)
ln(x) = natural logarithm or log base e sqrt(x) = square root
phi = the golden ratio, (1+sqrt(5))/2 A"/B = Ath root of B pi = 3.14159...
--LHS-- --RHS-- -Total-
max complexity: 67 62 129
dead-ends: 2747395 5223537 7970932 Time: 0.209
expressions: 196190 334648 530838
distinct: 102990 99793 202783 Memory: 14272KiB
Total equations tested: 10277681070 (1.028e+10)
For $1.155727349$, it gives
Your target value: T = 1.155727349 mrob.com/ries
e^x = pi for x = T - 0.0109975 {42}
x/pi = 1/e for x = T + 7.90922e-10 {57}
x-pi/e = 1/-(e^(e^pi)) for x = T + 7.01771e-10 {109}
pi-e x = 1/e^(e^3) for x = T + 9.48281e-11 {108}
pi 1/x-e = 1/e^(e^3) for x = T - 1.35727e-11 {109}
(e x)^(4^2) = pi^(4^2)-1 for x = T - 1.13793e-11 {126}
1/(pi-e x) = 6^(8+pi) for x = T + 4.12737e-12 {128}
(for more results, use the option '-l3')
e = base of natural logarithms, 2.71828... pi = 3.14159...
--LHS-- --RHS-- -Total-
max complexity: 67 62 129
dead-ends: 2754425 5223469 7977894 Time: 0.212
expressions: 196844 334628 531472
distinct: 102092 99784 201876 Memory: 14208KiB
Total equations tested: 10187148128 (1.019e+10)
It gives you a lot of examples to pick and choose from. Hope this helps!
Obligatory xkcd
Flavour text : Two tips: 1) $8675309$ is not just prime, it's a twin prime, and 2) if you ever find yourself raising $\log$(anything)^$e$ or taking the $\pi$-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.
