Average Error: 41.5 → 0.0
Time: 4.5s
Precision: binary64
Cost: 12992
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
↓
\[\sqrt{1 + e^{x}} \]
(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
↓
(FPCore (x) :precision binary64 (sqrt (+ 1.0 (exp x))))
double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}
↓
double code(double x) {
	return sqrt((1.0 + exp(x)));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + exp(x)))
end function

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}
↓
public static double code(double x) {
	return Math.sqrt((1.0 + Math.exp(x)));
}

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))
↓
def code(x):
	return math.sqrt((1.0 + math.exp(x)))

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end
↓
function code(x)
	return sqrt(Float64(1.0 + exp(x)))
end

function tmp = code(x)
	tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
end
↓
function tmp = code(x)
	tmp = sqrt((1.0 + exp(x)));
end

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Sqrt[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}
↓
\sqrt{1 + e^{x}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 41.5

    \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
    Proof
    (sqrt.f64 (+.f64 1 (exp.f64 x))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (exp.f64 x) 1))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 (exp.f64 x) 1) 1))): 0 points increase in error, 0 points decrease in error
    (sqrt.f64 (/.f64 (+.f64 (exp.f64 x) 1) (Rewrite<= *-inverses_binary64 (/.f64 (-.f64 (exp.f64 x) 1) (-.f64 (exp.f64 x) 1))))): 161 points increase in error, 0 points decrease in error
    (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (exp.f64 x) 1) (-.f64 (exp.f64 x) 1)) (-.f64 (exp.f64 x) 1)))): 2 points increase in error, 0 points decrease in error
    (sqrt.f64 (/.f64 (Rewrite<= difference-of-sqr-1_binary64 (-.f64 (*.f64 (exp.f64 x) (exp.f64 x)) 1)) (-.f64 (exp.f64 x) 1))): 10 points increase in error, 0 points decrease in error
    (sqrt.f64 (/.f64 (-.f64 (Rewrite<= exp-lft-sqr_binary64 (exp.f64 (*.f64 x 2))) 1) (-.f64 (exp.f64 x) 1))): 4 points increase in error, 0 points decrease in error
    (sqrt.f64 (/.f64 (-.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 x))) 1) (-.f64 (exp.f64 x) 1))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.0

    \[\leadsto \sqrt{1 + e^{x}} \]

Alternatives

Alternative 1
Error17.5
Cost6464
\[\sqrt{2} \]

Error

Reproduce

herbie shell --seed 2022291 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))