sqrtexp (problem 3.4.4)

Average Error: 63.81% → 0.05%

Time: 4.7s

Precision: binary64

Cost: 12992

Math

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

↓

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

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))

↓

(FPCore (x) :precision binary64 (sqrt (+ 1.0 (exp x))))

C

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)));
}

Fortran

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

Java

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)));
}

Python

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)))

Julia

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

MATLAB

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

Wolfram

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]

Derivation

1. Initial program 63.81

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

2. Simplified 0.05

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

   [Start] 63.81	\[ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   *-commutative [=>] 63.81	\[ \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]
   exp-lft-sqr [=>] 63.35	\[ \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]
   difference-of-sqr-1 [=>] 62.82	\[ \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]
   associate-/l* [=>] 62.81	\[ \sqrt{\color{blue}{\frac{e^{x} + 1}{\frac{e^{x} - 1}{e^{x} - 1}}}} \]
   *-inverses [=>] 0.05	\[ \sqrt{\frac{e^{x} + 1}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 0.05	\[ \sqrt{\color{blue}{e^{x} + 1}} \]
   +-commutative [=>] 0.05	\[ \sqrt{\color{blue}{1 + e^{x}}} \]

3. Final simplification 0.05

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

Alternatives

Alternative 1
Error	28.05%
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))