sqrtexp (problem 3.4.4)

Average Error: 41.2 → 0.0

Time: 7.3s

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 41.2

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
   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))))): 139 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)))): 1 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))): 13 points increase in error, 1 points decrease in error
   (sqrt.f64 (/.f64 (-.f64 (Rewrite<= exp-lft-sqr_binary64 (exp.f64 (*.f64 x 2))) 1) (-.f64 (exp.f64 x) 1))): 12 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 simplification 0.0

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

Alternatives

Alternative 1
Error	17.0
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x \leq -10.621655339931833:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.09375 + x \cdot 0.018229166666666668\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	17.1
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq -10.621655339931833:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + x \cdot 0.09375\right)\right)\\ \end{array} \]

Alternative 3
Error	17.7
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))