sqrtexp (problem 3.4.4)

Average Error: 41.5 → 0.3

Time: 6.0s

Precision: binary64

Cost: 19908

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4.2e-6)
   (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0)))
   (sqrt (+ x 2.0))))

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

↓

double code(double x) {
	double tmp;
	if (x <= -4.2e-6) {
		tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
	} else {
		tmp = sqrt((x + 2.0));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-4.2d-6)) then
        tmp = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
    else
        tmp = sqrt((x + 2.0d0))
    end if
    code = tmp
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) {
	double tmp;
	if (x <= -4.2e-6) {
		tmp = Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
	} else {
		tmp = Math.sqrt((x + 2.0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -4.2e-6:
		tmp = math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))
	else:
		tmp = math.sqrt((x + 2.0))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -4.2e-6)
		tmp = sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)));
	else
		tmp = sqrt(Float64(x + 2.0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2e-6)
		tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
	else
		tmp = sqrt((x + 2.0));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[x, -4.2e-6], 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], N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999996e-6

   1. Initial program 0.1

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

   if -4.1999999999999996e-6 < x

   1. Initial program 62.1

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

   2. Taylor expanded in x around 0 0.4

      \[\leadsto \sqrt{\color{blue}{2 + x}} \]

   3. Simplified 0.4

      \[\leadsto \sqrt{\color{blue}{x + 2}} \]
      Proof

      [Start] 0.4	\[ \sqrt{2 + x} \]
      rational_best.json-simplify-1 [=>] 0.4	\[ \sqrt{\color{blue}{x + 2}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.5
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))