sqrtexp (problem 3.4.4)

Percentage Accurate: 36.4% → 100.0%

Time: 7.1s

Alternatives: 5

Speedup: 3.1×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 36.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 + e^{x}} \end{array} \]

FPCore

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

C

double code(double x) {
	return sqrt((1.0 + exp(x)));
}

Fortran

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((1.0 + Math.exp(x)));
}

Python

def code(x):
	return math.sqrt((1.0 + math.exp(x)))

Julia

function code(x)
	return sqrt(Float64(1.0 + exp(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + exp(x)));
end

Wolfram

code[x_] := N[Sqrt[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Step-by-step derivation

   1. *-commutative 36.4%

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

   2. exp-lft-sqr 36.5%

      \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

   3. difference-of-sqr-1 37.1%

      \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

   4. associate-/l* 37.1%

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

   5. *-inverses 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 72.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + \left(2 + \frac{x \cdot \left(x \cdot 0.25\right)}{0.5 + x \cdot -0.16666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.0)
   (sqrt 2.0)
   (sqrt
    (+ x (+ 2.0 (/ (* x (* x 0.25)) (+ 0.5 (* x -0.16666666666666666))))))))

C

double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((x + (2.0 + ((x * (x * 0.25)) / (0.5 + (x * -0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-6.0d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt((x + (2.0d0 + ((x * (x * 0.25d0)) / (0.5d0 + (x * (-0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -6.0) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((x + (2.0 + ((x * (x * 0.25)) / (0.5 + (x * -0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -6.0:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((x + (2.0 + ((x * (x * 0.25)) / (0.5 + (x * -0.16666666666666666))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -6.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(x + Float64(2.0 + Float64(Float64(x * Float64(x * 0.25)) / Float64(0.5 + Float64(x * -0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -6.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((x + (2.0 + ((x * (x * 0.25)) / (0.5 + (x * -0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -6.0], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(x + N[(2.0 + N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6

   1. Initial program 100.0%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

      3. difference-of-sqr-1 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 20.7%

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

   if -6 < x

   1. Initial program 8.0%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Step-by-step derivation

      1. *-commutative 8.0%

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

      2. exp-lft-sqr 8.2%

         \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

      3. difference-of-sqr-1 9.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

      4. associate-/l* 9.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.1%

      \[\leadsto \sqrt{\color{blue}{2 + \left(x + \left(0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 97.1%

         \[\leadsto \sqrt{\color{blue}{\left(x + \left(0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right)\right) + 2}} \]

      2. associate-+l+ 97.1%

         \[\leadsto \sqrt{\color{blue}{x + \left(\left(0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right) + 2\right)}} \]

      3. +-commutative 97.1%

         \[\leadsto \sqrt{x + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + 0.16666666666666666 \cdot {x}^{3}\right)} + 2\right)} \]

      4. *-commutative 97.1%

         \[\leadsto \sqrt{x + \left(\left(\color{blue}{{x}^{2} \cdot 0.5} + 0.16666666666666666 \cdot {x}^{3}\right) + 2\right)} \]

      5. *-commutative 97.1%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{{x}^{3} \cdot 0.16666666666666666}\right) + 2\right)} \]

      6. unpow3 97.1%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.16666666666666666\right) + 2\right)} \]

      7. unpow2 97.1%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot 0.16666666666666666\right) + 2\right)} \]

      8. associate-*l* 97.1%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{{x}^{2} \cdot \left(x \cdot 0.16666666666666666\right)}\right) + 2\right)} \]

      9. distribute-lft-out 97.1%

         \[\leadsto \sqrt{x + \left(\color{blue}{{x}^{2} \cdot \left(0.5 + x \cdot 0.16666666666666666\right)} + 2\right)} \]

      10. unpow2 97.1%

          \[\leadsto \sqrt{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 2\right)} \]

   6. Simplified 97.1%

      \[\leadsto \sqrt{\color{blue}{x + \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 2\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \sqrt{x + \left(\color{blue}{\left(0.5 + x \cdot 0.16666666666666666\right) \cdot \left(x \cdot x\right)} + 2\right)} \]

      2. flip-+ 97.1%

         \[\leadsto \sqrt{x + \left(\color{blue}{\frac{0.5 \cdot 0.5 - \left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.16666666666666666\right)}{0.5 - x \cdot 0.16666666666666666}} \cdot \left(x \cdot x\right) + 2\right)} \]

      3. associate-*l/ 97.1%

         \[\leadsto \sqrt{x + \left(\color{blue}{\frac{\left(0.5 \cdot 0.5 - \left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot x\right)}{0.5 - x \cdot 0.16666666666666666}} + 2\right)} \]

      4. metadata-eval 97.1%

         \[\leadsto \sqrt{x + \left(\frac{\left(\color{blue}{0.25} - \left(x \cdot 0.16666666666666666\right) \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot x\right)}{0.5 - x \cdot 0.16666666666666666} + 2\right)} \]

      5. swap-sqr 97.1%

         \[\leadsto \sqrt{x + \left(\frac{\left(0.25 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 \cdot 0.16666666666666666\right)}\right) \cdot \left(x \cdot x\right)}{0.5 - x \cdot 0.16666666666666666} + 2\right)} \]

      6. metadata-eval 97.1%

         \[\leadsto \sqrt{x + \left(\frac{\left(0.25 - \left(x \cdot x\right) \cdot \color{blue}{0.027777777777777776}\right) \cdot \left(x \cdot x\right)}{0.5 - x \cdot 0.16666666666666666} + 2\right)} \]

      7. *-commutative 97.1%

         \[\leadsto \sqrt{x + \left(\frac{\left(0.25 - \left(x \cdot x\right) \cdot 0.027777777777777776\right) \cdot \left(x \cdot x\right)}{0.5 - \color{blue}{0.16666666666666666 \cdot x}} + 2\right)} \]

      8. cancel-sign-sub-inv 97.1%

         \[\leadsto \sqrt{x + \left(\frac{\left(0.25 - \left(x \cdot x\right) \cdot 0.027777777777777776\right) \cdot \left(x \cdot x\right)}{\color{blue}{0.5 + \left(-0.16666666666666666\right) \cdot x}} + 2\right)} \]

      9. metadata-eval 97.1%

         \[\leadsto \sqrt{x + \left(\frac{\left(0.25 - \left(x \cdot x\right) \cdot 0.027777777777777776\right) \cdot \left(x \cdot x\right)}{0.5 + \color{blue}{-0.16666666666666666} \cdot x} + 2\right)} \]

   8. Applied egg-rr 97.1%

      \[\leadsto \sqrt{x + \left(\color{blue}{\frac{\left(0.25 - \left(x \cdot x\right) \cdot 0.027777777777777776\right) \cdot \left(x \cdot x\right)}{0.5 + -0.16666666666666666 \cdot x}} + 2\right)} \]

   9. Taylor expanded in x around 0 97.3%

      \[\leadsto \sqrt{x + \left(\frac{\color{blue}{0.25 \cdot {x}^{2}}}{0.5 + -0.16666666666666666 \cdot x} + 2\right)} \]
   10. Step-by-step derivation

       1. unpow2 97.3%

          \[\leadsto \sqrt{x + \left(\frac{0.25 \cdot \color{blue}{\left(x \cdot x\right)}}{0.5 + -0.16666666666666666 \cdot x} + 2\right)} \]

       2. *-commutative 97.3%

          \[\leadsto \sqrt{x + \left(\frac{\color{blue}{\left(x \cdot x\right) \cdot 0.25}}{0.5 + -0.16666666666666666 \cdot x} + 2\right)} \]

       3. associate-*l* 97.3%

          \[\leadsto \sqrt{x + \left(\frac{\color{blue}{x \cdot \left(x \cdot 0.25\right)}}{0.5 + -0.16666666666666666 \cdot x} + 2\right)} \]

   11. Simplified 97.3%

       \[\leadsto \sqrt{x + \left(\frac{\color{blue}{x \cdot \left(x \cdot 0.25\right)}}{0.5 + -0.16666666666666666 \cdot x} + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + \left(2 + \frac{x \cdot \left(x \cdot 0.25\right)}{0.5 + x \cdot -0.16666666666666666}\right)}\\ \end{array} \]

Alternative 3: 72.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + \left(2 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.95)
   (sqrt 2.0)
   (sqrt (+ x (+ 2.0 (* (* x x) (+ 0.5 (* x 0.16666666666666666))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((x + (2.0 + ((x * x) * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.95d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt((x + (2.0d0 + ((x * x) * (0.5d0 + (x * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((x + (2.0 + ((x * x) * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.95:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((x + (2.0 + ((x * x) * (0.5 + (x * 0.16666666666666666))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.95)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(x + Float64(2.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.95)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((x + (2.0 + ((x * x) * (0.5 + (x * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.95], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(x + N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999996

   1. Initial program 100.0%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

      3. difference-of-sqr-1 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 20.7%

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

   if -1.94999999999999996 < x

   1. Initial program 7.5%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Step-by-step derivation

      1. *-commutative 7.5%

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

      2. exp-lft-sqr 7.7%

         \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

      3. difference-of-sqr-1 8.5%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

      4. associate-/l* 8.5%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.7%

      \[\leadsto \sqrt{\color{blue}{2 + \left(x + \left(0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 97.7%

         \[\leadsto \sqrt{\color{blue}{\left(x + \left(0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right)\right) + 2}} \]

      2. associate-+l+ 97.7%

         \[\leadsto \sqrt{\color{blue}{x + \left(\left(0.16666666666666666 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right) + 2\right)}} \]

      3. +-commutative 97.7%

         \[\leadsto \sqrt{x + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + 0.16666666666666666 \cdot {x}^{3}\right)} + 2\right)} \]

      4. *-commutative 97.7%

         \[\leadsto \sqrt{x + \left(\left(\color{blue}{{x}^{2} \cdot 0.5} + 0.16666666666666666 \cdot {x}^{3}\right) + 2\right)} \]

      5. *-commutative 97.7%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{{x}^{3} \cdot 0.16666666666666666}\right) + 2\right)} \]

      6. unpow3 97.7%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.16666666666666666\right) + 2\right)} \]

      7. unpow2 97.7%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \left(\color{blue}{{x}^{2}} \cdot x\right) \cdot 0.16666666666666666\right) + 2\right)} \]

      8. associate-*l* 97.7%

         \[\leadsto \sqrt{x + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{{x}^{2} \cdot \left(x \cdot 0.16666666666666666\right)}\right) + 2\right)} \]

      9. distribute-lft-out 97.7%

         \[\leadsto \sqrt{x + \left(\color{blue}{{x}^{2} \cdot \left(0.5 + x \cdot 0.16666666666666666\right)} + 2\right)} \]

      10. unpow2 97.7%

          \[\leadsto \sqrt{x + \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 2\right)} \]

   6. Simplified 97.7%

      \[\leadsto \sqrt{\color{blue}{x + \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + \left(2 + \left(x \cdot x\right) \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]

Alternative 4: 72.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + 2\right) + 0.5 \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0) (sqrt 2.0) (sqrt (+ (+ x 2.0) (* 0.5 (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt(((x + 2.0) + (0.5 * (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.0d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt(((x + 2.0d0) + (0.5d0 * (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(((x + 2.0) + (0.5 * (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt(((x + 2.0) + (0.5 * (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(Float64(x + 2.0) + Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(((x + 2.0) + (0.5 * (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(N[(x + 2.0), $MachinePrecision] + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 100.0%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

      3. difference-of-sqr-1 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 20.7%

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

   if -2 < x

   1. Initial program 7.5%

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   2. Step-by-step derivation

      1. *-commutative 7.5%

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

      2. exp-lft-sqr 7.7%

         \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

      3. difference-of-sqr-1 8.5%

         \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

      4. associate-/l* 8.5%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.1%

      \[\leadsto \sqrt{\color{blue}{2 + \left(x + 0.5 \cdot {x}^{2}\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+ 97.1%

         \[\leadsto \sqrt{\color{blue}{\left(2 + x\right) + 0.5 \cdot {x}^{2}}} \]

      2. unpow2 97.1%

         \[\leadsto \sqrt{\left(2 + x\right) + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   6. Simplified 97.1%

      \[\leadsto \sqrt{\color{blue}{\left(2 + x\right) + 0.5 \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + 2\right) + 0.5 \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 5: 71.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt 2.0))

C

double code(double x) {
	return sqrt(2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(2.0d0)
end function

Java

public static double code(double x) {
	return Math.sqrt(2.0);
}

Python

def code(x):
	return math.sqrt(2.0)

Julia

function code(x)
	return sqrt(2.0)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(2.0);
end

Wolfram

code[x_] := N[Sqrt[2.0], $MachinePrecision]

Derivation

1. Initial program 36.4%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Step-by-step derivation

   1. *-commutative 36.4%

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

   2. exp-lft-sqr 36.5%

      \[\leadsto \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]

   3. difference-of-sqr-1 37.1%

      \[\leadsto \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]

   4. associate-/l* 37.1%

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

   5. *-inverses 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 71.6%

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

5. Final simplification 71.6%

   \[\leadsto \sqrt{2} \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))