sqrtexp (problem 3.4.4)

Percentage Accurate: 35.0% → 100.0%

Time: 7.4s

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: 35.0% 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 38.9%

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

   1. *-commutative 38.9%

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

   2. exp-lft-sqr 39.7%

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

   3. difference-of-sqr-1 41.0%

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

   4. associate-*r/ 41.0%

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

   5. *-inverses 100.0%

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

   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. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \sqrt{1 + e^{x}} \]
6. Add Preprocessing

Alternative 2: 73.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   (sqrt 2.0)
   (sqrt (+ 2.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.9d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.9:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.9)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((2.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(2.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999

   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-*r/ 100.0%

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

      5. *-inverses 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -1.8999999999999999 < x

   1. Initial program 8.5%

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

      1. *-commutative 8.5%

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

      2. exp-lft-sqr 9.7%

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

      3. difference-of-sqr-1 11.7%

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

      4. associate-*r/ 11.7%

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

      5. *-inverses 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 99.0%

         \[\leadsto \sqrt{2 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)} \]

   7. Simplified 99.0%

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.0) (sqrt 2.0) (sqrt (+ 2.0 (* x (+ 1.0 (* x 0.5)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.0) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	}
	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((2.0d0 + (x * (1.0d0 + (x * 0.5d0)))))
    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((2.0 + (x * (1.0 + (x * 0.5)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.0:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(2.0 + Float64(x * Float64(1.0 + Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.0)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(2.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $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-*r/ 100.0%

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

      5. *-inverses 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -2 < x

   1. Initial program 8.5%

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

      1. *-commutative 8.5%

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

      2. exp-lft-sqr 9.7%

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

      3. difference-of-sqr-1 11.7%

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

      4. associate-*r/ 11.7%

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

      5. *-inverses 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 98.6%

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + 0.5 \cdot x\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \sqrt{2 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)} \]

   7. Simplified 98.6%

      \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + x \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.1) (sqrt 2.0) (sqrt (+ x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt((x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt((x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((x + 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt((x + 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = sqrt(2.0);
	else
		tmp = sqrt((x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001

   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-*r/ 100.0%

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

      5. *-inverses 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -1.1000000000000001 < x

   1. Initial program 8.5%

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

      1. *-commutative 8.5%

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

      2. exp-lft-sqr 9.7%

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

      3. difference-of-sqr-1 11.7%

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

      4. associate-*r/ 11.7%

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

      5. *-inverses 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 97.6%

      \[\leadsto \sqrt{\color{blue}{2 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.3% 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 38.9%

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

   1. *-commutative 38.9%

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

   2. exp-lft-sqr 39.7%

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

   3. difference-of-sqr-1 41.0%

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

   4. associate-*r/ 41.0%

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

   5. *-inverses 100.0%

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

   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. Add Preprocessing

5. Taylor expanded in x around 0 70.4%

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

6. Final simplification 70.4%

   \[\leadsto \sqrt{2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))