sqrtexp (problem 3.4.4)

Percentage Accurate: 35.5% → 100.0%

Time: 4.2s

Alternatives: 8

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: 35.5% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(1, \sqrt{e^{x}}\right) \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.hypot(1.0, Math.sqrt(Math.exp(x)));
}

Python

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

Julia

function code(x)
	return hypot(1.0, sqrt(exp(x)))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[1.0 ^ 2 + N[Sqrt[N[Exp[x], $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]

Derivation

1. Initial program 38.2%

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

   1. *-commutative 38.2%

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

   2. exp-lft-sqr 38.3%

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

   3. difference-of-sqr-1 38.7%

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

   4. associate-/l* 38.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. hypot-1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

6. Final simplification 100.0%

   \[\leadsto \mathsf{hypot}\left(1, \sqrt{e^{x}}\right) \]

Alternative 2: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(1, e^{x \cdot 0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot 1.0 (exp (* x 0.5))))

C

double code(double x) {
	return hypot(1.0, exp((x * 0.5)));
}

Java

public static double code(double x) {
	return Math.hypot(1.0, Math.exp((x * 0.5)));
}

Python

def code(x):
	return math.hypot(1.0, math.exp((x * 0.5)))

Julia

function code(x)
	return hypot(1.0, exp(Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = hypot(1.0, exp((x * 0.5)));
end

Wolfram

code[x_] := N[Sqrt[1.0 ^ 2 + N[Exp[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]

Derivation

1. Initial program 38.2%

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

   1. *-commutative 38.2%

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

   2. exp-lft-sqr 38.3%

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

   3. difference-of-sqr-1 38.7%

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

   4. associate-/l* 38.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. hypot-1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]
6. Step-by-step derivation

   1. pow1/2 100.0%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{{\left(e^{x}\right)}^{0.5}}\right) \]

   2. pow-exp 100.0%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{e^{x \cdot 0.5}}\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{e^{x \cdot 0.5}}\right) \]

8. Final simplification 100.0%

   \[\leadsto \mathsf{hypot}\left(1, e^{x \cdot 0.5}\right) \]

Alternative 3: 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 38.2%

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

   1. *-commutative 38.2%

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

   2. exp-lft-sqr 38.3%

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

   3. difference-of-sqr-1 38.7%

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

   4. associate-/l* 38.7%

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

   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 4: 71.9% 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 38.2%

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

   1. *-commutative 38.2%

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

   2. exp-lft-sqr 38.3%

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

   3. difference-of-sqr-1 38.7%

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

   4. associate-/l* 38.7%

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

   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 69.3%

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

5. Final simplification 69.3%

   \[\leadsto \sqrt{2} \]

Alternative 5: 15.6% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * -0.5;
	} else {
		tmp = 1.0 + (x * 0.5);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   6. Taylor expanded in x around 0 5.5%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   7. Taylor expanded in x around -inf 5.5%

      \[\leadsto \color{blue}{-0.5 \cdot x} \]
   8. Step-by-step derivation

      1. *-commutative 5.5%

         \[\leadsto \color{blue}{x \cdot -0.5} \]

   9. Simplified 5.5%

      \[\leadsto \color{blue}{x \cdot -0.5} \]

   if -1 < x

   1. Initial program 3.5%

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

      1. *-commutative 3.5%

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

      2. exp-lft-sqr 3.6%

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

      3. difference-of-sqr-1 4.2%

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

      4. associate-/l* 4.2%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   6. Taylor expanded in x around 0 97.4%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   7. Taylor expanded in x around inf 20.4%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \]

Alternative 6: 15.6% accurate, 43.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(Float64(x * -0.5) + -1.0);
	else
		tmp = 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 = (x * -0.5) + -1.0;
	else
		tmp = 1.0 + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.0], N[(N[(x * -0.5), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(x * 0.5), $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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   6. Taylor expanded in x around 0 5.5%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   7. Taylor expanded in x around -inf 5.6%

      \[\leadsto \color{blue}{-0.5 \cdot x - 1} \]

   if -2 < x

   1. Initial program 3.5%

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

      1. *-commutative 3.5%

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

      2. exp-lft-sqr 3.6%

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

      3. difference-of-sqr-1 4.2%

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

      4. associate-/l* 4.2%

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

      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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

   6. Taylor expanded in x around 0 97.4%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   7. Taylor expanded in x around inf 20.4%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;x \cdot -0.5 + -1\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \]

Alternative 7: 2.9% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

   1. *-commutative 38.2%

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

   2. exp-lft-sqr 38.3%

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

   3. difference-of-sqr-1 38.7%

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

   4. associate-/l* 38.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. hypot-1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

6. Taylor expanded in x around 0 64.4%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

7. Taylor expanded in x around inf 2.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

8. Final simplification 2.8%

   \[\leadsto x \cdot 0.5 \]

Alternative 8: 4.3% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.5))

C

double code(double x) {
	return x * -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.5d0)
end function

Java

public static double code(double x) {
	return x * -0.5;
}

Python

def code(x):
	return x * -0.5

Julia

function code(x)
	return Float64(x * -0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.5;
end

Wolfram

code[x_] := N[(x * -0.5), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

   1. *-commutative 38.2%

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

   2. exp-lft-sqr 38.3%

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

   3. difference-of-sqr-1 38.7%

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

   4. associate-/l* 38.7%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. hypot-1-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]

6. Taylor expanded in x around 0 64.4%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

7. Taylor expanded in x around -inf 4.4%

   \[\leadsto \color{blue}{-0.5 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 4.4%

      \[\leadsto \color{blue}{x \cdot -0.5} \]

9. Simplified 4.4%

   \[\leadsto \color{blue}{x \cdot -0.5} \]

10. Final simplification 4.4%

    \[\leadsto x \cdot -0.5 \]

Reproduce

herbie shell --seed 2023260 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))