sqrtexp (problem 3.4.4)

Percentage Accurate: 35.9% → 99.3%

Time: 6.6s

Alternatives: 8

Speedup: 309.0×

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: 35.9% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ (expm1 (* 2.0 x)) (expm1 x))))

C

double code(double x) {
	return sqrt((expm1((2.0 * x)) / expm1(x)));
}

Java

public static double code(double x) {
	return Math.sqrt((Math.expm1((2.0 * x)) / Math.expm1(x)));
}

Python

def code(x):
	return math.sqrt((math.expm1((2.0 * x)) / math.expm1(x)))

Julia

function code(x)
	return sqrt(Float64(expm1(Float64(2.0 * x)) / expm1(x)))
end

Wolfram

code[x_] := N[Sqrt[N[(N[(Exp[N[(2.0 * x), $MachinePrecision]] - 1), $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 34.3%

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

   1. expm1-define 35.1%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

   2. expm1-define 100.0%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 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 34.3%

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

   1. *-commutative 34.3%

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

   2. exp-lft-sqr 35.0%

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

   3. difference-of-sqr-1 35.5%

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

   4. associate-*r/ 35.5%

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

   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. Add Preprocessing

Alternative 3: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.6)
   1.0
   (sqrt (+ 2.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.6) {
		tmp = 1.0;
	} 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 <= (-1.6d0)) then
        tmp = 1.0d0
    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 <= -1.6) {
		tmp = 1.0;
	} 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 <= -1.6:
		tmp = 1.0
	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 <= -1.6)
		tmp = 1.0;
	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 <= -1.6)
		tmp = 1.0;
	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, -1.6], 1.0, 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 < -1.6000000000000001

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. exp-lft-sqr 100.0%

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

      5. *-commutative 100.0%

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

      6. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

      7. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      8. clear-num 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      9. sqrt-div 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

      11. clear-num 100.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

      12. expm1-undefine 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      13. *-commutative 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      14. exp-lft-sqr 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      16. expm1-undefine 100.0%

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

      17. flip-+ 100.0%

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

      18. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{1} \]

   if -1.6000000000000001 < x

   1. Initial program 3.8%

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

      1. *-commutative 3.8%

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

      2. exp-lft-sqr 4.9%

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

      3. difference-of-sqr-1 5.7%

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

      4. associate-*r/ 5.7%

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

      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 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

Alternative 4: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.35) 1.0 (sqrt (+ 2.0 (* x (+ 1.0 (* x 0.5)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = 1.0;
	} 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 <= (-1.35d0)) then
        tmp = 1.0d0
    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 <= -1.35) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.35:
		tmp = 1.0
	else:
		tmp = math.sqrt((2.0 + (x * (1.0 + (x * 0.5)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.35)
		tmp = 1.0;
	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 <= -1.35)
		tmp = 1.0;
	else
		tmp = sqrt((2.0 + (x * (1.0 + (x * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.35], 1.0, 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 < -1.3500000000000001

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. exp-lft-sqr 100.0%

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

      5. *-commutative 100.0%

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

      6. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

      7. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      8. clear-num 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      9. sqrt-div 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

      11. clear-num 100.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

      12. expm1-undefine 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      13. *-commutative 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      14. exp-lft-sqr 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      16. expm1-undefine 100.0%

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

      17. flip-+ 100.0%

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

      18. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{1} \]

   if -1.3500000000000001 < x

   1. Initial program 3.8%

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

      1. *-commutative 3.8%

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

      2. exp-lft-sqr 4.9%

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

      3. difference-of-sqr-1 5.7%

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

      4. associate-*r/ 5.7%

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

      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 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + x \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. exp-lft-sqr 100.0%

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

      5. *-commutative 100.0%

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

      6. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

      7. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      8. clear-num 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      9. sqrt-div 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

      11. clear-num 100.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

      12. expm1-undefine 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      13. *-commutative 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      14. exp-lft-sqr 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      16. expm1-undefine 100.0%

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

      17. flip-+ 100.0%

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

      18. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{1} \]

   if -1 < x

   1. Initial program 3.8%

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

      1. *-commutative 3.8%

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

      2. exp-lft-sqr 4.9%

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

      3. difference-of-sqr-1 5.7%

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

      4. associate-*r/ 5.7%

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

      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 98.8%

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

Alternative 6: 97.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -0.78) 1.0 (sqrt 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= -0.78) {
		tmp = 1.0;
	} else {
		tmp = sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.78d0)) then
        tmp = 1.0d0
    else
        tmp = sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.78) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.78:
		tmp = 1.0
	else:
		tmp = math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.78)
		tmp = 1.0;
	else
		tmp = sqrt(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.78)
		tmp = 1.0;
	else
		tmp = sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.78], 1.0, N[Sqrt[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.78000000000000003

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. exp-lft-sqr 100.0%

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

      5. *-commutative 100.0%

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

      6. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

      7. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      8. clear-num 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      9. sqrt-div 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

      11. clear-num 100.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

      12. expm1-undefine 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      13. *-commutative 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      14. exp-lft-sqr 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      16. expm1-undefine 100.0%

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

      17. flip-+ 100.0%

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

      18. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{1} \]

   if -0.78000000000000003 < x

   1. Initial program 3.8%

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

      1. *-commutative 3.8%

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

      2. exp-lft-sqr 4.9%

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

      3. difference-of-sqr-1 5.7%

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

      4. associate-*r/ 5.7%

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

      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 97.8%

      \[\leadsto \color{blue}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 48.9% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.56:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -0.56) 1.0 1.5))

C

double code(double x) {
	double tmp;
	if (x <= -0.56) {
		tmp = 1.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.56d0)) then
        tmp = 1.0d0
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.56) {
		tmp = 1.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.56:
		tmp = 1.0
	else:
		tmp = 1.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.56)
		tmp = 1.0;
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.56)
		tmp = 1.0;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.56], 1.0, 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -0.56000000000000005

   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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. exp-lft-sqr 100.0%

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

      5. *-commutative 100.0%

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

      6. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

      7. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      8. clear-num 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      9. sqrt-div 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      10. metadata-eval 100.0%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

      11. clear-num 100.0%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

      12. expm1-undefine 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      13. *-commutative 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      14. exp-lft-sqr 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      15. metadata-eval 100.0%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      16. expm1-undefine 100.0%

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

      17. flip-+ 100.0%

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

      18. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

   9. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{1} \]

   if -0.56000000000000005 < x

   1. Initial program 3.8%

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

      1. *-commutative 3.8%

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

      2. exp-lft-sqr 4.9%

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

      3. difference-of-sqr-1 5.7%

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

      4. associate-*r/ 5.7%

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

      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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. flip-+ 4.9%

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

      3. metadata-eval 4.9%

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

      4. exp-lft-sqr 3.8%

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

      5. *-commutative 3.8%

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

      6. expm1-undefine 5.1%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

      7. expm1-undefine 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      8. clear-num 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      9. sqrt-div 98.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

      10. metadata-eval 98.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

      11. clear-num 98.5%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

      12. expm1-undefine 5.3%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      13. *-commutative 5.3%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      14. exp-lft-sqr 5.1%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

      15. metadata-eval 5.1%

          \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

      16. expm1-undefine 4.9%

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

      17. flip-+ 98.5%

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

      18. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in x around 0 96.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

   9. Applied egg-rr 24.3%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 46.6% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 34.3%

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

   1. *-commutative 34.3%

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

   2. exp-lft-sqr 35.0%

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

   3. difference-of-sqr-1 35.5%

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

   4. associate-*r/ 35.5%

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

   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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. flip-+ 35.0%

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

   3. metadata-eval 35.0%

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

   4. exp-lft-sqr 34.3%

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

   5. *-commutative 34.3%

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

   6. expm1-undefine 35.1%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}} \]

   7. expm1-undefine 100.0%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

   8. clear-num 100.0%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

   9. sqrt-div 99.0%

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}}} \]

   10. metadata-eval 99.0%

       \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}} \]

   11. clear-num 99.0%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}}}}} \]

   12. expm1-undefine 35.2%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{2 \cdot x} - 1}}{\mathsf{expm1}\left(x\right)}}}} \]

   13. *-commutative 35.2%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{\color{blue}{x \cdot 2}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

   14. exp-lft-sqr 35.1%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}}} \]

   15. metadata-eval 35.1%

       \[\leadsto \frac{1}{\sqrt{\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}}} \]

   16. expm1-undefine 35.0%

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

   17. flip-+ 99.0%

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

   18. +-commutative 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around 0 72.4%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{0.5}}} \]

9. Applied egg-rr 45.5%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))