sqrtexp (problem 3.4.4)

Percentage Accurate: 35.6% → 100.0%

Time: 7.4s

Alternatives: 6

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: 35.6% 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 36.1%

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

   1. *-commutative 36.1%

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

   2. exp-lft-sqr 36.9%

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

   3. difference-of-sqr-1 37.9%

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

   4. associate-/l* 38.3%

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

   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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 72.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;\sqrt{2}\\ \mathbf{elif}\;x \leq 1.66:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot \left(x \cdot 0.09375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.4)
   (sqrt 2.0)
   (if (<= x 1.66)
     (pow (+ 0.5 (* x -0.25)) -0.5)
     (* (sqrt 2.0) (+ 1.0 (* x (* x 0.09375)))))))

C

double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = sqrt(2.0);
	} else if (x <= 1.66) {
		tmp = pow((0.5 + (x * -0.25)), -0.5);
	} else {
		tmp = sqrt(2.0) * (1.0 + (x * (x * 0.09375)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.4d0)) then
        tmp = sqrt(2.0d0)
    else if (x <= 1.66d0) then
        tmp = (0.5d0 + (x * (-0.25d0))) ** (-0.5d0)
    else
        tmp = sqrt(2.0d0) * (1.0d0 + (x * (x * 0.09375d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = Math.sqrt(2.0);
	} else if (x <= 1.66) {
		tmp = Math.pow((0.5 + (x * -0.25)), -0.5);
	} else {
		tmp = Math.sqrt(2.0) * (1.0 + (x * (x * 0.09375)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.4:
		tmp = math.sqrt(2.0)
	elif x <= 1.66:
		tmp = math.pow((0.5 + (x * -0.25)), -0.5)
	else:
		tmp = math.sqrt(2.0) * (1.0 + (x * (x * 0.09375)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.4)
		tmp = sqrt(2.0);
	elseif (x <= 1.66)
		tmp = Float64(0.5 + Float64(x * -0.25)) ^ -0.5;
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 + Float64(x * Float64(x * 0.09375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.4)
		tmp = sqrt(2.0);
	elseif (x <= 1.66)
		tmp = (0.5 + (x * -0.25)) ^ -0.5;
	else
		tmp = sqrt(2.0) * (1.0 + (x * (x * 0.09375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.4], N[Sqrt[2.0], $MachinePrecision], If[LessEqual[x, 1.66], N[Power[N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 + N[(x * N[(x * 0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000004

   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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -4.4000000000000004 < x < 1.65999999999999992

   1. Initial program 5.4%

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

      1. *-commutative 5.4%

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

      2. exp-lft-sqr 6.6%

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

      3. difference-of-sqr-1 8.1%

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

      4. associate-/l* 8.1%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. +-commutative 100.0%

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

      3. flip-+ 6.6%

         \[\leadsto {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]

      4. metadata-eval 6.6%

         \[\leadsto {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]

      5. exp-lft-sqr 5.4%

         \[\leadsto {\left(\frac{\color{blue}{e^{x \cdot 2}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      6. *-commutative 5.4%

         \[\leadsto {\left(\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      7. expm1-udef 7.8%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}\right)}^{0.5} \]

      8. expm1-udef 99.9%

         \[\leadsto {\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]

      9. div-inv 99.6%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(2 \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      10. div-inv 99.9%

          \[\leadsto {\color{blue}{\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      11. clear-num 99.9%

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}\right)}}^{0.5} \]

      12. inv-pow 99.9%

          \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{-1}\right)}}^{0.5} \]

      13. metadata-eval 99.9%

          \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

      14. pow-pow 100.0%

          \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto {\color{blue}{\left(0.5 + -0.25 \cdot x\right)}}^{-0.5} \]

   if 1.65999999999999992 < x

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. exp-lft-sqr 51.9%

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

      3. difference-of-sqr-1 51.9%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. +-commutative 100.0%

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

      3. flip-+ 51.9%

         \[\leadsto {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]

      4. metadata-eval 51.9%

         \[\leadsto {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]

      5. exp-lft-sqr 51.9%

         \[\leadsto {\left(\frac{\color{blue}{e^{x \cdot 2}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      6. *-commutative 51.9%

         \[\leadsto {\left(\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      7. expm1-udef 51.9%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}\right)}^{0.5} \]

      8. expm1-udef 51.9%

         \[\leadsto {\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]

      9. div-inv 51.9%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(2 \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      10. div-inv 51.9%

          \[\leadsto {\color{blue}{\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      11. clear-num 51.9%

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}\right)}}^{0.5} \]

      12. inv-pow 51.9%

          \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{-1}\right)}}^{0.5} \]

      13. metadata-eval 51.9%

          \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

      14. pow-pow 51.9%

          \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto {\color{blue}{\left(0.5 + -0.25 \cdot x\right)}}^{-0.5} \]

   8. Taylor expanded in x around 0 11.9%

      \[\leadsto \color{blue}{\sqrt{2} + \left(0.09375 \cdot \left({x}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \left(x \cdot \sqrt{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-rgt-identity 11.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot 1} + \left(0.09375 \cdot \left({x}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \left(x \cdot \sqrt{2}\right)\right) \]

      2. associate-*r* 11.9%

         \[\leadsto \sqrt{2} \cdot 1 + \left(\color{blue}{\left(0.09375 \cdot {x}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \left(x \cdot \sqrt{2}\right)\right) \]

      3. associate-*r* 11.9%

         \[\leadsto \sqrt{2} \cdot 1 + \left(\left(0.09375 \cdot {x}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(0.25 \cdot x\right) \cdot \sqrt{2}}\right) \]

      4. *-commutative 11.9%

         \[\leadsto \sqrt{2} \cdot 1 + \left(\left(0.09375 \cdot {x}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot 0.25\right)} \cdot \sqrt{2}\right) \]

      5. distribute-rgt-out 11.9%

         \[\leadsto \sqrt{2} \cdot 1 + \color{blue}{\sqrt{2} \cdot \left(0.09375 \cdot {x}^{2} + x \cdot 0.25\right)} \]

      6. distribute-lft-out 11.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \left(1 + \left(0.09375 \cdot {x}^{2} + x \cdot 0.25\right)\right)} \]

      7. +-commutative 11.9%

         \[\leadsto \sqrt{2} \cdot \left(1 + \color{blue}{\left(x \cdot 0.25 + 0.09375 \cdot {x}^{2}\right)}\right) \]

      8. *-commutative 11.9%

         \[\leadsto \sqrt{2} \cdot \left(1 + \left(\color{blue}{0.25 \cdot x} + 0.09375 \cdot {x}^{2}\right)\right) \]

      9. unpow2 11.9%

         \[\leadsto \sqrt{2} \cdot \left(1 + \left(0.25 \cdot x + 0.09375 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. associate-*r* 11.9%

          \[\leadsto \sqrt{2} \cdot \left(1 + \left(0.25 \cdot x + \color{blue}{\left(0.09375 \cdot x\right) \cdot x}\right)\right) \]

      11. distribute-rgt-out 11.9%

          \[\leadsto \sqrt{2} \cdot \left(1 + \color{blue}{x \cdot \left(0.25 + 0.09375 \cdot x\right)}\right) \]

      12. *-commutative 11.9%

          \[\leadsto \sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + \color{blue}{x \cdot 0.09375}\right)\right) \]

   10. Simplified 11.9%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + x \cdot 0.09375\right)\right)} \]

   11. Taylor expanded in x around inf 11.7%

       \[\leadsto \sqrt{2} \cdot \left(1 + x \cdot \color{blue}{\left(0.09375 \cdot x\right)}\right) \]
   12. Step-by-step derivation

       1. *-commutative 11.7%

          \[\leadsto \sqrt{2} \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.09375\right)}\right) \]

   13. Simplified 11.7%

       \[\leadsto \sqrt{2} \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.09375\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;\sqrt{2}\\ \mathbf{elif}\;x \leq 1.66:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot \left(x \cdot 0.09375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;\sqrt{2}\\ \mathbf{elif}\;x \leq 1.46:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.4)
   (sqrt 2.0)
   (if (<= x 1.46)
     (pow (+ 0.5 (* x -0.25)) -0.5)
     (* (sqrt 2.0) (+ 1.0 (* x 0.25))))))

C

double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = sqrt(2.0);
	} else if (x <= 1.46) {
		tmp = pow((0.5 + (x * -0.25)), -0.5);
	} else {
		tmp = sqrt(2.0) * (1.0 + (x * 0.25));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.4d0)) then
        tmp = sqrt(2.0d0)
    else if (x <= 1.46d0) then
        tmp = (0.5d0 + (x * (-0.25d0))) ** (-0.5d0)
    else
        tmp = sqrt(2.0d0) * (1.0d0 + (x * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = Math.sqrt(2.0);
	} else if (x <= 1.46) {
		tmp = Math.pow((0.5 + (x * -0.25)), -0.5);
	} else {
		tmp = Math.sqrt(2.0) * (1.0 + (x * 0.25));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.4:
		tmp = math.sqrt(2.0)
	elif x <= 1.46:
		tmp = math.pow((0.5 + (x * -0.25)), -0.5)
	else:
		tmp = math.sqrt(2.0) * (1.0 + (x * 0.25))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.4)
		tmp = sqrt(2.0);
	elseif (x <= 1.46)
		tmp = Float64(0.5 + Float64(x * -0.25)) ^ -0.5;
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 + Float64(x * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.4)
		tmp = sqrt(2.0);
	elseif (x <= 1.46)
		tmp = (0.5 + (x * -0.25)) ^ -0.5;
	else
		tmp = sqrt(2.0) * (1.0 + (x * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.4], N[Sqrt[2.0], $MachinePrecision], If[LessEqual[x, 1.46], N[Power[N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 + N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000004

   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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -4.4000000000000004 < x < 1.46

   1. Initial program 5.4%

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

      1. *-commutative 5.4%

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

      2. exp-lft-sqr 6.6%

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

      3. difference-of-sqr-1 8.1%

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

      4. associate-/l* 8.1%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. +-commutative 100.0%

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

      3. flip-+ 6.6%

         \[\leadsto {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]

      4. metadata-eval 6.6%

         \[\leadsto {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]

      5. exp-lft-sqr 5.4%

         \[\leadsto {\left(\frac{\color{blue}{e^{x \cdot 2}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      6. *-commutative 5.4%

         \[\leadsto {\left(\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      7. expm1-udef 7.8%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}\right)}^{0.5} \]

      8. expm1-udef 99.9%

         \[\leadsto {\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]

      9. div-inv 99.6%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(2 \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      10. div-inv 99.9%

          \[\leadsto {\color{blue}{\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      11. clear-num 99.9%

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}\right)}}^{0.5} \]

      12. inv-pow 99.9%

          \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{-1}\right)}}^{0.5} \]

      13. metadata-eval 99.9%

          \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

      14. pow-pow 100.0%

          \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto {\color{blue}{\left(0.5 + -0.25 \cdot x\right)}}^{-0.5} \]

   if 1.46 < x

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. exp-lft-sqr 51.9%

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

      3. difference-of-sqr-1 51.9%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. +-commutative 100.0%

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

      3. flip-+ 51.9%

         \[\leadsto {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]

      4. metadata-eval 51.9%

         \[\leadsto {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]

      5. exp-lft-sqr 51.9%

         \[\leadsto {\left(\frac{\color{blue}{e^{x \cdot 2}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      6. *-commutative 51.9%

         \[\leadsto {\left(\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      7. expm1-udef 51.9%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}\right)}^{0.5} \]

      8. expm1-udef 51.9%

         \[\leadsto {\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]

      9. div-inv 51.9%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(2 \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      10. div-inv 51.9%

          \[\leadsto {\color{blue}{\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      11. clear-num 51.9%

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}\right)}}^{0.5} \]

      12. inv-pow 51.9%

          \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{-1}\right)}}^{0.5} \]

      13. metadata-eval 51.9%

          \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

      14. pow-pow 51.9%

          \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]

   7. Taylor expanded in x around 0 11.2%

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

      1. associate-*r* 11.2%

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

      2. distribute-rgt1-in 11.2%

         \[\leadsto \color{blue}{\left(0.25 \cdot x + 1\right) \cdot \sqrt{2}} \]

      3. *-commutative 11.2%

         \[\leadsto \left(\color{blue}{x \cdot 0.25} + 1\right) \cdot \sqrt{2} \]

   9. Simplified 11.2%

      \[\leadsto \color{blue}{\left(x \cdot 0.25 + 1\right) \cdot \sqrt{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;\sqrt{2}\\ \mathbf{elif}\;x \leq 1.46:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot 0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;\sqrt{2}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.4)
   (sqrt 2.0)
   (if (<= x 1.55) (pow (+ 0.5 (* x -0.25)) -0.5) (sqrt (+ x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = sqrt(2.0);
	} else if (x <= 1.55) {
		tmp = pow((0.5 + (x * -0.25)), -0.5);
	} else {
		tmp = sqrt((x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.4d0)) then
        tmp = sqrt(2.0d0)
    else if (x <= 1.55d0) then
        tmp = (0.5d0 + (x * (-0.25d0))) ** (-0.5d0)
    else
        tmp = sqrt((x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = Math.sqrt(2.0);
	} else if (x <= 1.55) {
		tmp = Math.pow((0.5 + (x * -0.25)), -0.5);
	} else {
		tmp = Math.sqrt((x + 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.4:
		tmp = math.sqrt(2.0)
	elif x <= 1.55:
		tmp = math.pow((0.5 + (x * -0.25)), -0.5)
	else:
		tmp = math.sqrt((x + 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.4)
		tmp = sqrt(2.0);
	elseif (x <= 1.55)
		tmp = Float64(0.5 + Float64(x * -0.25)) ^ -0.5;
	else
		tmp = sqrt(Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.4)
		tmp = sqrt(2.0);
	elseif (x <= 1.55)
		tmp = (0.5 + (x * -0.25)) ^ -0.5;
	else
		tmp = sqrt((x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.4], N[Sqrt[2.0], $MachinePrecision], If[LessEqual[x, 1.55], N[Power[N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000004

   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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -4.4000000000000004 < x < 1.55000000000000004

   1. Initial program 5.4%

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

      1. *-commutative 5.4%

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

      2. exp-lft-sqr 6.6%

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

      3. difference-of-sqr-1 8.1%

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

      4. associate-/l* 8.1%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. +-commutative 100.0%

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

      3. flip-+ 6.6%

         \[\leadsto {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]

      4. metadata-eval 6.6%

         \[\leadsto {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]

      5. exp-lft-sqr 5.4%

         \[\leadsto {\left(\frac{\color{blue}{e^{x \cdot 2}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      6. *-commutative 5.4%

         \[\leadsto {\left(\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      7. expm1-udef 7.8%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}\right)}^{0.5} \]

      8. expm1-udef 99.9%

         \[\leadsto {\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]

      9. div-inv 99.6%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(2 \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      10. div-inv 99.9%

          \[\leadsto {\color{blue}{\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      11. clear-num 99.9%

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}\right)}}^{0.5} \]

      12. inv-pow 99.9%

          \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{-1}\right)}}^{0.5} \]

      13. metadata-eval 99.9%

          \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

      14. pow-pow 100.0%

          \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto {\color{blue}{\left(0.5 + -0.25 \cdot x\right)}}^{-0.5} \]

   if 1.55000000000000004 < x

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

      2. exp-lft-sqr 51.9%

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

      3. difference-of-sqr-1 51.9%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 11.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;\sqrt{2}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + x \cdot 0.09375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.7)
   (sqrt 2.0)
   (* (sqrt 2.0) (+ 1.0 (* x (+ 0.25 (* x 0.09375)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * (1.0 + (x * (0.25 + (x * 0.09375))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.7d0)) then
        tmp = sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * (1.0d0 + (x * (0.25d0 + (x * 0.09375d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.7) {
		tmp = Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * (1.0 + (x * (0.25 + (x * 0.09375))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.7:
		tmp = math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * (1.0 + (x * (0.25 + (x * 0.09375))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.7)
		tmp = sqrt(2.0);
	else
		tmp = Float64(sqrt(2.0) * Float64(1.0 + Float64(x * Float64(0.25 + Float64(x * 0.09375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.7)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(2.0) * (1.0 + (x * (0.25 + (x * 0.09375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.7], N[Sqrt[2.0], $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(1.0 + N[(x * N[(0.25 + N[(x * 0.09375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002

   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. Add Preprocessing

   5. Taylor expanded in x around 0 20.7%

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

   if -2.7000000000000002 < x

   1. Initial program 6.0%

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

      1. *-commutative 6.0%

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

      2. exp-lft-sqr 7.2%

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

      3. difference-of-sqr-1 8.6%

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

      4. associate-/l* 9.2%

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

      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. Add Preprocessing
   5. Step-by-step derivation

      1. pow1/2 100.0%

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

      2. +-commutative 100.0%

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

      3. flip-+ 7.2%

         \[\leadsto {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]

      4. metadata-eval 7.2%

         \[\leadsto {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]

      5. exp-lft-sqr 6.0%

         \[\leadsto {\left(\frac{\color{blue}{e^{x \cdot 2}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      6. *-commutative 6.0%

         \[\leadsto {\left(\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}\right)}^{0.5} \]

      7. expm1-udef 8.3%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{expm1}\left(2 \cdot x\right)}}{e^{x} - 1}\right)}^{0.5} \]

      8. expm1-udef 99.4%

         \[\leadsto {\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]

      9. div-inv 99.1%

         \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(2 \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      10. div-inv 99.4%

          \[\leadsto {\color{blue}{\left(\frac{\mathsf{expm1}\left(2 \cdot x\right)}{\mathsf{expm1}\left(x\right)}\right)}}^{0.5} \]

      11. clear-num 99.4%

          \[\leadsto {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}}\right)}}^{0.5} \]

      12. inv-pow 99.4%

          \[\leadsto {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{-1}\right)}}^{0.5} \]

      13. metadata-eval 99.4%

          \[\leadsto {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]

      14. pow-pow 99.4%

          \[\leadsto \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(2 \cdot x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]

   7. Taylor expanded in x around 0 98.4%

      \[\leadsto {\color{blue}{\left(0.5 + -0.25 \cdot x\right)}}^{-0.5} \]

   8. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{\sqrt{2} + \left(0.09375 \cdot \left({x}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \left(x \cdot \sqrt{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-rgt-identity 98.5%

         \[\leadsto \color{blue}{\sqrt{2} \cdot 1} + \left(0.09375 \cdot \left({x}^{2} \cdot \sqrt{2}\right) + 0.25 \cdot \left(x \cdot \sqrt{2}\right)\right) \]

      2. associate-*r* 98.5%

         \[\leadsto \sqrt{2} \cdot 1 + \left(\color{blue}{\left(0.09375 \cdot {x}^{2}\right) \cdot \sqrt{2}} + 0.25 \cdot \left(x \cdot \sqrt{2}\right)\right) \]

      3. associate-*r* 98.5%

         \[\leadsto \sqrt{2} \cdot 1 + \left(\left(0.09375 \cdot {x}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(0.25 \cdot x\right) \cdot \sqrt{2}}\right) \]

      4. *-commutative 98.5%

         \[\leadsto \sqrt{2} \cdot 1 + \left(\left(0.09375 \cdot {x}^{2}\right) \cdot \sqrt{2} + \color{blue}{\left(x \cdot 0.25\right)} \cdot \sqrt{2}\right) \]

      5. distribute-rgt-out 98.5%

         \[\leadsto \sqrt{2} \cdot 1 + \color{blue}{\sqrt{2} \cdot \left(0.09375 \cdot {x}^{2} + x \cdot 0.25\right)} \]

      6. distribute-lft-out 98.5%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \left(1 + \left(0.09375 \cdot {x}^{2} + x \cdot 0.25\right)\right)} \]

      7. +-commutative 98.5%

         \[\leadsto \sqrt{2} \cdot \left(1 + \color{blue}{\left(x \cdot 0.25 + 0.09375 \cdot {x}^{2}\right)}\right) \]

      8. *-commutative 98.5%

         \[\leadsto \sqrt{2} \cdot \left(1 + \left(\color{blue}{0.25 \cdot x} + 0.09375 \cdot {x}^{2}\right)\right) \]

      9. unpow2 98.5%

         \[\leadsto \sqrt{2} \cdot \left(1 + \left(0.25 \cdot x + 0.09375 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. associate-*r* 98.5%

          \[\leadsto \sqrt{2} \cdot \left(1 + \left(0.25 \cdot x + \color{blue}{\left(0.09375 \cdot x\right) \cdot x}\right)\right) \]

      11. distribute-rgt-out 98.5%

          \[\leadsto \sqrt{2} \cdot \left(1 + \color{blue}{x \cdot \left(0.25 + 0.09375 \cdot x\right)}\right) \]

      12. *-commutative 98.5%

          \[\leadsto \sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + \color{blue}{x \cdot 0.09375}\right)\right) \]

   10. Simplified 98.5%

       \[\leadsto \color{blue}{\sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + x \cdot 0.09375\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(1 + x \cdot \left(0.25 + x \cdot 0.09375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 36.1%

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

   1. *-commutative 36.1%

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

   2. exp-lft-sqr 36.9%

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

   3. difference-of-sqr-1 37.9%

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

   4. associate-/l* 38.3%

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

   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. Add Preprocessing

5. Taylor expanded in x around 0 72.0%

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

6. Final simplification 72.0%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))