sqrtexp (problem 3.4.4)

Percentage Accurate: 35.0% → 100.0%

Time: 8.4s

Alternatives: 3

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 35.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{1 + e^{x}}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (/ 1.0 (+ 1.0 (exp x))) -0.5))

C

double code(double x) {
	return pow((1.0 / (1.0 + exp(x))), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (1.0d0 + exp(x))) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow((1.0 / (1.0 + Math.exp(x))), -0.5);
}

Python

def code(x):
	return math.pow((1.0 / (1.0 + math.exp(x))), -0.5)

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + exp(x))) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (1.0 + exp(x))) ^ -0.5;
end

Wolfram

code[x_] := N[Power[N[(1.0 / N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 34.0%

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

   1. *-commutative 34.0%

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

   2. exp-lft-sqr 34.1%

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

   3. difference-of-sqr-1 34.3%

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

   4. associate-/l* 34.3%

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

   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. 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-+ 34.1%

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

   4. metadata-eval 34.1%

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

   5. exp-lft-sqr 34.0%

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

   6. *-commutative 34.0%

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

   7. expm1-udef 35.6%

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

   8. expm1-udef 99.2%

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

   9. div-inv 99.0%

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

   10. div-inv 99.2%

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

   11. clear-num 99.2%

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

   12. inv-pow 99.2%

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

   13. metadata-eval 99.2%

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

   14. pow-pow 99.2%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto {\left(\frac{1}{1 + e^{x}}\right)}^{-0.5} \]
8. 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 34.0%

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

   1. *-commutative 34.0%

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

   2. exp-lft-sqr 34.1%

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

   3. difference-of-sqr-1 34.3%

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

   4. associate-/l* 34.3%

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

   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 3: 72.2% 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 34.0%

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

   1. *-commutative 34.0%

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

   2. exp-lft-sqr 34.1%

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

   3. difference-of-sqr-1 34.3%

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

   4. associate-/l* 34.3%

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

   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 72.9%

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

6. Final simplification 72.9%

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

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))