sqrtexp (problem 3.4.4)

Percentage Accurate: 36.3% → 100.0%

Time: 3.3s

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: 36.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left({\left(1 + e^{x}\right)}^{1.5}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ 1.0 (exp x)) 1.5) 0.3333333333333333))

C

double code(double x) {
	return pow(pow((1.0 + exp(x)), 1.5), 0.3333333333333333);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + exp(x)) ** 1.5d0) ** 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow((1.0 + Math.exp(x)), 1.5), 0.3333333333333333);
}

Python

def code(x):
	return math.pow(math.pow((1.0 + math.exp(x)), 1.5), 0.3333333333333333)

Julia

function code(x)
	return (Float64(1.0 + exp(x)) ^ 1.5) ^ 0.3333333333333333
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + exp(x)) ^ 1.5) ^ 0.3333333333333333;
end

Wolfram

code[x_] := N[Power[N[Power[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 36.2%

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

   1. *-commutative 36.2%

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

   2. exp-lft-sqr 36.2%

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

   3. difference-of-sqr-1 36.7%

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

   4. associate-/l* 36.7%

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

   5. *-inverses 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 99.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{1 + e^{x}} \cdot \sqrt{1 + e^{x}}\right) \cdot \sqrt{1 + e^{x}}}} \]

   2. pow1/3 100.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{1 + e^{x}} \cdot \sqrt{1 + e^{x}}\right) \cdot \sqrt{1 + e^{x}}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto {\left(\color{blue}{\left(1 + e^{x}\right)} \cdot \sqrt{1 + e^{x}}\right)}^{0.3333333333333333} \]

   4. pow1 100.0%

      \[\leadsto {\left(\color{blue}{{\left(1 + e^{x}\right)}^{1}} \cdot \sqrt{1 + e^{x}}\right)}^{0.3333333333333333} \]

   5. pow1/2 100.0%

      \[\leadsto {\left({\left(1 + e^{x}\right)}^{1} \cdot \color{blue}{{\left(1 + e^{x}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

   6. pow-prod-up 100.0%

      \[\leadsto {\color{blue}{\left({\left(1 + e^{x}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

   7. metadata-eval 100.0%

      \[\leadsto {\left({\left(1 + e^{x}\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left({\left(1 + e^{x}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

6. Final simplification 100.0%

   \[\leadsto {\left({\left(1 + e^{x}\right)}^{1.5}\right)}^{0.3333333333333333} \]

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 36.2%

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

   1. *-commutative 36.2%

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

   2. exp-lft-sqr 36.2%

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

   3. difference-of-sqr-1 36.7%

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

   4. associate-/l* 36.7%

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

   5. *-inverses 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 71.1% 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 36.2%

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

   1. *-commutative 36.2%

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

   2. exp-lft-sqr 36.2%

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

   3. difference-of-sqr-1 36.7%

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

   4. associate-/l* 36.7%

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

   5. *-inverses 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 72.1%

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

5. Final simplification 72.1%

   \[\leadsto \sqrt{2} \]

Reproduce

herbie shell --seed 2023264 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))