sqrtexp (problem 3.4.4)

Percentage Accurate: 36.5% → 100.0%

Time: 7.4s

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: 36.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 + \frac{e^{1 + x}}{e}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ 1.0 (/ (exp (+ 1.0 x)) E))))

C

double code(double x) {
	return sqrt((1.0 + (exp((1.0 + x)) / ((double) M_E))));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 + (Math.exp((1.0 + x)) / Math.E)));
}

Python

def code(x):
	return math.sqrt((1.0 + (math.exp((1.0 + x)) / math.e)))

Julia

function code(x)
	return sqrt(Float64(1.0 + Float64(exp(Float64(1.0 + x)) / exp(1))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + (exp((1.0 + x)) / 2.71828182845904523536)));
end

Wolfram

code[x_] := N[Sqrt[N[(1.0 + N[(N[Exp[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] / E), $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 37.4%

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

   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-*r/ 37.9%

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

   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. *-un-lft-identity 100.0%

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

   2. exp-prod 100.0%

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

   3. expm1-log1p-u 66.0%

      \[\leadsto \sqrt{1 + {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)\right)}}} \]

   4. expm1-undefine 66.0%

      \[\leadsto \sqrt{1 + {\left(e^{1}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right)} - 1\right)}}} \]

   5. pow-sub 66.0%

      \[\leadsto \sqrt{1 + \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(x\right)}\right)}}{{\left(e^{1}\right)}^{1}}}} \]

6. Applied egg-rr 66.0%

   \[\leadsto \sqrt{1 + \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(x\right)}\right)}}{{\left(e^{1}\right)}^{1}}}} \]
7. Step-by-step derivation

   1. exp-1-e 66.0%

      \[\leadsto \sqrt{1 + \frac{{\color{blue}{e}}^{\left(e^{\mathsf{log1p}\left(x\right)}\right)}}{{\left(e^{1}\right)}^{1}}} \]

   2. log1p-undefine 66.0%

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

   3. rem-exp-log 100.0%

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

   4. +-commutative 100.0%

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

   5. unpow1 100.0%

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

   6. exp-1-e 100.0%

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

8. Simplified 100.0%

   \[\leadsto \sqrt{1 + \color{blue}{\frac{{e}^{\left(x + 1\right)}}{e}}} \]
9. Step-by-step derivation

   1. e-exp-1 100.0%

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

   2. pow-exp 100.0%

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

   3. *-un-lft-identity 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \sqrt{1 + \frac{e^{1 + x}}{e}} \]
12. 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 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 37.4%

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

   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-*r/ 37.9%

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

   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: 70.9% 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 37.4%

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

   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-*r/ 37.9%

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

   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 72.0%

   \[\leadsto \sqrt{\color{blue}{2}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))