sqrtexp (problem 3.4.4)

Percentage Accurate: 36.1% → 99.9%

Time: 6.7s

Alternatives: 4

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: 36.1% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (+ 2.0 (expm1 x))))

C

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

Java

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

Python

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

Julia

function code(x)
	return sqrt(Float64(2.0 + expm1(x)))
end

Wolfram

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

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

   2. exp-lft-sqr 34.3%

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

   3. difference-of-sqr-1 34.8%

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

   4. associate-*r/ 34.8%

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

   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. expm1-log1p-u 99.0%

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

   2. expm1-undefine 99.0%

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

6. Applied egg-rr 99.0%

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

   1. log1p-undefine 100.0%

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

   2. rem-exp-log 100.0%

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

   3. associate-+r+ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \sqrt{\left(\color{blue}{2} + e^{x}\right) - 1} \]

   5. associate--l+ 100.0%

      \[\leadsto \sqrt{\color{blue}{2 + \left(e^{x} - 1\right)}} \]

   6. expm1-define 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \sqrt{2 + \mathsf{expm1}\left(x\right)} \]
10. Add Preprocessing

Alternative 2: 99.9% 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 33.2%

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

   1. *-commutative 33.2%

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

   2. exp-lft-sqr 34.3%

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

   3. difference-of-sqr-1 34.8%

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

   4. associate-*r/ 34.8%

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

   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. Final simplification 100.0%

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

Alternative 3: 71.5% 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 33.2%

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

   1. *-commutative 33.2%

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

   2. exp-lft-sqr 34.3%

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

   3. difference-of-sqr-1 34.8%

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

   4. associate-*r/ 34.8%

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

   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 73.2%

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

6. Final simplification 73.2%

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

Alternative 4: 4.3% accurate, 154.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x))

C

double code(double x) {
	return -x;
}

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

function code(x)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 33.2%

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

   1. *-commutative 33.2%

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

   2. exp-lft-sqr 34.3%

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

   3. difference-of-sqr-1 34.8%

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

   4. associate-*r/ 34.8%

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

   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 67.4%

   \[\leadsto \sqrt{\color{blue}{2 + x}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 67.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 + x}\right)\right)} \]

   2. expm1-undefine 66.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 + x}\right)} - 1} \]

   3. log1p-undefine 66.4%

      \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{2 + x}\right)}} - 1 \]

   4. rem-exp-log 66.4%

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

7. Applied egg-rr 66.4%

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

   1. flip-+ 66.4%

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

   2. clear-num 66.4%

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

   3. metadata-eval 66.4%

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

   4. add-sqr-sqrt 65.4%

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

   5. associate--r+ 65.4%

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

   6. metadata-eval 65.4%

      \[\leadsto \frac{1}{\frac{1 - \sqrt{2 + x}}{\color{blue}{-1} - x}} - 1 \]

9. Applied egg-rr 65.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - \sqrt{2 + x}}{-1 - x}}} - 1 \]

10. Taylor expanded in x around -inf 4.1%

    \[\leadsto \color{blue}{-1 \cdot x} \]
11. Step-by-step derivation

    1. neg-mul-1 4.1%

       \[\leadsto \color{blue}{-x} \]

12. Simplified 4.1%

    \[\leadsto \color{blue}{-x} \]

13. Final simplification 4.1%

    \[\leadsto -x \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))