sqrtexp (problem 3.4.4)

Percentage Accurate: 36.2% → 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: 36.2% 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 N/A

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

   2. exp-lft-sqr N/A

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

   3. difference-of-sqr-1 N/A

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

   4. associate-*r/ N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(e^{x} + 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + e^{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right) \]

   10. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 72.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \sqrt{2 + x \cdot \frac{1}{1 + t\_0 \cdot \left(t\_0 + -1\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ 0.5 (* x 0.16666666666666666)))))
   (sqrt (+ 2.0 (* x (/ 1.0 (+ 1.0 (* t_0 (+ t_0 -1.0)))))))))

C

double code(double x) {
	double t_0 = x * (0.5 + (x * 0.16666666666666666));
	return sqrt((2.0 + (x * (1.0 / (1.0 + (t_0 * (t_0 + -1.0)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (0.5d0 + (x * 0.16666666666666666d0))
    code = sqrt((2.0d0 + (x * (1.0d0 / (1.0d0 + (t_0 * (t_0 + (-1.0d0))))))))
end function

Java

public static double code(double x) {
	double t_0 = x * (0.5 + (x * 0.16666666666666666));
	return Math.sqrt((2.0 + (x * (1.0 / (1.0 + (t_0 * (t_0 + -1.0)))))));
}

Python

def code(x):
	t_0 = x * (0.5 + (x * 0.16666666666666666))
	return math.sqrt((2.0 + (x * (1.0 / (1.0 + (t_0 * (t_0 + -1.0)))))))

Julia

function code(x)
	t_0 = Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))
	return sqrt(Float64(2.0 + Float64(x * Float64(1.0 / Float64(1.0 + Float64(t_0 * Float64(t_0 + -1.0)))))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (0.5 + (x * 0.16666666666666666));
	tmp = sqrt((2.0 + (x * (1.0 / (1.0 + (t_0 * (t_0 + -1.0)))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(2.0 + N[(x * N[(1.0 / N[(1.0 + N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 N/A

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

   2. exp-lft-sqr N/A

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

   3. difference-of-sqr-1 N/A

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

   4. associate-*r/ N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(e^{x} + 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + e^{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right) \]

   10. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 64.4%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

7. Simplified 64.4%

   \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) \cdot x\right)\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{{1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{\left({1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)}^{3}\right) \cdot x}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\left({1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)}^{3}\right) \cdot x\right) \cdot \frac{1}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\left({1}^{3} + {\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)}^{3}\right) \cdot x\right), \left(\frac{1}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right) - 1 \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\right)\right)}\right)\right)\right)\right) \]

9. Applied egg-rr 64.4%

   \[\leadsto \sqrt{2 + \color{blue}{\left(\left(1 + \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(\left(0.5 + x \cdot 0.16666666666666666\right) \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)\right) \cdot x\right) \cdot \frac{1}{1 + \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) - 1\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right), 1\right)\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

12. Simplified 71.2%

    \[\leadsto \sqrt{2 + \color{blue}{x} \cdot \frac{1}{1 + \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) - 1\right)}} \]

13. Final simplification 71.2%

    \[\leadsto \sqrt{2 + x \cdot \frac{1}{1 + \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right) + -1\right)}} \]
14. Add Preprocessing

Alternative 3: 71.2% 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 36.1%

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

   1. *-commutative N/A

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

   2. exp-lft-sqr N/A

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

   3. difference-of-sqr-1 N/A

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

   4. associate-*r/ N/A

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

   5. *-inverses N/A

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

   6. *-rgt-identity N/A

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

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(e^{x} + 1\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 + e^{x}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(e^{x}\right)\right)\right) \]

   10. exp-lowering-exp.f64 100.0%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(x\right)\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. sqrt-lowering-sqrt.f64 70.4%

      \[\leadsto \mathsf{sqrt.f64}\left(2\right) \]

7. Simplified 70.4%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))