ENA, Section 1.4, Mentioned, A

Percentage Accurate: 53.2% → 100.0%

Time: 10.9s

Alternatives: 5

Speedup: 11.4×

Specification

\[-0.01 \leq x \land x \leq 0.01\]

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (cos x)))

C

double code(double x) {
	return 1.0 - cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

public static double code(double x) {
	return 1.0 - Math.cos(x);
}

Python

def code(x):
	return 1.0 - math.cos(x)

Julia

function code(x)
	return Float64(1.0 - cos(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - cos(x);
end

Wolfram

code[x_] := N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (cos x)))

C

double code(double x) {
	return 1.0 - cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - cos(x)
end function

Java

public static double code(double x) {
	return 1.0 - Math.cos(x);
}

Python

def code(x):
	return 1.0 - math.cos(x)

Julia

function code(x)
	return Float64(1.0 - cos(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - cos(x);
end

Wolfram

code[x_] := N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sin x) (tan (/ x 2.0))))

C

double code(double x) {
	return sin(x) * tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sin(x) * tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.sin(x) * Math.tan((x / 2.0));
}

Python

def code(x):
	return math.sin(x) * math.tan((x / 2.0))

Julia

function code(x)
	return Float64(sin(x) * tan(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sin(x) * tan((x / 2.0));
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 53.5%

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

   2. div-inv 53.5%

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

   3. metadata-eval 53.5%

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

   4. 1-sub-cos 100.0%

      \[\leadsto \color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x} \]

   5. pow2 100.0%

      \[\leadsto \color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}} \]
5. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \color{blue}{\frac{{\sin x}^{2} \cdot 1}{1 + \cos x}} \]

   2. *-rgt-identity 100.0%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2}}}{1 + \cos x} \]

   3. unpow2 100.0%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x} \]

   4. associate-*r/ 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{1 + \cos x}} \]

   5. hang-0p-tan 100.0%

      \[\leadsto \sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)} \]

7. Final simplification 100.0%

   \[\leadsto \sin x \cdot \tan \left(\frac{x}{2}\right) \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x 2.0)))

C

double code(double x) {
	return 0.5 * pow(x, 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** 2.0d0)
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, 2.0);
}

Python

def code(x):
	return 0.5 * math.pow(x, 2.0)

Julia

function code(x)
	return Float64(0.5 * (x ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ 2.0);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[1 - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.5%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]

4. Final simplification 99.5%

   \[\leadsto 0.5 \cdot {x}^{2} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.16666666666666666 + \frac{-2}{x \cdot \left(-x\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ 0.16666666666666666 (/ -2.0 (* x (- x))))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (0.16666666666666666 + (-2.0 / (x * -x)));
}

Python

def code(x):
	return 1.0 / (0.16666666666666666 + (-2.0 / (x * -x)))

Julia

function code(x)
	return Float64(1.0 / Float64(0.16666666666666666 + Float64(-2.0 / Float64(x * Float64(-x)))))
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(0.16666666666666666 + N[(-2.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 53.5%

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

   2. clear-num 53.5%

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

   3. metadata-eval 53.5%

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

   4. 1-sub-cos 99.0%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{\sin x \cdot \sin x}}} \]

   5. pow2 99.0%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{{\sin x}^{2}}}} \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + \cos x}{{\sin x}^{2}}}} \]

5. Taylor expanded in x around 0 99.0%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + 2 \cdot \frac{1}{{x}^{2}}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}} \]

   2. metadata-eval 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \frac{\color{blue}{2}}{{x}^{2}}} \]

7. Simplified 99.0%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + \frac{2}{{x}^{2}}}} \]
8. Step-by-step derivation

   1. div-inv 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{2 \cdot \frac{1}{{x}^{2}}}} \]

   2. *-commutative 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{1}{{x}^{2}} \cdot 2}} \]

   3. add-sqr-sqrt 98.6%

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

   4. associate-*l* 98.6%

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

   5. sqrt-div 98.7%

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

   6. metadata-eval 98.7%

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

   7. unpow2 98.7%

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

   8. sqrt-prod 51.3%

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

   9. add-sqr-sqrt 76.5%

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

   10. sqrt-div 76.4%

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

   11. metadata-eval 76.4%

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

   12. unpow2 76.4%

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

   13. sqrt-prod 51.3%

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

   14. add-sqr-sqrt 98.6%

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

9. Applied egg-rr 98.6%

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

    1. frac-2neg 98.6%

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

    2. metadata-eval 98.6%

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

    3. associate-*l/ 98.6%

       \[\leadsto \frac{1}{0.16666666666666666 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 2}{x}}} \]

    4. metadata-eval 98.6%

       \[\leadsto \frac{1}{0.16666666666666666 + \frac{-1}{-x} \cdot \frac{\color{blue}{2}}{x}} \]

    5. frac-times 99.0%

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

    6. metadata-eval 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 4: 98.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.16666666666666666 + \frac{\frac{2}{x}}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ 0.16666666666666666 (/ (/ 2.0 x) x))))

C

double code(double x) {
	return 1.0 / (0.16666666666666666 + ((2.0 / x) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (0.16666666666666666d0 + ((2.0d0 / x) / x))
end function

Java

public static double code(double x) {
	return 1.0 / (0.16666666666666666 + ((2.0 / x) / x));
}

Python

def code(x):
	return 1.0 / (0.16666666666666666 + ((2.0 / x) / x))

Julia

function code(x)
	return Float64(1.0 / Float64(0.16666666666666666 + Float64(Float64(2.0 / x) / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (0.16666666666666666 + ((2.0 / x) / x));
end

Wolfram

code[x_] := N[(1.0 / N[(0.16666666666666666 + N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.5%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 53.5%

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

   2. clear-num 53.5%

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

   3. metadata-eval 53.5%

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

   4. 1-sub-cos 99.0%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{\sin x \cdot \sin x}}} \]

   5. pow2 99.0%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{{\sin x}^{2}}}} \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + \cos x}{{\sin x}^{2}}}} \]

5. Taylor expanded in x around 0 99.0%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + 2 \cdot \frac{1}{{x}^{2}}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}} \]

   2. metadata-eval 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \frac{\color{blue}{2}}{{x}^{2}}} \]

7. Simplified 99.0%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + \frac{2}{{x}^{2}}}} \]
8. Step-by-step derivation

   1. div-inv 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{2 \cdot \frac{1}{{x}^{2}}}} \]

   2. *-commutative 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{1}{{x}^{2}} \cdot 2}} \]

   3. add-sqr-sqrt 98.6%

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

   4. associate-*l* 98.6%

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

   5. sqrt-div 98.7%

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

   6. metadata-eval 98.7%

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

   7. unpow2 98.7%

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

   8. sqrt-prod 51.3%

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

   9. add-sqr-sqrt 76.5%

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

   10. sqrt-div 76.4%

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

   11. metadata-eval 76.4%

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

   12. unpow2 76.4%

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

   13. sqrt-prod 51.3%

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

   14. add-sqr-sqrt 98.6%

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

9. Applied egg-rr 98.6%

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

    1. associate-*l/ 98.8%

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

    2. *-un-lft-identity 98.8%

       \[\leadsto \frac{1}{0.16666666666666666 + \frac{\color{blue}{\frac{1}{x} \cdot 2}}{x}} \]

    3. associate-*l/ 98.8%

       \[\leadsto \frac{1}{0.16666666666666666 + \frac{\color{blue}{\frac{1 \cdot 2}{x}}}{x}} \]

    4. metadata-eval 98.8%

       \[\leadsto \frac{1}{0.16666666666666666 + \frac{\frac{\color{blue}{2}}{x}}{x}} \]

11. Applied egg-rr 98.8%

    \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{\frac{2}{x}}{x}}} \]

12. Final simplification 98.8%

    \[\leadsto \frac{1}{0.16666666666666666 + \frac{\frac{2}{x}}{x}} \]
13. Add Preprocessing

Alternative 5: 4.2% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

Derivation

1. Initial program 53.5%

   \[1 - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 53.5%

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

   2. clear-num 53.5%

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

   3. metadata-eval 53.5%

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

   4. 1-sub-cos 99.0%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{\sin x \cdot \sin x}}} \]

   5. pow2 99.0%

      \[\leadsto \frac{1}{\frac{1 + \cos x}{\color{blue}{{\sin x}^{2}}}} \]

4. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + \cos x}{{\sin x}^{2}}}} \]

5. Taylor expanded in x around 0 99.0%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + 2 \cdot \frac{1}{{x}^{2}}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}} \]

   2. metadata-eval 99.0%

      \[\leadsto \frac{1}{0.16666666666666666 + \frac{\color{blue}{2}}{{x}^{2}}} \]

7. Simplified 99.0%

   \[\leadsto \frac{1}{\color{blue}{0.16666666666666666 + \frac{2}{{x}^{2}}}} \]

8. Taylor expanded in x around inf 4.1%

   \[\leadsto \color{blue}{6} \]

9. Final simplification 4.1%

   \[\leadsto 6 \]
10. Add Preprocessing

Developer target: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sin x}{1 + \cos x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (sin x) (sin x)) (+ 1.0 (cos x))))

C

double code(double x) {
	return (sin(x) * sin(x)) / (1.0 + cos(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) * sin(x)) / (1.0d0 + cos(x))
end function

Java

public static double code(double x) {
	return (Math.sin(x) * Math.sin(x)) / (1.0 + Math.cos(x));
}

Python

def code(x):
	return (math.sin(x) * math.sin(x)) / (1.0 + math.cos(x))

Julia

function code(x)
	return Float64(Float64(sin(x) * sin(x)) / Float64(1.0 + cos(x)))
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) * sin(x)) / (1.0 + cos(x));
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, A"
  :precision binary64
  :pre (and (<= -0.01 x) (<= x 0.01))

  :herbie-target
  (/ (* (sin x) (sin x)) (+ 1.0 (cos x)))

  (- 1.0 (cos x)))