cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.8%

Time: 9.0s

Alternatives: 5

Speedup: 107.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (/ (tan (* x 0.5)) x) (/ (sin x) x)))

C

double code(double x) {
	return (tan((x * 0.5)) / x) * (sin(x) / x);
}

Fortran

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

Java

public static double code(double x) {
	return (Math.tan((x * 0.5)) / x) * (Math.sin(x) / x);
}

Python

def code(x):
	return (math.tan((x * 0.5)) / x) * (math.sin(x) / x)

Julia

function code(x)
	return Float64(Float64(tan(Float64(x * 0.5)) / x) * Float64(sin(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = (tan((x * 0.5)) / x) * (sin(x) / x);
end

Wolfram

code[x_] := N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

   1. flip-- 50.0%

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

   2. div-inv 50.0%

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

   3. metadata-eval 50.0%

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

   4. 1-sub-cos 75.1%

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

   5. pow2 75.1%

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

3. Applied egg-rr 75.1%

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

   1. unpow2 75.1%

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

   2. associate-*l* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. *-rgt-identity 75.0%

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

   5. hang-0p-tan 75.2%

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

5. Simplified 75.2%

   \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 75.2%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

      \[\leadsto \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]

8. Final simplification 99.8%

   \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \]

Alternative 2: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0045)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (+ (cos x) -1.0) x) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0045d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = ((cos(x) + (-1.0d0)) / x) / -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = ((Math.cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0045:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = ((math.cos(x) + -1.0) / x) / -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0045)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(cos(x) + -1.0) / x) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0045)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = ((cos(x) + -1.0) / x) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0045], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 32.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 70.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 70.0%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 70.0%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 0.00449999999999999966 < x

   1. Initial program 97.4%

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

      1. frac-2neg 97.4%

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

      2. div-inv 97.4%

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

      3. distribute-rgt-neg-in 97.4%

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

   3. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\left(-\left(1 - \cos x\right)\right) \cdot \frac{1}{x \cdot \left(-x\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.6%

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

      2. associate-*r/ 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

      4. div-inv 99.6%

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

      5. frac-2neg 99.6%

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

      6. clear-num 97.4%

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

      7. frac-2neg 97.4%

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

      8. metadata-eval 97.4%

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

      9. frac-2neg 97.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 51.1%

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

      12. sqr-neg 51.1%

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

      13. sqrt-prod 51.1%

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

      14. add-sqr-sqrt 51.1%

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

      15. distribute-frac-neg 51.1%

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

      16. frac-2neg 51.1%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in x around inf 97.4%

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

      1. unpow2 97.4%

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

      2. associate-*r/ 97.4%

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

      3. sub-neg 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

      6. associate-/l* 97.4%

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

      7. associate-*l/ 97.4%

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

      8. metadata-eval 97.4%

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

      9. associate-/r* 97.4%

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

      10. neg-mul-1 97.4%

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

      11. distribute-rgt-neg-in 97.4%

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

      12. associate-/r* 99.5%

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

      13. associate-/l* 99.6%

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

      14. *-lft-identity 99.6%

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

      15. +-commutative 99.6%

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

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\cos x + -1}{x}}{-x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]

Alternative 3: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0045)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0045d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0045) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0045:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0045)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0045)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0045], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 32.3%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 70.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. *-commutative 70.0%

         \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

      2. unpow2 70.0%

         \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664 \]

   4. Simplified 70.0%

      \[\leadsto \color{blue}{0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664} \]

   if 0.00449999999999999966 < x

   1. Initial program 97.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 4: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\sin x}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (/ (sin x) x)))

C

double code(double x) {
	return 0.5 * (sin(x) / x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 * (Math.sin(x) / x);
}

Python

def code(x):
	return 0.5 * (math.sin(x) / x)

Julia

function code(x)
	return Float64(0.5 * Float64(sin(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (sin(x) / x);
end

Wolfram

code[x_] := N[(0.5 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

   1. flip-- 50.0%

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

   2. div-inv 50.0%

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

   3. metadata-eval 50.0%

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

   4. 1-sub-cos 75.1%

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

   5. pow2 75.1%

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

3. Applied egg-rr 75.1%

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

   1. unpow2 75.1%

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

   2. associate-*l* 75.0%

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

   3. associate-*r/ 75.0%

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

   4. *-rgt-identity 75.0%

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

   5. hang-0p-tan 75.2%

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

5. Simplified 75.2%

   \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 75.2%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

      \[\leadsto \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]

8. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{0.5} \cdot \frac{\sin x}{x} \]

9. Final simplification 53.2%

   \[\leadsto 0.5 \cdot \frac{\sin x}{x} \]

Alternative 5: 51.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]

2. Taylor expanded in x around 0 52.3%

   \[\leadsto \color{blue}{0.5} \]

3. Final simplification 52.3%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023264 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))