cos2 (problem 3.4.1)

Percentage Accurate: 51.2% → 99.8%

Time: 11.4s

Alternatives: 9

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: 51.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. flip-- 52.5%

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

   2. div-inv 52.5%

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

   3. metadata-eval 52.5%

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

   4. 1-sub-cos 75.7%

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

   5. pow2 75.7%

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

3. Applied egg-rr 75.7%

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

   1. unpow2 75.7%

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

   2. associate-*l* 75.7%

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

   3. associate-*r/ 75.7%

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

   4. *-rgt-identity 75.7%

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

   5. hang-0p-tan 76.0%

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

5. Simplified 76.0%

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

   1. associate-/r* 76.7%

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

   2. div-inv 76.6%

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

   3. div-inv 76.6%

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

   4. metadata-eval 76.6%

      \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{1}{x} \]

7. Applied egg-rr 76.6%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot \frac{1}{x}} \]
8. Step-by-step derivation

   1. un-div-inv 76.7%

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

   2. associate-/l* 99.8%

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

   3. div-inv 99.7%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{1}{\frac{x}{\tan \left(x \cdot 0.5\right)}}}}{x} \]

   4. *-commutative 99.7%

      \[\leadsto \frac{\sin x \cdot \frac{1}{\frac{x}{\tan \color{blue}{\left(0.5 \cdot x\right)}}}}{x} \]

   5. clear-num 99.7%

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

   6. *-commutative 99.7%

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

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}}{x}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 99.7%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}\right)\right)}}{x} \]

    2. expm1-udef 26.9%

       \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}\right)} - 1}}{x} \]

    3. associate-*r/ 26.9%

       \[\leadsto \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}}\right)} - 1}{x} \]

11. Applied egg-rr 26.9%

    \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}\right)} - 1}}{x} \]
12. Step-by-step derivation

    1. expm1-def 76.7%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}\right)\right)}}{x} \]

    2. expm1-log1p 76.7%

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

    3. associate-*l/ 99.8%

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

    4. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- 1.0 (cos x)) (* x x)) 0.5)
   (* (sin x) (/ (tan (* x 0.5)) (* x x)))
   0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 x x)) < 0.5

   1. Initial program 68.4%

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

      1. flip-- 68.3%

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

      2. div-inv 68.2%

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

      3. metadata-eval 68.2%

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

      4. 1-sub-cos 98.4%

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

      5. pow2 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. unpow2 98.4%

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

      2. associate-*l* 98.4%

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

      3. associate-*r/ 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. hang-0p-tan 98.8%

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

   5. Simplified 98.8%

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

      1. expm1-log1p-u 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\right)\right)} \]

      2. expm1-udef 62.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x}\right)} - 1} \]

      3. div-inv 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot \frac{1}{x \cdot x}}\right)} - 1 \]

      4. pow2 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} - 1 \]

      5. pow-flip 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot \color{blue}{{x}^{\left(-2\right)}}\right)} - 1 \]

      6. metadata-eval 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right) \cdot {x}^{\color{blue}{-2}}\right)} - 1 \]

      7. associate-*l* 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(\tan \left(\frac{x}{2}\right) \cdot {x}^{-2}\right)}\right)} - 1 \]

      8. div-inv 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sin x \cdot \left(\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot {x}^{-2}\right)\right)} - 1 \]

      9. metadata-eval 61.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sin x \cdot \left(\tan \left(x \cdot \color{blue}{0.5}\right) \cdot {x}^{-2}\right)\right)} - 1 \]

   7. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 97.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)\right)\right)} \]

      2. expm1-log1p 97.6%

         \[\leadsto \color{blue}{\sin x \cdot \left(\tan \left(x \cdot 0.5\right) \cdot {x}^{-2}\right)} \]

      3. associate-*r* 97.5%

         \[\leadsto \color{blue}{\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{-2}} \]

      4. metadata-eval 97.5%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot {x}^{\color{blue}{\left(2 \cdot -1\right)}} \]

      5. pow-sqr 97.3%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \]

      6. unpow-1 97.3%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \]

      7. unpow-1 97.3%

         \[\leadsto \left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \]

      8. associate-*r* 98.5%

         \[\leadsto \color{blue}{\left(\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \frac{1}{x}\right) \cdot \frac{1}{x}} \]

      9. associate-*r/ 98.6%

         \[\leadsto \color{blue}{\frac{\left(\sin x \cdot \tan \left(x \cdot 0.5\right)\right) \cdot 1}{x}} \cdot \frac{1}{x} \]

      10. associate-*l/ 98.6%

          \[\leadsto \color{blue}{\left(\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot 1\right)} \cdot \frac{1}{x} \]

      11. *-rgt-identity 98.6%

          \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}} \cdot \frac{1}{x} \]

      12. associate-*r/ 98.7%

          \[\leadsto \color{blue}{\frac{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot 1}{x}} \]

      13. *-rgt-identity 98.7%

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

      14. associate-*r/ 99.6%

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

      15. associate-*r/ 99.6%

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

      16. associate-/r* 98.4%

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

      17. *-commutative 98.4%

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

   9. Simplified 98.4%

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

   if 0.5 < (/.f64 (-.f64 1 (cos.f64 x)) (*.f64 x x))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.000135) (not (<= x 0.000145)))
   (* (- 1.0 (cos x)) (pow x -2.0))
   0.5))

C

double code(double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.000145)) {
		tmp = (1.0 - cos(x)) * pow(x, -2.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.000135d0)) .or. (.not. (x <= 0.000145d0))) then
        tmp = (1.0d0 - cos(x)) * (x ** (-2.0d0))
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.000145)) {
		tmp = (1.0 - Math.cos(x)) * Math.pow(x, -2.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.000135) or not (x <= 0.000145):
		tmp = (1.0 - math.cos(x)) * math.pow(x, -2.0)
	else:
		tmp = 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.000135) || !(x <= 0.000145))
		tmp = Float64(Float64(1.0 - cos(x)) * (x ^ -2.0));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.000135) || ~((x <= 0.000145)))
		tmp = (1.0 - cos(x)) * (x ^ -2.0);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.000135], N[Not[LessEqual[x, 0.000145]], $MachinePrecision]], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000002e-4 or 1.45e-4 < x

   1. Initial program 98.4%

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

      1. clear-num 98.4%

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

      2. associate-/r/ 98.3%

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

      3. pow2 98.3%

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

      4. pow-flip 98.9%

         \[\leadsto \color{blue}{{x}^{\left(-2\right)}} \cdot \left(1 - \cos x\right) \]

      5. metadata-eval 98.9%

         \[\leadsto {x}^{\color{blue}{-2}} \cdot \left(1 - \cos x\right) \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{x}^{-2} \cdot \left(1 - \cos x\right)} \]

   if -1.35000000000000002e-4 < x < 1.45e-4

   1. Initial program 1.6%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. flip-- 52.5%

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

   2. div-inv 52.5%

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

   3. metadata-eval 52.5%

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

   4. 1-sub-cos 75.7%

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

   5. pow2 75.7%

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

3. Applied egg-rr 75.7%

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

   1. unpow2 75.7%

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

   2. associate-*l* 75.7%

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

   3. associate-*r/ 75.7%

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

   4. *-rgt-identity 75.7%

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

   5. hang-0p-tan 76.0%

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

5. Simplified 76.0%

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

   1. associate-/r* 76.7%

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

   2. div-inv 76.6%

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

   3. div-inv 76.6%

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

   4. metadata-eval 76.6%

      \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{1}{x} \]

7. Applied egg-rr 76.6%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot \frac{1}{x}} \]
8. Step-by-step derivation

   1. *-commutative 76.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x}} \]

   2. associate-/l* 99.6%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{\sin x}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \]

   3. frac-times 99.5%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin x}{x \cdot \frac{x}{\tan \left(x \cdot 0.5\right)}}} \]

   4. *-un-lft-identity 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 5: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. flip-- 52.5%

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

   2. div-inv 52.5%

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

   3. metadata-eval 52.5%

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

   4. 1-sub-cos 75.7%

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

   5. pow2 75.7%

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

3. Applied egg-rr 75.7%

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

   1. unpow2 75.7%

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

   2. associate-*l* 75.7%

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

   3. associate-*r/ 75.7%

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

   4. *-rgt-identity 75.7%

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

   5. hang-0p-tan 76.0%

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

5. Simplified 76.0%

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

   1. associate-/r* 76.7%

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

   2. div-inv 76.6%

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

   3. div-inv 76.6%

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

   4. metadata-eval 76.6%

      \[\leadsto \frac{\sin x \cdot \tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{1}{x} \]

7. Applied egg-rr 76.6%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(x \cdot 0.5\right)}{x} \cdot \frac{1}{x}} \]
8. Step-by-step derivation

   1. un-div-inv 76.7%

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

   2. associate-/l* 99.8%

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

   3. div-inv 99.7%

      \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{1}{\frac{x}{\tan \left(x \cdot 0.5\right)}}}}{x} \]

   4. *-commutative 99.7%

      \[\leadsto \frac{\sin x \cdot \frac{1}{\frac{x}{\tan \color{blue}{\left(0.5 \cdot x\right)}}}}{x} \]

   5. clear-num 99.7%

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

   6. *-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.000135) (not (<= x 0.000145)))
   (/ (- 1.0 (cos x)) (* x x))
   0.5))

C

double code(double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.000145)) {
		tmp = (1.0 - cos(x)) / (x * x);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.000135d0)) .or. (.not. (x <= 0.000145d0))) then
        tmp = (1.0d0 - cos(x)) / (x * x)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.000145)) {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.000135) or not (x <= 0.000145):
		tmp = (1.0 - math.cos(x)) / (x * x)
	else:
		tmp = 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.000135) || !(x <= 0.000145))
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.000135) || ~((x <= 0.000145)))
		tmp = (1.0 - cos(x)) / (x * x);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.000135], N[Not[LessEqual[x, 0.000145]], $MachinePrecision]], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000002e-4 or 1.45e-4 < x

   1. Initial program 98.4%

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

   if -1.35000000000000002e-4 < x < 1.45e-4

   1. Initial program 1.6%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.000135) (not (<= x 0.000145)))
   (/ (/ (- 1.0 (cos x)) x) x)
   0.5))

C

double code(double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.000145)) {
		tmp = ((1.0 - cos(x)) / x) / x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.000135d0)) .or. (.not. (x <= 0.000145d0))) then
        tmp = ((1.0d0 - cos(x)) / x) / x
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.000145)) {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.000135) or not (x <= 0.000145):
		tmp = ((1.0 - math.cos(x)) / x) / x
	else:
		tmp = 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.000135) || !(x <= 0.000145))
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.000135) || ~((x <= 0.000145)))
		tmp = ((1.0 - cos(x)) / x) / x;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.000135], N[Not[LessEqual[x, 0.000145]], $MachinePrecision]], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000002e-4 or 1.45e-4 < x

   1. Initial program 98.4%

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

      1. clear-num 98.4%

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

      2. associate-/r/ 98.3%

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

      3. pow2 98.3%

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

      4. pow-flip 98.9%

         \[\leadsto \color{blue}{{x}^{\left(-2\right)}} \cdot \left(1 - \cos x\right) \]

      5. metadata-eval 98.9%

         \[\leadsto {x}^{\color{blue}{-2}} \cdot \left(1 - \cos x\right) \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{x}^{-2} \cdot \left(1 - \cos x\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot {x}^{-2}} \]

      2. flip-- 98.6%

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

      3. metadata-eval 98.6%

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

      4. 1-sub-cos 99.1%

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

      5. unpow2 99.1%

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

      6. un-div-inv 99.0%

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

      7. metadata-eval 99.0%

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

      8. pow-flip 98.6%

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

      9. pow2 98.6%

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

      10. div-inv 98.5%

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

      11. associate-/r* 98.9%

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

      12. un-div-inv 99.0%

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

      13. unpow2 99.0%

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

      14. 1-sub-cos 98.6%

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

      15. metadata-eval 98.6%

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

      16. flip-- 98.9%

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

   5. Applied egg-rr 98.9%

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

   if -1.35000000000000002e-4 < x < 1.45e-4

   1. Initial program 1.6%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.000145\right):\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 78.3% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. frac-2neg 52.7%

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

   2. div-inv 52.6%

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

   3. distribute-rgt-neg-in 52.6%

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

3. Applied egg-rr 52.6%

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

   1. distribute-lft-neg-out 52.6%

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

   2. associate-/r* 52.8%

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

5. Simplified 52.8%

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

   1. associate-*r/ 53.6%

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

   2. add-sqr-sqrt 25.6%

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

   3. sqrt-unprod 39.0%

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

   4. sqr-neg 39.0%

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

   5. sqrt-prod 14.1%

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

   6. div-inv 14.1%

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

   7. add-sqr-sqrt 24.8%

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

   8. div-inv 24.8%

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

   9. clear-num 24.8%

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

   10. frac-2neg 24.8%

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

   11. metadata-eval 24.8%

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

   12. frac-times 24.8%

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

   13. metadata-eval 24.8%

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

   14. add-sqr-sqrt 10.7%

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

   15. sqrt-unprod 37.7%

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

   16. sqr-neg 37.7%

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

   17. sqrt-prod 27.7%

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

   18. add-sqr-sqrt 53.3%

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

7. Applied egg-rr 53.3%

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

8. Taylor expanded in x around 0 74.3%

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

9. Final simplification 74.3%

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

Alternative 9: 51.2% 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 52.7%

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

2. Taylor expanded in x around 0 49.6%

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

3. Final simplification 49.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))