cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.8%

Time: 12.6s

Alternatives: 8

Speedup: 9.7×

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: 51.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.0%

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

   1. flip-- 53.9%

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

   2. div-inv 53.9%

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

   3. metadata-eval 53.9%

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

   4. pow2 53.9%

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

3. Applied egg-rr 53.9%

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

   1. /-rgt-identity 53.9%

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

   2. associate-/r/ 53.9%

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

   3. unpow2 53.9%

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

   4. sqr-neg 53.9%

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

   5. remove-double-div 53.9%

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

   6. sqr-neg 53.9%

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

   7. 1-sub-cos 74.8%

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

   8. associate-*l/ 74.8%

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

   9. *-commutative 74.8%

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

   10. hang-0p-tan 75.0%

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

5. Simplified 75.0%

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

   1. times-frac 99.8%

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

   2. *-commutative 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. Step-by-step derivation

   1. clear-num 99.8%

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

   2. div-inv 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 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 54.0%

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

   1. flip-- 53.9%

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

   2. div-inv 53.9%

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

   3. metadata-eval 53.9%

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

   4. pow2 53.9%

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

3. Applied egg-rr 53.9%

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

   1. /-rgt-identity 53.9%

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

   2. associate-/r/ 53.9%

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

   3. unpow2 53.9%

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

   4. sqr-neg 53.9%

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

   5. remove-double-div 53.9%

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

   6. sqr-neg 53.9%

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

   7. 1-sub-cos 74.8%

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

   8. associate-*l/ 74.8%

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

   9. *-commutative 74.8%

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

   10. hang-0p-tan 75.0%

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

5. Simplified 75.0%

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

   1. times-frac 99.8%

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

   2. *-commutative 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 3: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (* (pow x -2.0) (- 1.0 (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0042d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    else
        tmp = (x ** (-2.0d0)) * (1.0d0 - cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = Math.pow(x, -2.0) * (1.0 - Math.cos(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = math.pow(x, -2.0) * (1.0 - math.cos(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = (x ^ -2.0) * (1.0 - cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.0%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. associate-*l/ 99.1%

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

      3. pow2 99.1%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{x}^{-2} \cdot \left(1 - \cos x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0042d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    else
        tmp = (1.0d0 / x) / (x / (1.0d0 - cos(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(x / Float64(1.0 - cos(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.0%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.1%

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

      1. *-un-lft-identity 99.1%

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

      2. associate-*l/ 99.1%

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

      3. pow2 99.1%

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

      4. pow-flip 99.4%

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

      5. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. sqr-pow 99.3%

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

      3. metadata-eval 99.3%

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

      4. inv-pow 99.3%

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

      5. metadata-eval 99.3%

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

      6. inv-pow 99.3%

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

      7. associate-*r* 99.3%

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

      8. div-inv 99.3%

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

      9. clear-num 99.3%

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

      10. associate-*l/ 99.3%

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

      11. div-inv 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} 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 <= 0.0042d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    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 <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	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 <= 0.0042)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $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 < 0.00419999999999999974

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.0%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 6: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} 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 <= 0.0042d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    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 <= 0.0042) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.0%

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

   if 0.00419999999999999974 < x

   1. Initial program 99.1%

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

      1. div-sub 98.9%

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

      2. pow2 98.9%

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

      3. pow-flip 99.1%

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

      4. metadata-eval 99.1%

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

      5. div-inv 99.0%

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

      6. pow2 99.0%

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

      7. pow-flip 99.4%

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

      8. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. distribute-rgt-out-- 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow-flip 99.1%

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

      5. unpow2 99.1%

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

      6. remove-double-div 99.0%

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

      7. associate-/r/ 98.9%

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

      8. associate-/l* 99.1%

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

      9. clear-num 99.4%

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

      10. remove-double-div 99.4%

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

   5. Applied egg-rr 99.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 7: 63.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9e+76) 0.5 (/ (+ (/ 1.0 x) (/ -1.0 x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 9e+76) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9d+76) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 / x) + ((-1.0d0) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9e+76) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9e+76:
		tmp = 0.5
	else:
		tmp = ((1.0 / x) + (-1.0 / x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(-1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9e+76)
		tmp = 0.5;
	else
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9e+76], 0.5, N[(N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.9999999999999995e76

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0 60.9%

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

   if 8.9999999999999995e76 < x

   1. Initial program 99.1%

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

      1. div-sub 98.9%

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

      2. pow2 98.9%

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

      3. pow-flip 99.0%

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

      4. metadata-eval 99.0%

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

      5. div-inv 99.0%

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

      6. pow2 99.0%

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

      7. pow-flip 99.4%

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

      8. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. distribute-rgt-out-- 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow-flip 99.0%

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

      5. unpow2 99.0%

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

      6. remove-double-div 99.0%

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

      7. associate-/r/ 98.9%

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

      8. associate-/l* 99.1%

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

      9. clear-num 99.4%

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

      10. remove-double-div 99.4%

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

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
   6. Step-by-step derivation

      1. div-sub 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in x around 0 56.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x}}{x}\\ \end{array} \]

Alternative 8: 51.1% 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 54.0%

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

2. Taylor expanded in x around 0 48.8%

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

3. Final simplification 48.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))