cos2 (problem 3.4.1)

Percentage Accurate: 50.2% → 99.8%

Time: 9.9s

Alternatives: 8

Speedup: 34.9×

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: 50.2% 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

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

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(sin(x) / Float64(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[Sin[x], $MachinePrecision] / N[(x / N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 51.8%

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

   1. flip-- 51.6%

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

   2. div-inv 51.6%

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

   3. metadata-eval 51.6%

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

   4. 1-sub-cos 76.8%

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

   5. pow2 76.8%

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

3. Applied egg-rr 76.8%

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

   1. associate-*r/ 76.8%

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

   2. *-rgt-identity 76.8%

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

5. Simplified 76.8%

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

6. Taylor expanded in x around inf 77.4%

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

   1. +-commutative 77.4%

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

   2. unpow2 77.4%

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

   3. associate-/r* 76.8%

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

   4. +-commutative 76.8%

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

   5. unpow2 76.8%

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

   6. associate-*r/ 76.8%

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

   7. associate-*l/ 77.4%

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

   8. hang-0p-tan 77.6%

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

8. Simplified 77.6%

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

   1. associate-*l/ 77.0%

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

   2. associate-/r* 78.0%

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

   3. associate-/l* 99.9%

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

   4. div-inv 99.9%

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

   5. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		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 <= 0.0052d0) 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 <= 0.0052) {
		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 <= 0.0052:
		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 <= 0.0052)
		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 <= 0.0052)
		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, 0.0052], 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 < 0.0051999999999999998

   1. Initial program 33.8%

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

   2. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. unpow2 67.9%

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

   4. Simplified 67.9%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 3: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		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 <= 0.0052d0) 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 <= 0.0052) {
		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 <= 0.0052:
		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 <= 0.0052)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	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 <= 0.0052)
		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, 0.0052], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $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 < 0.0051999999999999998

   1. Initial program 33.8%

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

   2. Taylor expanded in x around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. unpow2 67.9%

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

   4. Simplified 67.9%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. sqrt-div 98.4%

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

      4. sqrt-prod 99.1%

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

      5. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. unpow2 99.1%

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

      2. frac-times 98.5%

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

      3. add-sqr-sqrt 98.7%

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

      4. associate-/r* 99.4%

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

   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 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 4: 65.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \frac{2}{x \cdot x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.05)
   (+ 0.5 (* (* x x) -0.041666666666666664))
   (/ 1.0 (* (* x x) (+ 0.16666666666666666 (/ 2.0 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.05) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 1.0 / ((x * x) * (0.16666666666666666 + (2.0 / (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.05d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = 1.0d0 / ((x * x) * (0.16666666666666666d0 + (2.0d0 / (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.05) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 1.0 / ((x * x) * (0.16666666666666666 + (2.0 / (x * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.05:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = 1.0 / ((x * x) * (0.16666666666666666 + (2.0 / (x * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.05)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(1.0 / Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(2.0 / Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.05)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = 1.0 / ((x * x) * (0.16666666666666666 + (2.0 / (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.05], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999998

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 2.0499999999999998 < x

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. sqrt-div 98.5%

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

      4. sqrt-prod 99.1%

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

      5. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around 0 64.7%

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

      1. unpow2 64.7%

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

      2. associate-*r/ 64.7%

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

      3. metadata-eval 64.7%

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

   8. Simplified 64.7%

      \[\leadsto \frac{1}{\color{blue}{\left(0.16666666666666666 + \frac{2}{x \cdot x}\right)} \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + \frac{2}{x \cdot x}\right)}\\ \end{array} \]

Alternative 5: 65.1% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.25:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.25) (+ 0.5 (* (* x x) -0.041666666666666664)) (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.25) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.25d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.25) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.25:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.25)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.25)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.25], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.25

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

   4. Simplified 67.8%

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

   if 3.25 < x

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. sqrt-div 98.5%

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

      4. sqrt-prod 99.1%

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

      5. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around 0 64.7%

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

      1. unpow2 64.7%

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

      2. associate-*r/ 64.7%

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

      3. metadata-eval 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in x around inf 64.7%

      \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 64.7%

          \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   11. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.25:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]

Alternative 6: 65.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.5) 0.5 (/ 6.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = 0.5d0
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = 0.5
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = 0.5;
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], 0.5, N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 34.1%

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

   2. Taylor expanded in x around 0 68.3%

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

   if 3.5 < x

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. sqrt-div 98.5%

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

      4. sqrt-prod 99.1%

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

      5. add-sqr-sqrt 99.2%

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

   3. Applied egg-rr 99.2%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around 0 64.7%

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

      1. unpow2 64.7%

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

      2. associate-*r/ 64.7%

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

      3. metadata-eval 64.7%

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

   8. Simplified 64.7%

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

   9. Taylor expanded in x around inf 64.7%

      \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 64.7%

          \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   11. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]

Alternative 7: 64.3% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 9.5e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9.5d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.5e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.5e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e76

   1. Initial program 38.7%

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

   2. Taylor expanded in x around 0 64.0%

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

   if 9.5000000000000003e76 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in x around 0 74.3%

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

   3. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 27.0% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 51.8%

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

2. Taylor expanded in x around 0 28.3%

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

3. Taylor expanded in x around 0 29.0%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 29.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))