cos2 (problem 3.4.1)

Percentage Accurate: 49.8% → 99.8%

Time: 9.5s

Alternatives: 7

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: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. flip-- 50.1%

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

   2. div-inv 50.1%

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

   3. metadata-eval 50.1%

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

   4. 1-sub-cos 73.8%

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

   5. pow2 73.8%

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

3. Applied egg-rr 73.8%

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

   1. unpow2 73.8%

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

   2. associate-*l* 73.8%

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

   3. associate-*r/ 73.8%

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

   4. *-rgt-identity 73.8%

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

   5. hang-0p-tan 74.1%

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

5. Simplified 74.1%

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

   1. *-commutative 74.1%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (+
    0.5
    (+ (* 0.001388888888888889 (pow x 4.0)) (* x (* x -0.041666666666666664))))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = 0.5 + ((0.001388888888888889 * pow(x, 4.0)) + (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 <= 0.033d0) then
        tmp = 0.5d0 + ((0.001388888888888889d0 * (x ** 4.0d0)) + (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 <= 0.033) {
		tmp = 0.5 + ((0.001388888888888889 * Math.pow(x, 4.0)) + (x * (x * -0.041666666666666664)));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.033:
		tmp = 0.5 + ((0.001388888888888889 * math.pow(x, 4.0)) + (x * (x * -0.041666666666666664)))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = Float64(0.5 + Float64(Float64(0.001388888888888889 * (x ^ 4.0)) + Float64(x * Float64(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 <= 0.033)
		tmp = 0.5 + ((0.001388888888888889 * (x ^ 4.0)) + (x * (x * -0.041666666666666664)));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(0.5 + N[(N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * -0.041666666666666664), $MachinePrecision]), $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 < 0.033000000000000002

   1. Initial program 36.2%

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

   2. Taylor expanded in x around 0 65.8%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 65.8%

         \[\leadsto 0.5 + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + -0.041666666666666664 \cdot {x}^{2}\right)} \]

      2. fma-def 65.8%

         \[\leadsto 0.5 + \color{blue}{\mathsf{fma}\left(0.001388888888888889, {x}^{4}, -0.041666666666666664 \cdot {x}^{2}\right)} \]

      3. *-commutative 65.8%

         \[\leadsto 0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, \color{blue}{{x}^{2} \cdot -0.041666666666666664}\right) \]

      4. unpow2 65.8%

         \[\leadsto 0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, \color{blue}{\left(x \cdot x\right)} \cdot -0.041666666666666664\right) \]

      5. associate-*l* 65.8%

         \[\leadsto 0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, \color{blue}{x \cdot \left(x \cdot -0.041666666666666664\right)}\right) \]

   4. Simplified 65.8%

      \[\leadsto \color{blue}{0.5 + \mathsf{fma}\left(0.001388888888888889, {x}^{4}, x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]
   5. Step-by-step derivation

      1. fma-udef 65.8%

         \[\leadsto 0.5 + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]

   6. Applied egg-rr 65.8%

      \[\leadsto 0.5 + \color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)} \]

   if 0.033000000000000002 < x

   1. Initial program 95.6%

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

      1. associate-/r* 99.0%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. un-div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;0.5 + \left(0.001388888888888889 \cdot {x}^{4} + x \cdot \left(x \cdot -0.041666666666666664\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 3: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 <= 0.005d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    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 <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	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 <= 0.005)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $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 < 0.0050000000000000001

   1. Initial program 36.2%

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

   2. Taylor expanded in x around 0 65.6%

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

      1. unpow2 65.6%

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

   4. Simplified 65.6%

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

   if 0.0050000000000000001 < x

   1. Initial program 95.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 4: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (* x x)))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 <= 0.005d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x * x))
    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 <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x * x)));
	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 <= 0.005)
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(-0.041666666666666664 * N[(x * x), $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 < 0.0050000000000000001

   1. Initial program 36.2%

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

   2. Taylor expanded in x around 0 65.6%

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

      1. unpow2 65.6%

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

   4. Simplified 65.6%

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

   if 0.0050000000000000001 < x

   1. Initial program 95.6%

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

      1. associate-/r* 99.0%

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

      2. div-inv 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. un-div-inv 99.0%

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

   5. Applied egg-rr 99.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 5: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)} \end{array} \]

FPCore

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

C

double code(double x) {
	return 1.0 / fma(0.16666666666666666, (x * x), 2.0);
}

Julia

function code(x)
	return Float64(1.0 / fma(0.16666666666666666, Float64(x * x), 2.0))
end

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. associate-/r* 52.2%

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

   2. div-inv 52.2%

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

3. Applied egg-rr 52.2%

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

   1. *-commutative 52.2%

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

   2. clear-num 52.2%

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

   3. frac-times 51.1%

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

   4. metadata-eval 51.1%

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

5. Applied egg-rr 51.1%

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

6. Taylor expanded in x around 0 77.4%

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

   1. fma-def 77.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, {x}^{2}, 2\right)}} \]

   2. unpow2 77.4%

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

8. Simplified 77.4%

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

9. Final simplification 77.4%

   \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)} \]

Alternative 6: 78.9% accurate, 8.2× 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(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 50.4%

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

   1. associate-/r* 52.2%

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

   2. div-inv 52.2%

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

3. Applied egg-rr 52.2%

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

   1. *-commutative 52.2%

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

   2. clear-num 52.2%

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

   3. frac-times 51.1%

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

   4. metadata-eval 51.1%

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

5. Applied egg-rr 51.1%

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

6. Taylor expanded in x around 0 77.2%

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

7. Final simplification 77.2%

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

Alternative 7: 52.6% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 50.4%

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

2. Taylor expanded in x around 0 51.4%

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

3. Final simplification 51.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))