cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.8%

Time: 9.0s

Alternatives: 9

Speedup: 17.8×

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

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

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 53.5%

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

   1. expm1-log1p-u 53.4%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x \cdot x} \]

4. Applied egg-rr 53.4%

   \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x \cdot x} \]
5. Step-by-step derivation

   1. expm1-undefine 53.3%

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

   2. sub-neg 53.3%

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

   3. log1p-undefine 53.3%

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

   4. rem-exp-log 53.3%

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

   5. associate-+r- 53.3%

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

   6. metadata-eval 53.3%

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

   7. metadata-eval 53.3%

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

6. Simplified 53.3%

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

   1. +-commutative 53.3%

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

   2. associate-+r- 53.5%

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

   3. metadata-eval 53.5%

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

   4. flip-- 53.3%

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

   5. metadata-eval 53.3%

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

   6. 1-sub-cos 78.2%

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

   7. div-inv 78.2%

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

   8. pow2 78.2%

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

8. Applied egg-rr 78.2%

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

   1. associate-*r/ 78.2%

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

   2. *-rgt-identity 78.2%

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

   3. unpow2 78.2%

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

   4. associate-*r/ 78.2%

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

   5. hang-0p-tan 78.4%

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

10. Simplified 78.4%

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

    1. *-commutative 78.4%

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

    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%

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

12. Applied egg-rr 99.8%

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

Alternative 2: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.03)
   (+
    0.5
    (* (pow x 2.0) (- (* 0.001388888888888889 (* x x)) 0.041666666666666664)))
   (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = 0.5 + (pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	} 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 <= 0.03d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * ((0.001388888888888889d0 * (x * x)) - 0.041666666666666664d0))
    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 <= 0.03) {
		tmp = 0.5 + (Math.pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.03:
		tmp = 0.5 + (math.pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664))
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.03)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * Float64(Float64(0.001388888888888889 * Float64(x * x)) - 0.041666666666666664)));
	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 <= 0.03)
		tmp = 0.5 + ((x ^ 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.03], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $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 < 0.029999999999999999

   1. Initial program 38.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.1%

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

      1. unpow2 63.1%

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

   5. Applied egg-rr 63.1%

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

   if 0.029999999999999999 < x

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

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

      3. sqrt-div 99.2%

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

      4. sqrt-prod 98.9%

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

      5. add-sqr-sqrt 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. frac-times 99.2%

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

      3. add-sqr-sqrt 99.3%

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

      4. *-un-lft-identity 99.3%

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

      5. frac-times 99.4%

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

      6. clear-num 99.3%

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

      7. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

Alternative 3: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (/ (/ 1.0 x) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} 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 <= 0.00016d0) then
        tmp = 0.5d0
    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 <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) / (x / (1.0 - Math.cos(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = (1.0 / x) / (x / (1.0 - math.cos(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		tmp = 0.5;
	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 <= 0.00016)
		tmp = 0.5;
	else
		tmp = (1.0 / x) / (x / (1.0 - cos(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 0.5, 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 < 1.60000000000000013e-4

   1. Initial program 38.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

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

      3. sqrt-div 99.2%

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

      4. sqrt-prod 98.9%

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

      5. add-sqr-sqrt 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. frac-times 99.2%

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

      3. add-sqr-sqrt 99.3%

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

      4. *-un-lft-identity 99.3%

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

      5. frac-times 99.4%

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

      6. clear-num 99.3%

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

      7. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

Alternative 4: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (* (/ 1.0 x) (/ (- 1.0 (cos x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 - cos(x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00016d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x) * ((1.0d0 - cos(x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * ((1.0 - Math.cos(x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = (1.0 / x) * ((1.0 - math.cos(x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(1.0 - cos(x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00016)
		tmp = 0.5;
	else
		tmp = (1.0 / x) * ((1.0 - cos(x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 0.5, N[(N[(1.0 / x), $MachinePrecision] * N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.60000000000000013e-4

   1. Initial program 38.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1 - \cos x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} 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 <= 0.00016d0) then
        tmp = 0.5d0
    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 <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		tmp = 0.5;
	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 <= 0.00016)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 0.5, N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.60000000000000013e-4

   1. Initial program 38.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

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

      3. sqrt-div 99.2%

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

      4. sqrt-prod 98.9%

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

      5. add-sqr-sqrt 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. unpow2 99.2%

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

      2. frac-times 99.2%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

Alternative 6: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} 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 <= 0.00016d0) then
        tmp = 0.5d0
    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 <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		tmp = 0.5;
	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 <= 0.00016)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 0.5, N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.60000000000000013e-4

   1. Initial program 38.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 99.3%

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

Alternative 7: 62.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.05e+76) 0.5 (+ (/ 1.0 (* x x)) (/ (/ -1.0 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.05e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / (x * x)) + ((-1.0 / x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.05d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / (x * x)) + (((-1.0d0) / x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.05e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / (x * x)) + ((-1.0 / x) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.05e+76:
		tmp = 0.5
	else:
		tmp = (1.0 / (x * x)) + ((-1.0 / x) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.05e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / Float64(x * x)) + Float64(Float64(-1.0 / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.05e+76)
		tmp = 0.5;
	else
		tmp = (1.0 / (x * x)) + ((-1.0 / x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.05e+76], 0.5, N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000003e76

   1. Initial program 44.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 58.8%

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

   if 1.05000000000000003e76 < x

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. sqrt-div 99.4%

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

      4. sqrt-prod 99.3%

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

      5. add-sqr-sqrt 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. unpow2 99.4%

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

      2. frac-times 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. associate-/r* 99.5%

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

      5. div-sub 99.5%

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

      6. div-sub 99.4%

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

      7. associate-/r* 99.4%

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

      8. unpow2 99.4%

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

      9. pow-flip 99.4%

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

      10. metadata-eval 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. metadata-eval 99.4%

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

      2. pow-prod-up 99.3%

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

      3. inv-pow 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

      6. inv-pow 99.3%

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

      7. frac-2neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. frac-times 99.4%

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

      10. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around 0 72.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.5e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.5e+77) {
		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 <= 1.5d+77) 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 <= 1.5e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e77

   1. Initial program 44.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 58.3%

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

   if 1.4999999999999999e77 < x

   1. Initial program 99.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.7%

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

   4. Taylor expanded in x around 0 75.7%

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

Alternative 9: 27.8% 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 53.5%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 29.5%

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

4. Taylor expanded in x around 0 30.2%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024172 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))