cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.8%

Time: 9.9s

Alternatives: 6

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: 50.7% 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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-8}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2e-8) 0.5 (* (pow x_m -2.0) (* (sin x_m) (tan (/ x_m 2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2e-8) {
		tmp = 0.5;
	} else {
		tmp = pow(x_m, -2.0) * (sin(x_m) * tan((x_m / 2.0)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2d-8) then
        tmp = 0.5d0
    else
        tmp = (x_m ** (-2.0d0)) * (sin(x_m) * tan((x_m / 2.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2e-8) {
		tmp = 0.5;
	} else {
		tmp = Math.pow(x_m, -2.0) * (Math.sin(x_m) * Math.tan((x_m / 2.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2e-8:
		tmp = 0.5
	else:
		tmp = math.pow(x_m, -2.0) * (math.sin(x_m) * math.tan((x_m / 2.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2e-8)
		tmp = 0.5;
	else
		tmp = Float64((x_m ^ -2.0) * Float64(sin(x_m) * tan(Float64(x_m / 2.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2e-8)
		tmp = 0.5;
	else
		tmp = (x_m ^ -2.0) * (sin(x_m) * tan((x_m / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2e-8], 0.5, N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-8

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.5%

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

   if 2e-8 < x

   1. Initial program 97.8%

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

      1. clear-num 97.7%

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

      2. inv-pow 97.7%

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

      3. flip-- 97.3%

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

      4. associate-/r/ 97.2%

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

      5. unpow-prod-down 97.1%

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

      6. pow2 97.1%

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

      7. metadata-eval 97.1%

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

      8. pow2 97.1%

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

      9. inv-pow 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. associate-*r/ 97.2%

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

      2. *-rgt-identity 97.2%

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

      3. unpow-1 97.2%

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

      4. associate-/r/ 97.2%

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

   6. Simplified 97.2%

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

      1. unpow2 97.2%

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

      2. 1-sub-cos 98.2%

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

   8. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. add-sqr-sqrt 98.1%

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

      3. times-frac 98.1%

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

      4. pow2 98.1%

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

      5. pow-flip 99.1%

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

      6. metadata-eval 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. *-commutative 99.1%

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

       2. associate-*l/ 99.0%

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

       3. associate-/l* 99.0%

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

       4. associate-/l/ 99.0%

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

       5. unpow2 99.0%

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

       6. rem-square-sqrt 99.1%

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

       7. associate-*r/ 99.1%

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

       8. hang-0p-tan 99.6%

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

   12. Simplified 99.6%

       \[\leadsto \color{blue}{{x}^{-2} \cdot \left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{{\left(\frac{\sin x\_m}{x\_m}\right)}^{2}}{1 + \cos x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (pow (/ (sin x_m) x_m) 2.0) (+ 1.0 (cos x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow((sin(x_m) / x_m), 2.0) / (1.0 + cos(x_m));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = ((sin(x_m) / x_m) ** 2.0d0) / (1.0d0 + cos(x_m))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow((Math.sin(x_m) / x_m), 2.0) / (1.0 + Math.cos(x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow((math.sin(x_m) / x_m), 2.0) / (1.0 + math.cos(x_m))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((Float64(sin(x_m) / x_m) ^ 2.0) / Float64(1.0 + cos(x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((sin(x_m) / x_m) ^ 2.0) / (1.0 + cos(x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision], 2.0], $MachinePrecision] / N[(1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

   1. clear-num 52.8%

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

   2. inv-pow 52.8%

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

   3. flip-- 52.6%

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

   4. associate-/r/ 52.6%

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

   5. unpow-prod-down 52.6%

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

   6. pow2 52.6%

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

   7. metadata-eval 52.6%

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

   8. pow2 52.6%

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

   9. inv-pow 52.6%

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

4. Applied egg-rr 52.6%

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

   1. associate-*r/ 52.6%

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

   2. *-rgt-identity 52.6%

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

   3. unpow-1 52.6%

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

   4. associate-/r/ 52.6%

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

6. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. 1-sub-cos 75.5%

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

8. Applied egg-rr 75.5%

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

9. Taylor expanded in x around inf 76.0%

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

    1. unpow2 76.0%

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

    2. unpow2 76.0%

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

    3. times-frac 99.5%

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

    4. unpow2 99.5%

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

11. Simplified 99.5%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0052d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((1.0d0 - cos(x_m)) / x_m) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0052:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0052)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0052)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0052], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.3%

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

      1. associate-/r* 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0052d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (1.0d0 - cos(x_m)) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0052:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0052)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0052)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0052], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.3%

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

Alternative 5: 75.5% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.12e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.12e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.12d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.12e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.12e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.12e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.12e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.12e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1199999999999999e77

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 60.4%

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

   if 1.1199999999999999e77 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 69.4%

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

   4. Taylor expanded in x around 0 69.4%

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

Alternative 6: 27.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 52.8%

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

3. Taylor expanded in x around 0 25.1%

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

4. Taylor expanded in x around 0 25.9%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))