cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.8%

Time: 10.5s

Alternatives: 8

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: 50.4% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)}{x\_m} \cdot \frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0004)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (/ (* (sin x_m) (tan (/ x_m 2.0))) x_m) (/ 1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0004) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / x_m) * (1.0 / 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 <= 0.0004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((sin(x_m) * tan((x_m / 2.0d0))) / x_m) * (1.0d0 / 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 <= 0.0004) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((Math.sin(x_m) * Math.tan((x_m / 2.0))) / x_m) * (1.0 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0004:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((math.sin(x_m) * math.tan((x_m / 2.0))) / x_m) * (1.0 / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(sin(x_m) * tan(Float64(x_m / 2.0))) / x_m) * Float64(1.0 / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0004)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / x_m) * (1.0 / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0004], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000019e-4

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 4.00000000000000019e-4 < x

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. div-inv 99.6%

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

      2. flip-- 99.4%

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

      3. metadata-eval 99.4%

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

      4. unpow2 99.4%

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

      5. div-inv 99.4%

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

      6. unpow2 99.4%

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

      7. 1-sub-cos 99.3%

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

      8. associate-*l* 99.3%

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

      9. pow2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.3%

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

      2. *-lft-identity 99.3%

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

      3. associate-/r* 99.3%

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

      4. associate-/l/ 99.3%

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

      5. associate-*r/ 99.3%

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

      6. associate-*r/ 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. unpow2 99.3%

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

      9. associate-*r/ 99.3%

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

      10. hang-0p-tan 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return ((sin(x_m) / x_m) / (x_m / sin(x_m))) / (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) / (x_m / sin(x_m))) / (1.0d0 + cos(x_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.7%

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

   1. clear-num 45.7%

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

   2. inv-pow 45.7%

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

   3. flip-- 45.6%

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

   4. associate-/r/ 45.6%

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

   5. unpow-prod-down 45.6%

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

   6. pow2 45.6%

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

   7. metadata-eval 45.6%

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

   8. pow2 45.6%

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

   9. inv-pow 45.6%

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

4. Applied egg-rr 45.6%

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

   1. associate-*r/ 45.6%

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

   2. *-rgt-identity 45.6%

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

   3. unpow-1 45.6%

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

6. Simplified 45.6%

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

   1. unpow2 45.6%

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

   2. 1-sub-cos 71.7%

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

8. Applied egg-rr 71.7%

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

9. Taylor expanded in x around inf 71.7%

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

    1. *-lft-identity 71.7%

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

    2. associate-*l/ 69.4%

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

    3. unpow2 69.4%

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

    4. associate-/l/ 69.5%

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

    5. *-rgt-identity 69.5%

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

    6. associate-*r/ 69.3%

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

    7. unpow2 69.3%

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

    8. swap-sqr 99.3%

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

    9. unpow2 99.3%

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

    10. associate-*l/ 99.6%

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

    11. *-lft-identity 99.6%

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

11. Simplified 99.6%

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

    1. unpow2 99.6%

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

    2. clear-num 99.6%

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

    3. un-div-inv 99.6%

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

13. Applied egg-rr 99.6%

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

14. Final simplification 99.6%

    \[\leadsto \frac{\frac{\frac{\sin x}{x}}{\frac{x}{\sin x}}}{1 + \cos x} \]
15. Add Preprocessing

Alternative 3: 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 45.7%

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

   1. clear-num 45.7%

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

   2. inv-pow 45.7%

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

   3. flip-- 45.6%

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

   4. associate-/r/ 45.6%

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

   5. unpow-prod-down 45.6%

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

   6. pow2 45.6%

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

   7. metadata-eval 45.6%

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

   8. pow2 45.6%

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

   9. inv-pow 45.6%

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

4. Applied egg-rr 45.6%

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

   1. associate-*r/ 45.6%

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

   2. *-rgt-identity 45.6%

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

   3. unpow-1 45.6%

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

6. Simplified 45.6%

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

   1. unpow2 45.6%

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

   2. 1-sub-cos 71.7%

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

8. Applied egg-rr 71.7%

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

9. Taylor expanded in x around inf 71.7%

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

    1. *-lft-identity 71.7%

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

    2. associate-*l/ 69.4%

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

    3. unpow2 69.4%

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

    4. associate-/l/ 69.5%

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

    5. *-rgt-identity 69.5%

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

    6. associate-*r/ 69.3%

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

    7. unpow2 69.3%

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

    8. swap-sqr 99.3%

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

    9. unpow2 99.3%

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

    10. associate-*l/ 99.6%

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

    11. *-lft-identity 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0038)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (pow x_m -2.0) (- 1.0 (cos x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = pow(x_m, -2.0) * (1.0 - cos(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 <= 0.0038d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (x_m ** (-2.0d0)) * (1.0d0 - cos(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 <= 0.0038) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = Math.pow(x_m, -2.0) * (1.0 - Math.cos(x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0038:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = math.pow(x_m, -2.0) * (1.0 - math.cos(x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0038)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64((x_m ^ -2.0) * Float64(1.0 - cos(x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0038)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (x_m ^ -2.0) * (1.0 - cos(x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0038], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00379999999999999999

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 0.00379999999999999999 < x

   1. Initial program 99.6%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.7%

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

      3. pow2 99.7%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0038)
   (+ 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 <= 0.0038) {
		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 <= 0.0038d0) 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 <= 0.0038) {
		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 <= 0.0038:
		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 <= 0.0038)
		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 <= 0.0038)
		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, 0.0038], 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 < 0.00379999999999999999

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 0.00379999999999999999 < x

   1. Initial program 99.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0038)
   (+ 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 <= 0.0038) {
		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 <= 0.0038d0) 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 <= 0.0038) {
		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 <= 0.0038:
		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 <= 0.0038)
		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 <= 0.0038)
		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, 0.0038], 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 < 0.00379999999999999999

   1. Initial program 30.0%

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

   3. Taylor expanded in x around 0 71.7%

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

   if 0.00379999999999999999 < x

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(\frac{1}{x\_m} + \frac{-1}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.8e+75) 0.5 (* (/ 1.0 x_m) (+ (/ 1.0 x_m) (/ -1.0 x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e+75) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) * ((1.0 / x_m) + (-1.0 / 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 <= 2.8d+75) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x_m) * ((1.0d0 / x_m) + ((-1.0d0) / 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 <= 2.8e+75) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) * ((1.0 / x_m) + (-1.0 / x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.8e+75:
		tmp = 0.5
	else:
		tmp = (1.0 / x_m) * ((1.0 / x_m) + (-1.0 / x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.8e+75)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(Float64(1.0 / x_m) + Float64(-1.0 / x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.8e+75)
		tmp = 0.5;
	else
		tmp = (1.0 / x_m) * ((1.0 / x_m) + (-1.0 / x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e+75], 0.5, N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.80000000000000012e75

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0 68.1%

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

   if 2.80000000000000012e75 < x

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. div-sub 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0 79.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} + \frac{-1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 45.7%

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

3. Taylor expanded in x around 0 56.7%

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

4. Final simplification 56.7%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))