cos2 (problem 3.4.1)

Percentage Accurate: 51.2% → 99.6%

Time: 10.7s

Alternatives: 9

Speedup: 9.7×

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: 51.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.1:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x\_m}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x\_m}^{6} + \log \left(1 + \mathsf{expm1}\left(0.001388888888888889 \cdot {x\_m}^{4}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{x\_m}{1 - \cos x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (+
      (* -2.48015873015873e-5 (pow x_m 6.0))
      (log (+ 1.0 (expm1 (* 0.001388888888888889 (pow x_m 4.0))))))))
   (/ (/ 1.0 x_m) (/ x_m (- 1.0 (cos x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.1) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + ((-2.48015873015873e-5 * pow(x_m, 6.0)) + log((1.0 + expm1((0.001388888888888889 * pow(x_m, 4.0)))))));
	} else {
		tmp = (1.0 / x_m) / (x_m / (1.0 - cos(x_m)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.1) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * Math.pow(x_m, 6.0)) + Math.log((1.0 + Math.expm1((0.001388888888888889 * Math.pow(x_m, 4.0)))))));
	} else {
		tmp = (1.0 / x_m) / (x_m / (1.0 - Math.cos(x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.1:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * math.pow(x_m, 6.0)) + math.log((1.0 + math.expm1((0.001388888888888889 * math.pow(x_m, 4.0)))))))
	else:
		tmp = (1.0 / x_m) / (x_m / (1.0 - math.cos(x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.1)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(Float64(-2.48015873015873e-5 * (x_m ^ 6.0)) + log(Float64(1.0 + expm1(Float64(0.001388888888888889 * (x_m ^ 4.0))))))));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m / Float64(1.0 - cos(x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.1], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[Log[N[(1.0 + N[(Exp[N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0 67.7%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 67.8%

         \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.001388888888888889 \cdot {x}^{4}\right)\right)}\right)\right) \]

      2. log1p-undefine 67.8%

         \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.001388888888888889 \cdot {x}^{4}\right)\right)}\right)\right) \]

   5. Applied egg-rr 67.8%

      \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \color{blue}{\log \left(1 + \mathsf{expm1}\left(0.001388888888888889 \cdot {x}^{4}\right)\right)}\right)\right) \]

   if 0.10000000000000001 < x

   1. Initial program 98.6%

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

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

      1. *-commutative 99.4%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \log \left(1 + \mathsf{expm1}\left(0.001388888888888889 \cdot {x}^{4}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.1:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x\_m}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x\_m}^{6} + 0.001388888888888889 \cdot {x\_m}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{x\_m}{1 - \cos x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (+
      (* -2.48015873015873e-5 (pow x_m 6.0))
      (* 0.001388888888888889 (pow x_m 4.0)))))
   (/ (/ 1.0 x_m) (/ x_m (- 1.0 (cos x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.1) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + ((-2.48015873015873e-5 * pow(x_m, 6.0)) + (0.001388888888888889 * pow(x_m, 4.0))));
	} else {
		tmp = (1.0 / x_m) / (x_m / (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 <= 0.1d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x_m ** 2.0d0)) + (((-2.48015873015873d-5) * (x_m ** 6.0d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0))))
    else
        tmp = (1.0d0 / x_m) / (x_m / (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 <= 0.1) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * Math.pow(x_m, 6.0)) + (0.001388888888888889 * Math.pow(x_m, 4.0))));
	} else {
		tmp = (1.0 / x_m) / (x_m / (1.0 - Math.cos(x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.1:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * math.pow(x_m, 6.0)) + (0.001388888888888889 * math.pow(x_m, 4.0))))
	else:
		tmp = (1.0 / x_m) / (x_m / (1.0 - math.cos(x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.1)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(Float64(-2.48015873015873e-5 * (x_m ^ 6.0)) + Float64(0.001388888888888889 * (x_m ^ 4.0)))));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m / 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 <= 0.1)
		tmp = 0.5 + ((-0.041666666666666664 * (x_m ^ 2.0)) + ((-2.48015873015873e-5 * (x_m ^ 6.0)) + (0.001388888888888889 * (x_m ^ 4.0))));
	else
		tmp = (1.0 / x_m) / (x_m / (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, 0.1], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0 67.7%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]

   if 0.10000000000000001 < x

   1. Initial program 98.6%

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

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

      1. *-commutative 99.4%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.078:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{0.00033068783068783067 \cdot {x\_m}^{5} + \left(0.008333333333333333 \cdot {x\_m}^{3} + \left(2 \cdot \frac{1}{x\_m} + x\_m \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{x\_m}{1 - \cos x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.078)
   (/
    (/ 1.0 x_m)
    (+
     (* 0.00033068783068783067 (pow x_m 5.0))
     (+
      (* 0.008333333333333333 (pow x_m 3.0))
      (+ (* 2.0 (/ 1.0 x_m)) (* x_m 0.16666666666666666)))))
   (/ (/ 1.0 x_m) (/ x_m (- 1.0 (cos x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.078) {
		tmp = (1.0 / x_m) / ((0.00033068783068783067 * pow(x_m, 5.0)) + ((0.008333333333333333 * pow(x_m, 3.0)) + ((2.0 * (1.0 / x_m)) + (x_m * 0.16666666666666666))));
	} else {
		tmp = (1.0 / x_m) / (x_m / (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 <= 0.078d0) then
        tmp = (1.0d0 / x_m) / ((0.00033068783068783067d0 * (x_m ** 5.0d0)) + ((0.008333333333333333d0 * (x_m ** 3.0d0)) + ((2.0d0 * (1.0d0 / x_m)) + (x_m * 0.16666666666666666d0))))
    else
        tmp = (1.0d0 / x_m) / (x_m / (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 <= 0.078) {
		tmp = (1.0 / x_m) / ((0.00033068783068783067 * Math.pow(x_m, 5.0)) + ((0.008333333333333333 * Math.pow(x_m, 3.0)) + ((2.0 * (1.0 / x_m)) + (x_m * 0.16666666666666666))));
	} else {
		tmp = (1.0 / x_m) / (x_m / (1.0 - Math.cos(x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.078:
		tmp = (1.0 / x_m) / ((0.00033068783068783067 * math.pow(x_m, 5.0)) + ((0.008333333333333333 * math.pow(x_m, 3.0)) + ((2.0 * (1.0 / x_m)) + (x_m * 0.16666666666666666))))
	else:
		tmp = (1.0 / x_m) / (x_m / (1.0 - math.cos(x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.078)
		tmp = Float64(Float64(1.0 / x_m) / Float64(Float64(0.00033068783068783067 * (x_m ^ 5.0)) + Float64(Float64(0.008333333333333333 * (x_m ^ 3.0)) + Float64(Float64(2.0 * Float64(1.0 / x_m)) + Float64(x_m * 0.16666666666666666)))));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m / 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 <= 0.078)
		tmp = (1.0 / x_m) / ((0.00033068783068783067 * (x_m ^ 5.0)) + ((0.008333333333333333 * (x_m ^ 3.0)) + ((2.0 * (1.0 / x_m)) + (x_m * 0.16666666666666666))));
	else
		tmp = (1.0 / x_m) / (x_m / (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, 0.078], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(0.00033068783068783067 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.008333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0779999999999999999

   1. Initial program 34.5%

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

      1. associate-/r* 35.9%

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

      2. div-inv 35.8%

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

   4. Applied egg-rr 35.8%

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

      1. *-commutative 35.8%

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

      2. clear-num 35.8%

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

      3. un-div-inv 35.9%

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

   6. Applied egg-rr 35.9%

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

   7. Taylor expanded in x around 0 82.8%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.00033068783068783067 \cdot {x}^{5} + \left(0.008333333333333333 \cdot {x}^{3} + \left(0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)\right)}} \]

   if 0.0779999999999999999 < x

   1. Initial program 98.6%

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

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

      1. *-commutative 99.4%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.078:\\ \;\;\;\;\frac{\frac{1}{x}}{0.00033068783068783067 \cdot {x}^{5} + \left(0.008333333333333333 \cdot {x}^{3} + \left(2 \cdot \frac{1}{x} + x \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{2}{x\_m} + x\_m \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{x\_m}{1 - \cos x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0038)
   (/ (/ 1.0 x_m) (+ (/ 2.0 x_m) (* x_m 0.16666666666666666)))
   (/ (/ 1.0 x_m) (/ x_m (- 1.0 (cos x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0038) {
		tmp = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	} else {
		tmp = (1.0 / x_m) / (x_m / (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 = (1.0d0 / x_m) / ((2.0d0 / x_m) + (x_m * 0.16666666666666666d0))
    else
        tmp = (1.0d0 / x_m) / (x_m / (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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	} else {
		tmp = (1.0 / x_m) / (x_m / (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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666))
	else:
		tmp = (1.0 / x_m) / (x_m / (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(Float64(1.0 / x_m) / Float64(Float64(2.0 / x_m) + Float64(x_m * 0.16666666666666666)));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m / 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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	else
		tmp = (1.0 / x_m) / (x_m / (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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(2.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00379999999999999999

   1. Initial program 34.5%

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

      1. associate-/r* 35.9%

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

      2. div-inv 35.8%

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

   4. Applied egg-rr 35.8%

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

      1. *-commutative 35.8%

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

      2. clear-num 35.8%

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

      3. un-div-inv 35.9%

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

   6. Applied egg-rr 35.9%

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

   7. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \frac{1}{x} + 0.16666666666666666 \cdot x}} \]

      2. *-un-lft-identity 84.1%

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

      3. fma-define 84.1%

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

      4. un-div-inv 84.1%

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

   9. Applied egg-rr 84.1%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(1, \frac{2}{x}, 0.16666666666666666 \cdot x\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 \cdot \frac{2}{x} + 0.16666666666666666 \cdot x}} \]

       2. *-lft-identity 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x}} + 0.16666666666666666 \cdot x} \]

       3. *-commutative 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\frac{2}{x} + \color{blue}{x \cdot 0.16666666666666666}} \]

   11. Simplified 84.1%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x} + x \cdot 0.16666666666666666}} \]

   if 0.00379999999999999999 < x

   1. Initial program 98.6%

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

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

      1. *-commutative 99.4%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{2}{x\_m} + x\_m \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0038)
   (/ (/ 1.0 x_m) (+ (/ 2.0 x_m) (* x_m 0.16666666666666666)))
   (/ (- 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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	} 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 = (1.0d0 / x_m) / ((2.0d0 / x_m) + (x_m * 0.16666666666666666d0))
    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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	} 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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666))
	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(Float64(1.0 / x_m) / Float64(Float64(2.0 / x_m) + Float64(x_m * 0.16666666666666666)));
	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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(2.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.16666666666666666), $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 34.5%

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

      1. associate-/r* 35.9%

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

      2. div-inv 35.8%

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

   4. Applied egg-rr 35.8%

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

      1. *-commutative 35.8%

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

      2. clear-num 35.8%

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

      3. un-div-inv 35.9%

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

   6. Applied egg-rr 35.9%

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

   7. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \frac{1}{x} + 0.16666666666666666 \cdot x}} \]

      2. *-un-lft-identity 84.1%

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

      3. fma-define 84.1%

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

      4. un-div-inv 84.1%

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

   9. Applied egg-rr 84.1%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(1, \frac{2}{x}, 0.16666666666666666 \cdot x\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 \cdot \frac{2}{x} + 0.16666666666666666 \cdot x}} \]

       2. *-lft-identity 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x}} + 0.16666666666666666 \cdot x} \]

       3. *-commutative 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\frac{2}{x} + \color{blue}{x \cdot 0.16666666666666666}} \]

   11. Simplified 84.1%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x} + x \cdot 0.16666666666666666}} \]

   if 0.00379999999999999999 < x

   1. Initial program 98.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0038:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\frac{2}{x\_m} + x\_m \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0038)
   (/ (/ 1.0 x_m) (+ (/ 2.0 x_m) (* x_m 0.16666666666666666)))
   (/ (/ (- 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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	} 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 = (1.0d0 / x_m) / ((2.0d0 / x_m) + (x_m * 0.16666666666666666d0))
    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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	} 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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666))
	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(Float64(1.0 / x_m) / Float64(Float64(2.0 / x_m) + Float64(x_m * 0.16666666666666666)));
	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 = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
	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[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(2.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.16666666666666666), $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 34.5%

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

      1. associate-/r* 35.9%

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

      2. div-inv 35.8%

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

   4. Applied egg-rr 35.8%

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

      1. *-commutative 35.8%

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

      2. clear-num 35.8%

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

      3. un-div-inv 35.9%

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

   6. Applied egg-rr 35.9%

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

   7. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \frac{1}{x} + 0.16666666666666666 \cdot x}} \]

      2. *-un-lft-identity 84.1%

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

      3. fma-define 84.1%

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

      4. un-div-inv 84.1%

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

   9. Applied egg-rr 84.1%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(1, \frac{2}{x}, 0.16666666666666666 \cdot x\right)}} \]
   10. Step-by-step derivation

       1. fma-undefine 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 \cdot \frac{2}{x} + 0.16666666666666666 \cdot x}} \]

       2. *-lft-identity 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x}} + 0.16666666666666666 \cdot x} \]

       3. *-commutative 84.1%

          \[\leadsto \frac{\frac{1}{x}}{\frac{2}{x} + \color{blue}{x \cdot 0.16666666666666666}} \]

   11. Simplified 84.1%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x} + x \cdot 0.16666666666666666}} \]

   if 0.00379999999999999999 < x

   1. Initial program 98.6%

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

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

      1. un-div-inv 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. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0038:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{x\_m \cdot 0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5) 0.5 (/ (/ 1.0 x_m) (* x_m 0.16666666666666666))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666);
	}
	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 <= 3.5d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x_m) / (x_m * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.5:
		tmp = 0.5
	else:
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m * 0.16666666666666666));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], 0.5, N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0 67.9%

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

   if 3.5 < x

   1. Initial program 98.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. clear-num 99.4%

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

      3. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around 0 54.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in x around inf 54.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.16666666666666666 \cdot x}} \]
   9. Step-by-step derivation

      1. *-commutative 54.4%

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]

   10. Simplified 54.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot 0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{x\_m}}{\frac{2}{x\_m} + x\_m \cdot 0.16666666666666666} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (/ 1.0 x_m) (+ (/ 2.0 x_m) (* x_m 0.16666666666666666))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (1.0d0 / x_m) / ((2.0d0 / x_m) + (x_m * 0.16666666666666666d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / x_m) / Float64(Float64(2.0 / x_m) + Float64(x_m * 0.16666666666666666)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / x_m) / ((2.0 / x_m) + (x_m * 0.16666666666666666));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(2.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

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

   1. associate-/r* 53.7%

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

   2. div-inv 53.7%

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

4. Applied egg-rr 53.7%

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

   1. *-commutative 53.7%

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

   2. clear-num 53.7%

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

   3. un-div-inv 53.7%

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

6. Applied egg-rr 53.7%

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

7. Taylor expanded in x around 0 75.7%

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

   1. +-commutative 75.7%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{2 \cdot \frac{1}{x} + 0.16666666666666666 \cdot x}} \]

   2. *-un-lft-identity 75.7%

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

   3. fma-define 75.7%

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

   4. un-div-inv 75.7%

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

9. Applied egg-rr 75.7%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{fma}\left(1, \frac{2}{x}, 0.16666666666666666 \cdot x\right)}} \]
10. Step-by-step derivation

    1. fma-undefine 75.7%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{1 \cdot \frac{2}{x} + 0.16666666666666666 \cdot x}} \]

    2. *-lft-identity 75.7%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x}} + 0.16666666666666666 \cdot x} \]

    3. *-commutative 75.7%

       \[\leadsto \frac{\frac{1}{x}}{\frac{2}{x} + \color{blue}{x \cdot 0.16666666666666666}} \]

11. Simplified 75.7%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2}{x} + x \cdot 0.16666666666666666}} \]

12. Final simplification 75.7%

    \[\leadsto \frac{\frac{1}{x}}{\frac{2}{x} + x \cdot 0.16666666666666666} \]
13. Add Preprocessing

Alternative 9: 51.1% 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 52.5%

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

3. Taylor expanded in x around 0 50.3%

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

4. Final simplification 50.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))