mixedcos

Percentage Accurate: 66.6% → 96.8%

Time: 12.8s

Alternatives: 9

Speedup: 24.1×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;{\left(c \cdot \left(s\_m \cdot x\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{x\_m \cdot \left(s\_m \cdot c\right)}}{s\_m \cdot \left(x\_m \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (if (<= x_m 2.7e-176)
   (pow (* c (* s_m x_m)) -2.0)
   (/ (/ (cos (* x_m 2.0)) (* x_m (* s_m c))) (* s_m (* x_m c)))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 2.7e-176) {
		tmp = pow((c * (s_m * x_m)), -2.0);
	} else {
		tmp = (cos((x_m * 2.0)) / (x_m * (s_m * c))) / (s_m * (x_m * c));
	}
	return tmp;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.7d-176) then
        tmp = (c * (s_m * x_m)) ** (-2.0d0)
    else
        tmp = (cos((x_m * 2.0d0)) / (x_m * (s_m * c))) / (s_m * (x_m * c))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 2.7e-176) {
		tmp = Math.pow((c * (s_m * x_m)), -2.0);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / (x_m * (s_m * c))) / (s_m * (x_m * c));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	tmp = 0
	if x_m <= 2.7e-176:
		tmp = math.pow((c * (s_m * x_m)), -2.0)
	else:
		tmp = (math.cos((x_m * 2.0)) / (x_m * (s_m * c))) / (s_m * (x_m * c))
	return tmp

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	tmp = 0.0
	if (x_m <= 2.7e-176)
		tmp = Float64(c * Float64(s_m * x_m)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / Float64(x_m * Float64(s_m * c))) / Float64(s_m * Float64(x_m * c)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp_2 = code(x_m, c, s_m)
	tmp = 0.0;
	if (x_m <= 2.7e-176)
		tmp = (c * (s_m * x_m)) ^ -2.0;
	else
		tmp = (cos((x_m * 2.0)) / (x_m * (s_m * c))) / (s_m * (x_m * c));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := If[LessEqual[x$95$m, 2.7e-176], N[Power[N[(c * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(s$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6999999999999998e-176

   1. Initial program 63.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 62.6%

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

      2. *-commutative 62.6%

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

      3. unpow2 62.6%

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

      4. sqr-neg 62.6%

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

      5. unpow2 62.6%

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

      6. cos-neg 62.6%

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

      7. *-commutative 62.6%

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

      8. distribute-rgt-neg-in 62.6%

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

      9. metadata-eval 62.6%

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

      10. unpow2 62.6%

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

      11. sqr-neg 62.6%

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

      12. unpow2 62.6%

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

      13. associate-*r* 59.3%

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

      14. unpow2 59.3%

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

      15. *-commutative 59.3%

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

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.1%

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

      1. associate-/r* 54.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 54.4%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 54.4%

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

      4. unpow2 54.4%

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

      5. swap-sqr 66.1%

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

      6. unpow2 66.1%

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

      7. associate-/r* 66.8%

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

      8. unpow2 66.8%

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

      9. unpow2 66.8%

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

      10. swap-sqr 82.6%

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

      11. unpow2 82.6%

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

   7. Simplified 82.6%

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

      1. *-un-lft-identity 82.6%

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

      2. pow-flip 83.0%

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

      3. *-commutative 83.0%

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

      4. *-commutative 83.0%

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

      5. associate-*l* 82.7%

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

      6. metadata-eval 82.7%

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

   9. Applied egg-rr 82.7%

      \[\leadsto \color{blue}{1 \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}} \]
   10. Step-by-step derivation

       1. *-lft-identity 82.7%

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

       2. associate-*r* 83.0%

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

       3. *-commutative 83.0%

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

       4. *-commutative 83.0%

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

       5. *-commutative 83.0%

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

   11. Simplified 83.0%

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

   if 2.6999999999999998e-176 < x

   1. Initial program 69.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 65.4%

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

      1. associate-/r* 65.3%

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

      2. *-commutative 65.3%

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

      3. unpow2 65.3%

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

      4. unpow2 65.3%

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

      5. swap-sqr 77.0%

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

      6. unpow2 77.0%

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

      7. associate-/r* 77.1%

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

      8. *-commutative 77.1%

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

      9. unpow2 77.1%

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

      10. unpow2 77.1%

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

      11. swap-sqr 93.9%

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      12. unpow2 93.9%

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

      13. *-commutative 93.9%

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

      14. associate-*l* 98.6%

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

   5. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*r* 93.9%

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

      3. *-commutative 93.9%

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

      4. div-inv 93.8%

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

      5. *-commutative 93.8%

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

      6. pow-flip 94.5%

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

      7. *-commutative 94.5%

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

      8. *-commutative 94.5%

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

      9. associate-*l* 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)} \]
   8. Step-by-step derivation

      1. metadata-eval 99.7%

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

      2. pow-flip 98.6%

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

      3. unpow2 98.6%

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

      4. associate-/r* 99.7%

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

      5. associate-*l/ 99.7%

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

      6. associate-/r* 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in s around 0 94.5%

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

       1. associate-*r* 99.8%

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot s\right) \cdot x}}}{s \cdot \left(x \cdot c\right)} \]

       2. *-commutative 99.8%

          \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(c \cdot s\right)}}}{s \cdot \left(x \cdot c\right)} \]

   12. Simplified 99.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;{\left(c \cdot \left(s \cdot x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(s \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(s\_m \cdot x\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{s\_m \cdot \left(\left(x\_m \cdot c\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (if (<= x_m 1e-9)
   (pow (* c (* s_m x_m)) -2.0)
   (/ (cos (* x_m 2.0)) (* s_m (* (* x_m c) (* s_m (* x_m c)))))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 1e-9) {
		tmp = pow((c * (s_m * x_m)), -2.0);
	} else {
		tmp = cos((x_m * 2.0)) / (s_m * ((x_m * c) * (s_m * (x_m * c))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 1d-9) then
        tmp = (c * (s_m * x_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / (s_m * ((x_m * c) * (s_m * (x_m * c))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 1e-9) {
		tmp = Math.pow((c * (s_m * x_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (s_m * ((x_m * c) * (s_m * (x_m * c))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	tmp = 0
	if x_m <= 1e-9:
		tmp = math.pow((c * (s_m * x_m)), -2.0)
	else:
		tmp = math.cos((x_m * 2.0)) / (s_m * ((x_m * c) * (s_m * (x_m * c))))
	return tmp

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	tmp = 0.0
	if (x_m <= 1e-9)
		tmp = Float64(c * Float64(s_m * x_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(Float64(x_m * c) * Float64(s_m * Float64(x_m * c)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp_2 = code(x_m, c, s_m)
	tmp = 0.0;
	if (x_m <= 1e-9)
		tmp = (c * (s_m * x_m)) ^ -2.0;
	else
		tmp = cos((x_m * 2.0)) / (s_m * ((x_m * c) * (s_m * (x_m * c))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := If[LessEqual[x$95$m, 1e-9], N[Power[N[(c * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x$95$m * c), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000006e-9

   1. Initial program 64.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 63.9%

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

      2. *-commutative 63.9%

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

      3. unpow2 63.9%

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

      4. sqr-neg 63.9%

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

      5. unpow2 63.9%

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

      6. cos-neg 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. metadata-eval 63.9%

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

      10. unpow2 63.9%

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

      11. sqr-neg 63.9%

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

      12. unpow2 63.9%

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

      13. associate-*r* 61.2%

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

      14. unpow2 61.2%

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

      15. *-commutative 61.2%

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

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.7%

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

      1. associate-/r* 57.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 57.1%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 57.1%

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

      4. unpow2 57.1%

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

      5. swap-sqr 68.3%

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

      6. unpow2 68.3%

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

      7. associate-/r* 68.8%

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

      8. unpow2 68.8%

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

      9. unpow2 68.8%

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

      10. swap-sqr 85.4%

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

      11. unpow2 85.4%

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

   7. Simplified 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. pow-flip 85.9%

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

      3. *-commutative 85.9%

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

      4. *-commutative 85.9%

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

      5. associate-*l* 85.6%

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

      6. metadata-eval 85.6%

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

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{1 \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}} \]
   10. Step-by-step derivation

       1. *-lft-identity 85.6%

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

       2. associate-*r* 85.9%

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

       3. *-commutative 85.9%

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

       4. *-commutative 85.9%

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

       5. *-commutative 85.9%

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

   11. Simplified 85.9%

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

   if 1.00000000000000006e-9 < x

   1. Initial program 68.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 62.9%

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

      1. associate-/r* 62.9%

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

      2. *-commutative 62.9%

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

      3. unpow2 62.9%

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

      4. unpow2 62.9%

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

      5. swap-sqr 76.1%

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

      6. unpow2 76.1%

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

      7. associate-/r* 76.2%

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

      8. *-commutative 76.2%

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

      9. unpow2 76.2%

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

      10. unpow2 76.2%

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

      11. swap-sqr 91.3%

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      12. unpow2 91.3%

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

      13. *-commutative 91.3%

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

      14. associate-*l* 98.5%

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

   5. Simplified 98.5%

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

      1. associate-*r* 91.3%

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

      2. *-commutative 91.3%

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

      3. unpow2 91.3%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      4. associate-*r* 91.3%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}} \]

      5. associate-*r* 88.5%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot x\right)\right) \cdot s}} \]

      6. *-commutative 88.5%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(\left(x \cdot s\right) \cdot c\right)} \cdot \left(c \cdot x\right)\right) \cdot s} \]

      7. *-commutative 88.5%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(\color{blue}{\left(s \cdot x\right)} \cdot c\right) \cdot \left(c \cdot x\right)\right) \cdot s} \]

      8. associate-*l* 95.7%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(s \cdot \left(x \cdot c\right)\right)} \cdot \left(c \cdot x\right)\right) \cdot s} \]

      9. *-commutative 95.7%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(s \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot c\right)}\right) \cdot s} \]

   7. Applied egg-rr 95.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(s \cdot x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;{\left(c \cdot \left(s\_m \cdot x\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{\left(x\_m \cdot c\right) \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (if (<= x_m 2.3e-12)
   (pow (* c (* s_m x_m)) -2.0)
   (/ (cos (* x_m 2.0)) (* (* x_m c) (* s_m (* s_m (* x_m c)))))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 2.3e-12) {
		tmp = pow((c * (s_m * x_m)), -2.0);
	} else {
		tmp = cos((x_m * 2.0)) / ((x_m * c) * (s_m * (s_m * (x_m * c))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 2.3d-12) then
        tmp = (c * (s_m * x_m)) ** (-2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / ((x_m * c) * (s_m * (s_m * (x_m * c))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double tmp;
	if (x_m <= 2.3e-12) {
		tmp = Math.pow((c * (s_m * x_m)), -2.0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / ((x_m * c) * (s_m * (s_m * (x_m * c))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	tmp = 0
	if x_m <= 2.3e-12:
		tmp = math.pow((c * (s_m * x_m)), -2.0)
	else:
		tmp = math.cos((x_m * 2.0)) / ((x_m * c) * (s_m * (s_m * (x_m * c))))
	return tmp

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	tmp = 0.0
	if (x_m <= 2.3e-12)
		tmp = Float64(c * Float64(s_m * x_m)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(Float64(x_m * c) * Float64(s_m * Float64(s_m * Float64(x_m * c)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp_2 = code(x_m, c, s_m)
	tmp = 0.0;
	if (x_m <= 2.3e-12)
		tmp = (c * (s_m * x_m)) ^ -2.0;
	else
		tmp = cos((x_m * 2.0)) / ((x_m * c) * (s_m * (s_m * (x_m * c))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := If[LessEqual[x$95$m, 2.3e-12], N[Power[N[(c * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c), $MachinePrecision] * N[(s$95$m * N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.29999999999999989e-12

   1. Initial program 64.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 64.2%

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

      2. *-commutative 64.2%

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

      3. unpow2 64.2%

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

      4. sqr-neg 64.2%

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

      5. unpow2 64.2%

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

      6. cos-neg 64.2%

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

      7. *-commutative 64.2%

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

      8. distribute-rgt-neg-in 64.2%

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

      9. metadata-eval 64.2%

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

      10. unpow2 64.2%

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

      11. sqr-neg 64.2%

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

      12. unpow2 64.2%

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

      13. associate-*r* 61.5%

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

      14. unpow2 61.5%

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

      15. *-commutative 61.5%

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

   3. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.9%

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

      1. associate-/r* 57.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 57.4%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 57.4%

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

      4. unpow2 57.4%

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

      5. swap-sqr 68.6%

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

      6. unpow2 68.6%

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

      7. associate-/r* 69.2%

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

      8. unpow2 69.2%

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

      9. unpow2 69.2%

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

      10. swap-sqr 85.3%

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

      11. unpow2 85.3%

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

   7. Simplified 85.3%

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

      1. *-un-lft-identity 85.3%

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

      2. pow-flip 85.8%

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

      3. *-commutative 85.8%

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

      4. *-commutative 85.8%

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

      5. associate-*l* 85.5%

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

      6. metadata-eval 85.5%

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

   9. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{1 \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}} \]
   10. Step-by-step derivation

       1. *-lft-identity 85.5%

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

       2. associate-*r* 85.8%

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

       3. *-commutative 85.8%

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

       4. *-commutative 85.8%

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

       5. *-commutative 85.8%

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

   11. Simplified 85.8%

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

   if 2.29999999999999989e-12 < x

   1. Initial program 67.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 62.0%

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

      1. associate-/r* 62.0%

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

      2. *-commutative 62.0%

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

      3. unpow2 62.0%

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

      4. unpow2 62.0%

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

      5. swap-sqr 75.0%

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

      6. unpow2 75.0%

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

      7. associate-/r* 75.1%

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

      8. *-commutative 75.1%

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

      9. unpow2 75.1%

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

      10. unpow2 75.1%

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

      11. swap-sqr 91.4%

          \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      12. unpow2 91.4%

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

      13. *-commutative 91.4%

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

      14. associate-*l* 98.5%

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

   5. Simplified 98.5%

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

      1. associate-*r* 91.4%

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

      2. *-commutative 91.4%

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

      3. unpow2 91.4%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

      4. associate-*r* 91.5%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right)} \cdot \left(c \cdot \left(x \cdot s\right)\right)} \]

      5. associate-*l* 91.4%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}} \]

      6. *-commutative 91.4%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot c\right)} \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]

      7. *-commutative 91.4%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot c\right)}\right)} \]

      8. *-commutative 91.4%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot c\right)\right)} \]

      9. associate-*l* 95.7%

         \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}\right)} \]

   7. Applied egg-rr 95.7%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \frac{\frac{\frac{1}{s\_m}}{x\_m \cdot c} \cdot \cos \left(x\_m \cdot 2\right)}{s\_m \cdot \left(x\_m \cdot c\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (/ (* (/ (/ 1.0 s_m) (* x_m c)) (cos (* x_m 2.0))) (* s_m (* x_m c))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	return (((1.0 / s_m) / (x_m * c)) * cos((x_m * 2.0))) / (s_m * (x_m * c));
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = (((1.0d0 / s_m) / (x_m * c)) * cos((x_m * 2.0d0))) / (s_m * (x_m * c))
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	return (((1.0 / s_m) / (x_m * c)) * Math.cos((x_m * 2.0))) / (s_m * (x_m * c));
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	return (((1.0 / s_m) / (x_m * c)) * math.cos((x_m * 2.0))) / (s_m * (x_m * c))

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	return Float64(Float64(Float64(Float64(1.0 / s_m) / Float64(x_m * c)) * cos(Float64(x_m * 2.0))) / Float64(s_m * Float64(x_m * c)))
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	tmp = (((1.0 / s_m) / (x_m * c)) * cos((x_m * 2.0))) / (s_m * (x_m * c));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := N[(N[(N[(N[(1.0 / s$95$m), $MachinePrecision] / N[(x$95$m * c), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.0%

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

   1. associate-/r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. associate-/r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

   10. unpow2 74.6%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   11. swap-sqr 95.0%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   12. unpow2 95.0%

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

   13. *-commutative 95.0%

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

   14. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 95.0%

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

   3. *-commutative 95.0%

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

   4. div-inv 95.0%

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

   5. *-commutative 95.0%

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

   6. pow-flip 95.5%

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

   7. *-commutative 95.5%

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

   8. *-commutative 95.5%

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

   9. associate-*l* 97.5%

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

   10. metadata-eval 97.5%

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

   11. *-commutative 97.5%

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

7. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)} \]
8. Step-by-step derivation

   1. metadata-eval 97.5%

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

   2. pow-flip 96.9%

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

   3. unpow2 96.9%

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

   4. associate-/r* 97.6%

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

   5. associate-*l/ 97.6%

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

   6. associate-/r* 97.6%

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

9. Applied egg-rr 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{s}}{x \cdot c} \cdot \cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}} \]
10. Add Preprocessing

Alternative 5: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\right)\\ \frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c)))) (/ (/ (cos (* x_m 2.0)) t_0) t_0)))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double t_0 = s_m * (x_m * c);
	return (cos((x_m * 2.0)) / t_0) / t_0;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = s_m * (x_m * c)
    code = (cos((x_m * 2.0d0)) / t_0) / t_0
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double t_0 = s_m * (x_m * c);
	return (Math.cos((x_m * 2.0)) / t_0) / t_0;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	t_0 = s_m * (x_m * c)
	return (math.cos((x_m * 2.0)) / t_0) / t_0

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	t_0 = Float64(s_m * Float64(x_m * c))
	return Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) / t_0)
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	t_0 = s_m * (x_m * c);
	tmp = (cos((x_m * 2.0)) / t_0) / t_0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.0%

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

   1. associate-/r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. associate-/r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

   10. unpow2 74.6%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   11. swap-sqr 95.0%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   12. unpow2 95.0%

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

   13. *-commutative 95.0%

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

   14. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 95.0%

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

   3. *-commutative 95.0%

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

   4. div-inv 95.0%

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

   5. *-commutative 95.0%

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

   6. pow-flip 95.5%

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

   7. *-commutative 95.5%

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

   8. *-commutative 95.5%

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

   9. associate-*l* 97.5%

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

   10. metadata-eval 97.5%

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

   11. *-commutative 97.5%

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

7. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)} \]
8. Step-by-step derivation

   1. metadata-eval 97.5%

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

   2. pow-pow 97.5%

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

   3. inv-pow 97.5%

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

   4. pow2 97.5%

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

   5. frac-times 96.9%

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

   6. metadata-eval 96.9%

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

   7. unpow2 96.9%

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

   8. associate-*l/ 96.9%

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

   9. unpow2 96.9%

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

   10. *-un-lft-identity 96.9%

       \[\leadsto \frac{\color{blue}{\cos \left(x \cdot 2\right)}}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \]

   11. associate-/r* 97.5%

       \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}} \]

9. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}} \]
10. Add Preprocessing

Alternative 6: 79.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ {\left(c \cdot \left(s\_m \cdot x\_m\right)\right)}^{-2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m) :precision binary64 (pow (* c (* s_m x_m)) -2.0))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	return pow((c * (s_m * x_m)), -2.0);
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = (c * (s_m * x_m)) ** (-2.0d0)
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	return Math.pow((c * (s_m * x_m)), -2.0);
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	return math.pow((c * (s_m * x_m)), -2.0)

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	return Float64(c * Float64(s_m * x_m)) ^ -2.0
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	tmp = (c * (s_m * x_m)) ^ -2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := N[Power[N[(c * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 65.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-/r* 65.1%

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

   2. *-commutative 65.1%

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

   3. unpow2 65.1%

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

   4. sqr-neg 65.1%

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

   5. unpow2 65.1%

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

   6. cos-neg 65.1%

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

   7. *-commutative 65.1%

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

   8. distribute-rgt-neg-in 65.1%

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

   9. metadata-eval 65.1%

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

   10. unpow2 65.1%

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

   11. sqr-neg 65.1%

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

   12. unpow2 65.1%

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

   13. associate-*r* 61.6%

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

   14. unpow2 61.6%

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

   15. *-commutative 61.6%

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

3. Simplified 61.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 57.7%

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

   1. associate-/r* 57.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 57.3%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 57.3%

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

   4. unpow2 57.3%

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

   5. swap-sqr 66.3%

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

   6. unpow2 66.3%

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

   7. associate-/r* 66.7%

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

   8. unpow2 66.7%

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

   9. unpow2 66.7%

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

   10. swap-sqr 80.9%

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

   11. unpow2 80.9%

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

7. Simplified 80.9%

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

   1. *-un-lft-identity 80.9%

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

   2. pow-flip 81.2%

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

   3. *-commutative 81.2%

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

   4. *-commutative 81.2%

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

   5. associate-*l* 81.2%

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

   6. metadata-eval 81.2%

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

9. Applied egg-rr 81.2%

   \[\leadsto \color{blue}{1 \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2}} \]
10. Step-by-step derivation

    1. *-lft-identity 81.2%

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

    2. associate-*r* 81.2%

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

    3. *-commutative 81.2%

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

    4. *-commutative 81.2%

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

    5. *-commutative 81.2%

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

11. Simplified 81.2%

    \[\leadsto \color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{-2}} \]
12. Add Preprocessing

Alternative 7: 78.8% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \frac{\frac{\frac{1}{s\_m}}{x\_m \cdot c}}{s\_m \cdot \left(x\_m \cdot c\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (/ (/ (/ 1.0 s_m) (* x_m c)) (* s_m (* x_m c))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	return ((1.0 / s_m) / (x_m * c)) / (s_m * (x_m * c));
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = ((1.0d0 / s_m) / (x_m * c)) / (s_m * (x_m * c))
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	return ((1.0 / s_m) / (x_m * c)) / (s_m * (x_m * c));
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	return ((1.0 / s_m) / (x_m * c)) / (s_m * (x_m * c))

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	return Float64(Float64(Float64(1.0 / s_m) / Float64(x_m * c)) / Float64(s_m * Float64(x_m * c)))
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	tmp = ((1.0 / s_m) / (x_m * c)) / (s_m * (x_m * c));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := N[(N[(N[(1.0 / s$95$m), $MachinePrecision] / N[(x$95$m * c), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.0%

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

   1. associate-/r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. associate-/r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

   10. unpow2 74.6%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   11. swap-sqr 95.0%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   12. unpow2 95.0%

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

   13. *-commutative 95.0%

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

   14. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 95.0%

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

   3. *-commutative 95.0%

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

   4. div-inv 95.0%

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

   5. *-commutative 95.0%

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

   6. pow-flip 95.5%

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

   7. *-commutative 95.5%

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

   8. *-commutative 95.5%

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

   9. associate-*l* 97.5%

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

   10. metadata-eval 97.5%

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

   11. *-commutative 97.5%

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

7. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)} \]
8. Step-by-step derivation

   1. metadata-eval 97.5%

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

   2. pow-flip 96.9%

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

   3. unpow2 96.9%

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

   4. associate-/r* 97.6%

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

   5. associate-*l/ 97.6%

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

   6. associate-/r* 97.6%

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

9. Applied egg-rr 97.6%

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

10. Taylor expanded in x around 0 80.7%

    \[\leadsto \frac{\color{blue}{\frac{1}{c \cdot \left(s \cdot x\right)}}}{s \cdot \left(x \cdot c\right)} \]
11. Step-by-step derivation

    1. *-commutative 80.7%

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

    2. associate-*r* 81.1%

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

    3. associate-/r* 81.2%

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

12. Simplified 81.2%

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{s}}{x \cdot c}}}{s \cdot \left(x \cdot c\right)} \]
13. Add Preprocessing

Alternative 8: 78.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \frac{\frac{1}{c \cdot \left(s\_m \cdot x\_m\right)}}{s\_m \cdot \left(x\_m \cdot c\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (/ (/ 1.0 (* c (* s_m x_m))) (* s_m (* x_m c))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	return (1.0 / (c * (s_m * x_m))) / (s_m * (x_m * c));
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c * (s_m * x_m))) / (s_m * (x_m * c))
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	return (1.0 / (c * (s_m * x_m))) / (s_m * (x_m * c));
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	return (1.0 / (c * (s_m * x_m))) / (s_m * (x_m * c))

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	return Float64(Float64(1.0 / Float64(c * Float64(s_m * x_m))) / Float64(s_m * Float64(x_m * c)))
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	tmp = (1.0 / (c * (s_m * x_m))) / (s_m * (x_m * c));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := N[(N[(1.0 / N[(c * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.0%

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

   1. associate-/r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. associate-/r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

   10. unpow2 74.6%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   11. swap-sqr 95.0%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   12. unpow2 95.0%

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

   13. *-commutative 95.0%

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

   14. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*r* 95.0%

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

   3. *-commutative 95.0%

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

   4. div-inv 95.0%

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

   5. *-commutative 95.0%

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

   6. pow-flip 95.5%

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

   7. *-commutative 95.5%

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

   8. *-commutative 95.5%

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

   9. associate-*l* 97.5%

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

   10. metadata-eval 97.5%

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

   11. *-commutative 97.5%

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

7. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)} \]
8. Step-by-step derivation

   1. metadata-eval 97.5%

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

   2. pow-flip 96.9%

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

   3. unpow2 96.9%

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

   4. associate-/r* 97.6%

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

   5. associate-*l/ 97.6%

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

   6. associate-/r* 97.6%

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

9. Applied egg-rr 97.6%

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

10. Taylor expanded in x around 0 80.7%

    \[\leadsto \frac{\color{blue}{\frac{1}{c \cdot \left(s \cdot x\right)}}}{s \cdot \left(x \cdot c\right)} \]
11. Step-by-step derivation

    1. *-commutative 80.7%

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

12. Simplified 80.7%

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

13. Final simplification 80.7%

    \[\leadsto \frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{s \cdot \left(x \cdot c\right)} \]
14. Add Preprocessing

Alternative 9: 76.5% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \frac{1}{s\_m \cdot \left(\left(x\_m \cdot c\right) \cdot \left(s\_m \cdot \left(x\_m \cdot c\right)\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (/ 1.0 (* s_m (* (* x_m c) (* s_m (* x_m c))))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	return 1.0 / (s_m * ((x_m * c) * (s_m * (x_m * c))));
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = 1.0d0 / (s_m * ((x_m * c) * (s_m * (x_m * c))))
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	return 1.0 / (s_m * ((x_m * c) * (s_m * (x_m * c))));
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	return 1.0 / (s_m * ((x_m * c) * (s_m * (x_m * c))))

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	return Float64(1.0 / Float64(s_m * Float64(Float64(x_m * c) * Float64(s_m * Float64(x_m * c)))))
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp = code(x_m, c, s_m)
	tmp = 1.0 / (s_m * ((x_m * c) * (s_m * (x_m * c))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := N[(1.0 / N[(s$95$m * N[(N[(x$95$m * c), $MachinePrecision] * N[(s$95$m * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 62.0%

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

   1. associate-/r* 61.6%

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

   2. *-commutative 61.6%

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

   3. unpow2 61.6%

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

   4. unpow2 61.6%

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

   5. swap-sqr 74.1%

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

   6. unpow2 74.1%

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

   7. associate-/r* 74.6%

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

   8. *-commutative 74.6%

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

   9. unpow2 74.6%

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

   10. unpow2 74.6%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}} \]

   11. swap-sqr 95.0%

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   12. unpow2 95.0%

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

   13. *-commutative 95.0%

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

   14. associate-*l* 96.7%

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

5. Simplified 96.7%

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

   1. associate-*r* 95.0%

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

   2. *-commutative 95.0%

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

   3. unpow2 95.0%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   4. associate-*r* 93.7%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}} \]

   5. associate-*r* 90.3%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot x\right)\right) \cdot s}} \]

   6. *-commutative 90.3%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(\left(x \cdot s\right) \cdot c\right)} \cdot \left(c \cdot x\right)\right) \cdot s} \]

   7. *-commutative 90.3%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(\color{blue}{\left(s \cdot x\right)} \cdot c\right) \cdot \left(c \cdot x\right)\right) \cdot s} \]

   8. associate-*l* 93.4%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(s \cdot \left(x \cdot c\right)\right)} \cdot \left(c \cdot x\right)\right) \cdot s} \]

   9. *-commutative 93.4%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(s \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot c\right)}\right) \cdot s} \]

7. Applied egg-rr 93.4%

   \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(x \cdot c\right)\right) \cdot s}} \]

8. Taylor expanded in x around 0 79.0%

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

9. Final simplification 79.0%

   \[\leadsto \frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))