mixedcos

Percentage Accurate: 66.8% → 97.0%

Time: 15.9s

Alternatives: 10

Speedup: 24.1×

• 32.0% 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: 66.8% 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: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{t_0}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s \cdot c}}{s \cdot c} \cdot \frac{t_0}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= x 2.7e+60)
     (/ (/ (/ t_0 (* x s)) c) (* c (* x s)))
     (* (/ (/ (/ 1.0 x) (* s c)) (* s c)) (/ t_0 x)))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (x <= 2.7e+60) {
		tmp = ((t_0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = (((1.0 / x) / (s * c)) / (s * c)) * (t_0 / x);
	}
	return tmp;
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (x <= 2.7d+60) then
        tmp = ((t_0 / (x * s)) / c) / (c * (x * s))
    else
        tmp = (((1.0d0 / x) / (s * c)) / (s * c)) * (t_0 / x)
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 2.7e+60) {
		tmp = ((t_0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = (((1.0 / x) / (s * c)) / (s * c)) * (t_0 / x);
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 2.7e+60:
		tmp = ((t_0 / (x * s)) / c) / (c * (x * s))
	else:
		tmp = (((1.0 / x) / (s * c)) / (s * c)) * (t_0 / x)
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 2.7e+60)
		tmp = Float64(Float64(Float64(t_0 / Float64(x * s)) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / x) / Float64(s * c)) / Float64(s * c)) * Float64(t_0 / x));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 2.7e+60)
		tmp = ((t_0 / (x * s)) / c) / (c * (x * s));
	else
		tmp = (((1.0 / x) / (s * c)) / (s * c)) * (t_0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.7e+60], N[(N[(N[(t$95$0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / x), $MachinePrecision] / N[(s * c), $MachinePrecision]), $MachinePrecision] / N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6999999999999999e60

   1. Initial program 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. add-sqr-sqrt 72.5%

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

      3. times-frac 72.5%

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

      4. sqrt-prod 72.5%

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

      5. unpow2 72.5%

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

      6. sqrt-prod 41.6%

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

      7. add-sqr-sqrt 51.2%

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

      8. *-commutative 51.2%

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

      9. associate-*r* 46.1%

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

      10. unpow2 46.1%

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

      11. pow-prod-down 51.2%

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

      12. sqrt-prod 51.2%

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

   3. Applied egg-rr 89.8%

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

      1. associate-*l/ 89.9%

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

      2. *-lft-identity 89.9%

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

      3. unpow2 89.9%

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

      4. rem-sqrt-square 89.9%

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

      5. *-commutative 89.9%

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

      6. unpow2 89.9%

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

      7. rem-sqrt-square 98.7%

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

      8. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around inf 80.3%

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

      1. associate-/r* 80.6%

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

      2. *-commutative 80.6%

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

      3. unpow2 80.6%

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

      4. sqr-abs 80.6%

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

      5. unpow2 80.6%

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

      6. associate-/l/ 80.3%

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

      7. *-commutative 80.3%

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

      8. unpow2 80.3%

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

      9. unpow2 80.3%

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

      10. swap-sqr 98.4%

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

      11. associate-/l/ 98.7%

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

      12. *-rgt-identity 98.7%

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

      13. associate-*r/ 98.7%

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

      14. associate-*r/ 98.7%

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

      15. *-rgt-identity 98.7%

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

      16. associate-*r/ 98.7%

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

   8. Simplified 98.3%

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

      1. metadata-eval 98.3%

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

      2. pow-flip 98.0%

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

      3. unpow2 98.0%

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

      4. associate-/r* 98.3%

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

      5. associate-*r/ 98.3%

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

      6. div-inv 98.3%

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

      7. associate-*r* 97.4%

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

      8. *-commutative 97.4%

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

      9. associate-/r* 97.1%

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

      10. associate-*r* 98.4%

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

   10. Applied egg-rr 98.4%

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

   if 2.6999999999999999e60 < x

   1. Initial program 72.4%

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

      1. *-un-lft-identity 72.4%

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

      2. associate-*r* 72.3%

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

      3. times-frac 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r* 72.5%

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

      6. *-commutative 72.5%

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

      7. pow-prod-down 92.6%

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

   3. Applied egg-rr 92.6%

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

      1. unpow2 92.6%

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

      2. *-commutative 92.6%

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

      3. *-commutative 92.6%

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

   5. Applied egg-rr 92.6%

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

      1. inv-pow 92.6%

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

      2. associate-*l* 97.9%

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

      3. unpow-prod-down 98.0%

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

      4. inv-pow 98.0%

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

      5. associate-*r* 88.3%

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

   7. Applied egg-rr 88.3%

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

      1. unpow-1 88.3%

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

      2. associate-*l/ 88.4%

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

      3. *-lft-identity 88.4%

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

      4. associate-*r* 98.1%

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

      5. associate-/l/ 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s \cdot c}}{s \cdot c} \cdot \frac{\cos \left(x \cdot 2\right)}{x}\\ \end{array} \]

Alternative 2: 97.2% accurate, 2.7× speedup

Math

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

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (* (/ (cos (* x 2.0)) (* x (* s c))) (/ (/ -1.0 x) (* s (- c)))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return (cos((x * 2.0)) / (x * (s * c))) * ((-1.0 / x) / (s * -c));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 / x), $MachinePrecision] / N[(s * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. add-sqr-sqrt 72.5%

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

   3. times-frac 72.5%

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

   4. sqrt-prod 72.5%

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

   5. unpow2 72.5%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-*r* 48.5%

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

   10. unpow2 48.5%

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

   11. pow-prod-down 53.4%

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

   12. sqrt-prod 53.4%

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

3. Applied egg-rr 89.5%

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

   1. associate-*l/ 89.5%

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

   2. *-lft-identity 89.5%

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

   3. unpow2 89.5%

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

   4. rem-sqrt-square 89.5%

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

   5. *-commutative 89.5%

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

   6. unpow2 89.5%

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

   7. rem-sqrt-square 96.9%

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

   8. *-commutative 96.9%

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

5. Simplified 96.9%

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

   1. frac-2neg 96.9%

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

   2. div-inv 96.9%

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

   3. associate-/r* 96.9%

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

   4. *-commutative 96.9%

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

   5. add-sqr-sqrt 55.3%

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

   6. fabs-sqr 55.3%

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

   7. add-sqr-sqrt 67.0%

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

   8. *-commutative 67.0%

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

   9. distribute-rgt-neg-in 67.0%

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

7. Applied egg-rr 96.9%

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

   1. *-commutative 96.9%

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

   2. distribute-rgt-neg-out 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r* 95.5%

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

   5. associate-/l/ 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 98.3%

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

9. Simplified 98.3%

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

    1. inv-pow 98.3%

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

    2. distribute-lft-neg-in 98.3%

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

    3. unpow-prod-down 98.2%

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

    4. distribute-rgt-neg-in 98.2%

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

    5. inv-pow 98.2%

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

11. Applied egg-rr 98.2%

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

    1. *-commutative 98.2%

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

    2. unpow-1 98.2%

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

    3. associate-*r/ 98.3%

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

    4. *-rgt-identity 98.3%

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

    5. distribute-rgt-neg-out 98.3%

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

    6. *-commutative 98.3%

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

    7. distribute-rgt-neg-in 98.3%

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

13. Simplified 98.3%

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

14. Final simplification 98.3%

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

Alternative 3: 97.2% accurate, 2.7× speedup

Math

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

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (* (/ (cos (* x 2.0)) (* x (* s c))) (/ -1.0 (* x (* s (- c))))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return (cos((x * 2.0)) / (x * (s * c))) * (-1.0 / (x * (s * -c)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(x * N[(s * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. add-sqr-sqrt 72.5%

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

   3. times-frac 72.5%

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

   4. sqrt-prod 72.5%

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

   5. unpow2 72.5%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-*r* 48.5%

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

   10. unpow2 48.5%

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

   11. pow-prod-down 53.4%

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

   12. sqrt-prod 53.4%

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

3. Applied egg-rr 89.5%

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

   1. associate-*l/ 89.5%

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

   2. *-lft-identity 89.5%

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

   3. unpow2 89.5%

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

   4. rem-sqrt-square 89.5%

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

   5. *-commutative 89.5%

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

   6. unpow2 89.5%

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

   7. rem-sqrt-square 96.9%

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

   8. *-commutative 96.9%

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

5. Simplified 96.9%

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

   1. frac-2neg 96.9%

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

   2. div-inv 96.9%

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

   3. associate-/r* 96.9%

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

   4. *-commutative 96.9%

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

   5. add-sqr-sqrt 55.3%

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

   6. fabs-sqr 55.3%

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

   7. add-sqr-sqrt 67.0%

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

   8. *-commutative 67.0%

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

   9. distribute-rgt-neg-in 67.0%

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

7. Applied egg-rr 96.9%

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

   1. *-commutative 96.9%

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

   2. distribute-rgt-neg-out 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r* 95.5%

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

   5. associate-/l/ 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 98.3%

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

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 4: 87.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot c}}{x \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= x 4e-152)
     (/ (/ 1.0 t_0) t_0)
     (/ (/ (cos (* x 2.0)) (* s c)) (* x t_0)))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 4e-152) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (cos((x * 2.0)) / (s * c)) / (x * t_0);
	}
	return tmp;
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s)
    if (x <= 4d-152) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = (cos((x * 2.0d0)) / (s * c)) / (x * t_0)
    end if
    code = tmp
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 4e-152) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (Math.cos((x * 2.0)) / (s * c)) / (x * t_0);
	}
	return tmp;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 4e-152:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = (math.cos((x * 2.0)) / (s * c)) / (x * t_0)
	return tmp

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 4e-152)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / Float64(s * c)) / Float64(x * t_0));
	end
	return tmp
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (x <= 4e-152)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = (cos((x * 2.0)) / (s * c)) / (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e-152], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s * c), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000026e-152

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0 58.5%

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

      1. *-commutative 58.5%

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

      2. unpow2 58.5%

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

      3. unpow2 58.5%

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

      4. swap-sqr 73.3%

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

      5. unpow2 73.3%

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

      6. unpow2 73.3%

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

      7. rem-square-sqrt 73.3%

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

      8. swap-sqr 82.1%

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

      9. unpow2 82.1%

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

      10. unpow2 82.1%

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

      11. rem-sqrt-square 88.5%

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

      12. *-commutative 88.5%

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

   4. Simplified 88.5%

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

      1. add-sqr-sqrt 66.5%

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

      2. sqrt-prod 88.5%

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

      3. unpow2 88.5%

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

      4. pow2 88.5%

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

      5. add-sqr-sqrt 88.5%

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

      6. pow-flip 88.5%

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

      7. add-sqr-sqrt 53.3%

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

      8. fabs-sqr 53.3%

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

      9. add-sqr-sqrt 88.5%

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

      10. metadata-eval 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. metadata-eval 88.5%

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

      2. pow-prod-up 88.5%

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

      3. pow-prod-down 88.5%

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

      4. sqr-neg 88.5%

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

      5. associate-*r* 87.3%

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

      6. associate-*r* 87.9%

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

      7. pow-prod-down 87.9%

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

      8. inv-pow 87.9%

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

      9. inv-pow 87.9%

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

      10. un-div-inv 87.9%

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

   8. Applied egg-rr 88.5%

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

   if 4.00000000000000026e-152 < x

   1. Initial program 75.6%

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

      1. *-un-lft-identity 75.6%

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

      2. add-sqr-sqrt 75.6%

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

      3. times-frac 75.6%

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

      4. sqrt-prod 75.6%

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

      5. unpow2 75.6%

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

      6. sqrt-prod 40.9%

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

      7. add-sqr-sqrt 56.5%

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

      8. *-commutative 56.5%

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

      9. associate-*r* 54.5%

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

      10. unpow2 54.5%

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

      11. pow-prod-down 56.5%

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

      12. sqrt-prod 56.5%

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

   3. Applied egg-rr 88.2%

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

      1. associate-*l/ 88.2%

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

      2. *-lft-identity 88.2%

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

      3. unpow2 88.2%

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

      4. rem-sqrt-square 88.2%

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

      5. *-commutative 88.2%

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

      6. unpow2 88.2%

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

      7. rem-sqrt-square 93.6%

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

      8. *-commutative 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in x around inf 83.0%

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

      1. associate-/r* 83.0%

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

      2. *-commutative 83.0%

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

      3. unpow2 83.0%

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

      4. sqr-abs 83.0%

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

      5. unpow2 83.0%

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

      6. associate-/l/ 83.0%

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

      7. *-commutative 83.0%

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

      8. unpow2 83.0%

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

      9. unpow2 83.0%

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

      10. swap-sqr 93.1%

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

      11. associate-/l/ 93.6%

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

      12. *-rgt-identity 93.6%

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

      13. associate-*r/ 93.7%

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

      14. associate-*r/ 93.6%

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

      15. *-rgt-identity 93.6%

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

      16. associate-*r/ 93.7%

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

   8. Simplified 98.7%

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

      1. metadata-eval 98.7%

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

      2. pow-flip 97.5%

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

      3. unpow2 97.5%

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

      4. associate-/r* 98.7%

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

      5. associate-*r/ 98.7%

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

      6. div-inv 98.7%

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

      7. add-sqr-sqrt 68.9%

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

      8. sqrt-unprod 76.6%

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

      9. sqr-neg 76.6%

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

      10. sqrt-unprod 30.7%

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

      11. add-sqr-sqrt 38.0%

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

      12. remove-double-neg 38.0%

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

      13. distribute-rgt-neg-out 38.0%

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

   10. Applied egg-rr 90.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot c}}{x \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]

Alternative 5: 97.2% accurate, 2.7× speedup

Math

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

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (/ (- (/ (cos (* x 2.0)) (* x (* s c)))) (* x (* s (- c)))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return -(cos((x * 2.0)) / (x * (s * c))) / (x * (s * -c));
}

Fortran

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

Java

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

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	return -(math.cos((x * 2.0)) / (x * (s * c))) / (x * (s * -c))

Julia

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

MATLAB

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

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[((-N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / N[(x * N[(s * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. add-sqr-sqrt 72.5%

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

   3. times-frac 72.5%

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

   4. sqrt-prod 72.5%

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

   5. unpow2 72.5%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-*r* 48.5%

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

   10. unpow2 48.5%

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

   11. pow-prod-down 53.4%

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

   12. sqrt-prod 53.4%

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

3. Applied egg-rr 89.5%

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

   1. associate-*l/ 89.5%

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

   2. *-lft-identity 89.5%

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

   3. unpow2 89.5%

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

   4. rem-sqrt-square 89.5%

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

   5. *-commutative 89.5%

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

   6. unpow2 89.5%

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

   7. rem-sqrt-square 96.9%

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

   8. *-commutative 96.9%

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

5. Simplified 96.9%

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

   1. frac-2neg 96.9%

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

   2. div-inv 96.9%

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

   3. associate-/r* 96.9%

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

   4. *-commutative 96.9%

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

   5. add-sqr-sqrt 55.3%

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

   6. fabs-sqr 55.3%

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

   7. add-sqr-sqrt 67.0%

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

   8. *-commutative 67.0%

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

   9. distribute-rgt-neg-in 67.0%

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

7. Applied egg-rr 96.9%

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

   1. associate-*r/ 96.9%

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

   2. *-rgt-identity 96.9%

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

   3. associate-/l/ 96.9%

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

   4. *-commutative 96.9%

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

   5. associate-*r* 95.5%

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

   6. distribute-rgt-neg-out 95.5%

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

   7. *-commutative 95.5%

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

   8. associate-*r* 98.2%

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

   9. distribute-rgt-neg-in 98.2%

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

9. Simplified 98.2%

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

10. Final simplification 98.2%

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

Alternative 6: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. add-sqr-sqrt 72.5%

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

   3. times-frac 72.5%

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

   4. sqrt-prod 72.5%

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

   5. unpow2 72.5%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-*r* 48.5%

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

   10. unpow2 48.5%

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

   11. pow-prod-down 53.4%

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

   12. sqrt-prod 53.4%

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

3. Applied egg-rr 89.5%

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

   1. associate-*l/ 89.5%

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

   2. *-lft-identity 89.5%

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

   3. unpow2 89.5%

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

   4. rem-sqrt-square 89.5%

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

   5. *-commutative 89.5%

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

   6. unpow2 89.5%

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

   7. rem-sqrt-square 96.9%

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

   8. *-commutative 96.9%

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

5. Simplified 96.9%

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

   1. frac-2neg 96.9%

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

   2. div-inv 96.9%

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

   3. associate-/r* 96.9%

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

   4. *-commutative 96.9%

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

   5. add-sqr-sqrt 55.3%

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

   6. fabs-sqr 55.3%

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

   7. add-sqr-sqrt 67.0%

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

   8. *-commutative 67.0%

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

   9. distribute-rgt-neg-in 67.0%

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

7. Applied egg-rr 96.9%

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

   1. *-commutative 96.9%

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

   2. distribute-rgt-neg-out 96.9%

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

   3. *-commutative 96.9%

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

   4. associate-*r* 95.5%

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

   5. associate-/l/ 95.5%

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

   6. *-commutative 95.5%

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

   7. associate-*r* 98.3%

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

9. Simplified 98.3%

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

    1. add-sqr-sqrt 55.8%

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

    2. sqrt-unprod 63.7%

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

    3. sqr-neg 63.7%

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

    4. sqrt-unprod 27.8%

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

    5. add-sqr-sqrt 33.7%

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

    6. associate-/r* 33.7%

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

    7. frac-times 33.8%

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

    8. /-rgt-identity 33.8%

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

11. Applied egg-rr 96.9%

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

12. Final simplification 96.9%

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

Alternative 7: 97.3% accurate, 2.7× speedup

Math

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

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (/ (/ (/ (cos (* x 2.0)) (* x s)) c) (* c (* x s))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return ((cos((x * 2.0)) / (x * s)) / c) / (c * (x * s));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. add-sqr-sqrt 72.5%

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

   3. times-frac 72.5%

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

   4. sqrt-prod 72.5%

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

   5. unpow2 72.5%

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

   6. sqrt-prod 41.8%

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

   7. add-sqr-sqrt 53.4%

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

   8. *-commutative 53.4%

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

   9. associate-*r* 48.5%

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

   10. unpow2 48.5%

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

   11. pow-prod-down 53.4%

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

   12. sqrt-prod 53.4%

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

3. Applied egg-rr 89.5%

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

   1. associate-*l/ 89.5%

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

   2. *-lft-identity 89.5%

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

   3. unpow2 89.5%

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

   4. rem-sqrt-square 89.5%

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

   5. *-commutative 89.5%

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

   6. unpow2 89.5%

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

   7. rem-sqrt-square 96.9%

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

   8. *-commutative 96.9%

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

5. Simplified 96.9%

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

6. Taylor expanded in x around inf 81.2%

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

   1. associate-/r* 81.2%

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

   2. *-commutative 81.2%

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

   3. unpow2 81.2%

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

   4. sqr-abs 81.2%

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

   5. unpow2 81.2%

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

   6. associate-/l/ 81.2%

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

   7. *-commutative 81.2%

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

   8. unpow2 81.2%

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

   9. unpow2 81.2%

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

   10. swap-sqr 96.6%

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

   11. associate-/l/ 96.9%

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

   12. *-rgt-identity 96.9%

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

   13. associate-*r/ 96.9%

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

   14. associate-*r/ 96.9%

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

   15. *-rgt-identity 96.9%

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

   16. associate-*r/ 96.9%

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

8. Simplified 98.3%

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

   1. metadata-eval 98.3%

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

   2. pow-flip 97.7%

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

   3. unpow2 97.7%

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

   4. associate-/r* 98.2%

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

   5. associate-*r/ 98.2%

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

   6. div-inv 98.2%

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

   7. associate-*r* 95.5%

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

   8. *-commutative 95.5%

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

   9. associate-/r* 95.2%

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

   10. associate-*r* 96.6%

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

10. Applied egg-rr 96.6%

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

11. Final simplification 96.6%

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

Alternative 8: 79.6% accurate, 3.0× speedup

Math

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

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (pow (* c (* x s)) -2.0))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	return pow((c * (x * s)), -2.0);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 72.5%

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

2. Taylor expanded in x around 0 60.1%

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

   1. *-commutative 60.1%

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

   2. unpow2 60.1%

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

   3. unpow2 60.1%

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

   4. swap-sqr 72.4%

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

   5. unpow2 72.4%

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

   6. unpow2 72.4%

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

   7. rem-square-sqrt 72.4%

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

   8. swap-sqr 79.0%

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

   9. unpow2 79.0%

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

   10. unpow2 79.0%

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

   11. rem-sqrt-square 83.6%

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

   12. *-commutative 83.6%

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

4. Simplified 83.6%

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

   1. add-sqr-sqrt 57.5%

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

   2. sqrt-prod 83.6%

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

   3. unpow2 83.6%

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

   4. pow2 83.6%

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

   5. add-sqr-sqrt 83.6%

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

   6. pow-flip 83.9%

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

   7. add-sqr-sqrt 48.3%

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

   8. fabs-sqr 48.3%

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

   9. add-sqr-sqrt 83.9%

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

   10. metadata-eval 83.9%

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

6. Applied egg-rr 83.9%

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

7. Final simplification 83.9%

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

Alternative 9: 79.4% accurate, 24.1× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{1}{t_0 \cdot t_0} \end{array} \end{array} \]

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return 1.0 / (t_0 * t_0)

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.5%

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

2. Taylor expanded in x around 0 60.1%

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

   1. *-commutative 60.1%

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

   2. unpow2 60.1%

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

   3. unpow2 60.1%

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

   4. swap-sqr 72.4%

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

   5. unpow2 72.4%

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

   6. unpow2 72.4%

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

   7. rem-square-sqrt 72.4%

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

   8. swap-sqr 79.0%

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

   9. unpow2 79.0%

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

   10. unpow2 79.0%

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

   11. rem-sqrt-square 83.6%

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

   12. *-commutative 83.6%

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

4. Simplified 83.6%

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

   1. add-sqr-sqrt 57.5%

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

   2. sqrt-prod 83.6%

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

   3. unpow2 83.6%

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

   4. pow2 83.6%

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

   5. sqrt-pow1 62.9%

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

   6. metadata-eval 62.9%

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

   7. pow1 62.9%

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

   8. add-sqr-sqrt 34.8%

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

   9. fabs-sqr 34.8%

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

   10. add-sqr-sqrt 61.8%

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

   11. sqrt-pow1 62.5%

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

   12. metadata-eval 62.5%

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

   13. pow1 62.5%

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

   14. add-sqr-sqrt 42.4%

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

   15. fabs-sqr 42.4%

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

   16. add-sqr-sqrt 83.6%

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

6. Applied egg-rr 83.6%

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

7. Final simplification 83.6%

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

Alternative 10: 79.6% accurate, 24.1× speedup

Math

\[\begin{array}{l} s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\frac{1}{t_0}}{t_0} \end{array} \end{array} \]

FPCore

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ (/ 1.0 t_0) t_0)))

C

s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) / t_0;
}

Fortran

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = (1.0d0 / t_0) / t_0
end function

Java

s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) / t_0;
}

Python

s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return (1.0 / t_0) / t_0

Julia

s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 72.5%

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

2. Taylor expanded in x around 0 60.1%

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

   1. *-commutative 60.1%

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

   2. unpow2 60.1%

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

   3. unpow2 60.1%

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

   4. swap-sqr 72.4%

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

   5. unpow2 72.4%

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

   6. unpow2 72.4%

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

   7. rem-square-sqrt 72.4%

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

   8. swap-sqr 79.0%

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

   9. unpow2 79.0%

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

   10. unpow2 79.0%

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

   11. rem-sqrt-square 83.6%

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

   12. *-commutative 83.6%

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

4. Simplified 83.6%

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

   1. add-sqr-sqrt 57.5%

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

   2. sqrt-prod 83.6%

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

   3. unpow2 83.6%

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

   4. pow2 83.6%

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

   5. add-sqr-sqrt 83.6%

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

   6. pow-flip 83.9%

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

   7. add-sqr-sqrt 48.3%

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

   8. fabs-sqr 48.3%

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

   9. add-sqr-sqrt 83.9%

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

   10. metadata-eval 83.9%

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

6. Applied egg-rr 83.9%

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

   1. metadata-eval 83.9%

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

   2. pow-prod-up 83.8%

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

   3. pow-prod-down 83.6%

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

   4. sqr-neg 83.6%

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

   5. associate-*r* 82.8%

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

   6. associate-*r* 83.6%

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

   7. pow-prod-down 83.9%

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

   8. inv-pow 83.9%

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

   9. inv-pow 83.9%

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

   10. un-div-inv 83.9%

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

8. Applied egg-rr 83.9%

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

9. Final simplification 83.9%

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

Reproduce

herbie shell --seed 2023300 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))