mixedcos

Percentage Accurate: 66.5% → 99.3%

Time: 13.2s

Alternatives: 8

Speedup: 24.1×

• 31.4% 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: 66.5% 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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} c = |c|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;\frac{t_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1 \cdot \frac{-1}{x}}{c \cdot s}}{x \cdot \left(c \cdot \left(-s\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: c 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))) (t_1 (cos (* 2.0 x))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ (/ t_1 t_0) t_0)
     (/ (/ (* t_1 (/ -1.0 x)) (* c s)) (* x (* c (- s)))))))

C

c = abs(c);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = ((t_1 * (-1.0 / x)) / (c * s)) / (x * (c * -s));
	}
	return tmp;
}

Java

c = Math.abs(c);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((2.0 * x));
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = ((t_1 * (-1.0 / x)) / (c * s)) / (x * (c * -s));
	}
	return tmp;
}

Python

c = abs(c)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((2.0 * x))
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (t_1 / t_0) / t_0
	else:
		tmp = ((t_1 * (-1.0 / x)) / (c * s)) / (x * (c * -s))
	return tmp

Julia

c = abs(c)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	else
		tmp = Float64(Float64(Float64(t_1 * Float64(-1.0 / x)) / Float64(c * s)) / Float64(x * Float64(c * Float64(-s))));
	end
	return tmp
end

MATLAB

c = abs(c)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((2.0 * x));
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (t_1 / t_0) / t_0;
	else
		tmp = ((t_1 * (-1.0 / x)) / (c * s)) / (x * (c * -s));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c 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]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(t$95$1 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] / N[(x * N[(c * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x))) < +inf.0

   1. Initial program 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. add-sqr-sqrt 79.2%

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

      3. times-frac 79.2%

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

      4. sqrt-prod 79.2%

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

      5. unpow2 79.2%

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

      6. sqrt-prod 38.3%

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

      7. add-sqr-sqrt 58.9%

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

      8. *-commutative 58.9%

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

      9. associate-*r* 54.8%

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

      10. unpow2 54.8%

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

      11. pow-prod-down 59.0%

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

      12. sqrt-prod 58.9%

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

   3. Applied egg-rr 95.3%

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

      1. associate-*l/ 95.4%

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

      2. *-lft-identity 95.4%

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

      3. unpow2 95.4%

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

      4. rem-sqrt-square 95.4%

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

      5. *-commutative 95.4%

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

      6. unpow2 95.4%

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

      7. rem-sqrt-square 99.7%

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

      8. *-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. associate-/r* 96.9%

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

      2. add-sqr-sqrt 51.4%

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

      3. fabs-sqr 51.4%

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

      4. add-sqr-sqrt 68.2%

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

      5. associate-/l/ 69.5%

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

      6. div-inv 69.5%

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

      7. times-frac 68.2%

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

      8. *-commutative 68.2%

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

      9. add-sqr-sqrt 48.1%

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

      10. fabs-sqr 48.1%

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

      11. add-sqr-sqrt 96.9%

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

      12. associate-*r* 96.0%

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

   7. Applied egg-rr 96.0%

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

      1. frac-times 98.0%

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

      2. div-inv 98.0%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 98.0%

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

      6. *-commutative 98.0%

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

      7. *-commutative 98.0%

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

      8. associate-*l* 99.7%

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

   9. Applied egg-rr 99.7%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))

   1. Initial program 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. sqrt-prod 0.0%

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

      5. unpow2 0.0%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r* 0.0%

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

      10. unpow2 0.0%

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

      11. pow-prod-down 0.0%

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

      12. sqrt-prod 0.0%

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

   3. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.3%

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

      2. *-lft-identity 44.3%

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

      3. unpow2 44.3%

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

      4. rem-sqrt-square 44.3%

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

      5. *-commutative 44.3%

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

      6. unpow2 44.3%

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

      7. rem-sqrt-square 78.3%

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

      8. *-commutative 78.3%

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

   5. Simplified 78.3%

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

      1. frac-2neg 78.3%

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

      2. distribute-frac-neg 78.3%

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

      3. associate-/r* 78.4%

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

      4. add-sqr-sqrt 39.4%

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

      5. fabs-sqr 39.4%

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

      6. add-sqr-sqrt 40.8%

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

      7. associate-/r* 40.8%

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

      8. *-commutative 40.8%

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

      9. associate-*r* 40.8%

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

      10. add-sqr-sqrt 30.4%

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

      11. fabs-sqr 30.4%

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

      12. add-sqr-sqrt 76.4%

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

      13. associate-*r* 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. associate-/r* 97.0%

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

      2. div-inv 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. associate-*l/ 97.2%

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

   11. Applied egg-rr 97.2%

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

4. Final simplification 99.2%

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

Alternative 2: 96.9% accurate, 2.7× speedup

Math

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

FPCore

NOTE: c 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 (* 2.0 x)) (* c s)) (/ -1.0 x)) (* x (* c (- s)))))

C

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

Fortran

NOTE: c 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((2.0d0 * x)) / (c * s)) * ((-1.0d0) / x)) / (x * (c * -s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c 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[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(c * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

   1. *-un-lft-identity 64.9%

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

   2. add-sqr-sqrt 64.9%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. unpow2 65.0%

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

   6. sqrt-prod 31.4%

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

   7. add-sqr-sqrt 48.4%

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

   8. *-commutative 48.4%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 48.4%

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

   12. sqrt-prod 48.3%

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

3. Applied egg-rr 86.1%

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

   1. associate-*l/ 86.2%

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

   2. *-lft-identity 86.2%

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

   3. unpow2 86.2%

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

   4. rem-sqrt-square 86.2%

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

   5. *-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. rem-sqrt-square 95.9%

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

   8. *-commutative 95.9%

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

5. Simplified 95.9%

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

   1. frac-2neg 95.9%

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

   2. distribute-frac-neg 95.9%

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

   3. associate-/r* 95.9%

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

   4. add-sqr-sqrt 50.3%

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

   5. fabs-sqr 50.3%

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

   6. add-sqr-sqrt 64.3%

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

   7. associate-/r* 64.3%

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

   8. *-commutative 64.3%

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

   9. associate-*r* 63.3%

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

   10. add-sqr-sqrt 44.9%

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

   11. fabs-sqr 44.9%

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

   12. add-sqr-sqrt 94.1%

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

   13. associate-*r* 97.8%

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

7. Applied egg-rr 97.8%

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

   1. associate-/r* 97.9%

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

   2. div-inv 97.9%

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

9. Applied egg-rr 97.9%

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

10. Final simplification 97.9%

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

Alternative 3: 97.1% accurate, 2.7× speedup

Math

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

FPCore

NOTE: c 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 (* 2.0 x))) (* x (* c s))) (* x (* c (- s)))))

C

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

Fortran

NOTE: c 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((2.0d0 * x)) / (x * (c * s))) / (x * (c * -s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c 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[(2.0 * x), $MachinePrecision]], $MachinePrecision]) / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(c * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

   1. *-un-lft-identity 64.9%

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

   2. add-sqr-sqrt 64.9%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. unpow2 65.0%

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

   6. sqrt-prod 31.4%

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

   7. add-sqr-sqrt 48.4%

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

   8. *-commutative 48.4%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 48.4%

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

   12. sqrt-prod 48.3%

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

3. Applied egg-rr 86.1%

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

   1. associate-*l/ 86.2%

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

   2. *-lft-identity 86.2%

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

   3. unpow2 86.2%

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

   4. rem-sqrt-square 86.2%

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

   5. *-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. rem-sqrt-square 95.9%

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

   8. *-commutative 95.9%

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

5. Simplified 95.9%

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

   1. frac-2neg 95.9%

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

   2. distribute-frac-neg 95.9%

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

   3. associate-/r* 95.9%

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

   4. add-sqr-sqrt 50.3%

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

   5. fabs-sqr 50.3%

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

   6. add-sqr-sqrt 64.3%

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

   7. associate-/r* 64.3%

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

   8. *-commutative 64.3%

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

   9. associate-*r* 63.3%

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

   10. add-sqr-sqrt 44.9%

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

   11. fabs-sqr 44.9%

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

   12. add-sqr-sqrt 94.1%

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

   13. associate-*r* 97.8%

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

7. Applied egg-rr 97.8%

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

8. Final simplification 97.8%

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

Alternative 4: 96.9% accurate, 2.7× speedup

Math

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

FPCore

NOTE: c 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 (* 2.0 x))) (* c s)) x) (* x (* c (- s)))))

C

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

Fortran

NOTE: c 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((2.0d0 * x)) / (c * s)) / x) / (x * (c * -s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c 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[(2.0 * x), $MachinePrecision]], $MachinePrecision]) / N[(c * s), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(x * N[(c * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

   1. *-un-lft-identity 64.9%

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

   2. add-sqr-sqrt 64.9%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. unpow2 65.0%

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

   6. sqrt-prod 31.4%

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

   7. add-sqr-sqrt 48.4%

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

   8. *-commutative 48.4%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 48.4%

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

   12. sqrt-prod 48.3%

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

3. Applied egg-rr 86.1%

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

   1. associate-*l/ 86.2%

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

   2. *-lft-identity 86.2%

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

   3. unpow2 86.2%

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

   4. rem-sqrt-square 86.2%

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

   5. *-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. rem-sqrt-square 95.9%

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

   8. *-commutative 95.9%

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

5. Simplified 95.9%

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

   1. frac-2neg 95.9%

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

   2. distribute-frac-neg 95.9%

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

   3. associate-/r* 95.9%

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

   4. add-sqr-sqrt 50.3%

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

   5. fabs-sqr 50.3%

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

   6. add-sqr-sqrt 64.3%

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

   7. associate-/r* 64.3%

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

   8. *-commutative 64.3%

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

   9. associate-*r* 63.3%

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

   10. add-sqr-sqrt 44.9%

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

   11. fabs-sqr 44.9%

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

   12. add-sqr-sqrt 94.1%

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

   13. associate-*r* 97.8%

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

7. Applied egg-rr 97.8%

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

   1. associate-/r* 97.9%

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

   2. div-inv 97.9%

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

9. Applied egg-rr 97.9%

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

    1. un-div-inv 97.9%

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

11. Applied egg-rr 97.9%

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

12. Final simplification 97.9%

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

Alternative 5: 97.2% accurate, 2.7× speedup

Math

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

FPCore

NOTE: c 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 (* 2.0 x)) t_0) t_0)))

C

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

Fortran

NOTE: c 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((2.0d0 * x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c 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[(2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.9%

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

   1. *-un-lft-identity 64.9%

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

   2. add-sqr-sqrt 64.9%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. unpow2 65.0%

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

   6. sqrt-prod 31.4%

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

   7. add-sqr-sqrt 48.4%

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

   8. *-commutative 48.4%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 48.4%

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

   12. sqrt-prod 48.3%

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

3. Applied egg-rr 86.1%

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

   1. associate-*l/ 86.2%

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

   2. *-lft-identity 86.2%

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

   3. unpow2 86.2%

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

   4. rem-sqrt-square 86.2%

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

   5. *-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. rem-sqrt-square 95.9%

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

   8. *-commutative 95.9%

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

5. Simplified 95.9%

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

   1. associate-/r* 90.7%

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

   2. add-sqr-sqrt 48.9%

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

   3. fabs-sqr 48.9%

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

   4. add-sqr-sqrt 62.9%

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

   5. associate-/l/ 64.3%

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

   6. div-inv 64.3%

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

   7. times-frac 62.9%

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

   8. *-commutative 62.9%

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

   9. add-sqr-sqrt 44.5%

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

   10. fabs-sqr 44.5%

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

   11. add-sqr-sqrt 90.4%

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

   12. associate-*r* 89.7%

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

7. Applied egg-rr 89.7%

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

   1. frac-times 94.1%

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

   2. div-inv 94.1%

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

   3. associate-*l* 95.9%

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

   4. *-commutative 95.9%

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

   5. associate-*r* 94.1%

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

   6. *-commutative 94.1%

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

   7. *-commutative 94.1%

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

   8. associate-*l* 95.9%

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

9. Applied egg-rr 95.9%

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

10. Final simplification 95.9%

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

Alternative 6: 76.5% accurate, 24.1× speedup

Math

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

FPCore

NOTE: c 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 (/ 1.0 (* (* x s) (* c (* s (* x c))))))

C

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

Fortran

NOTE: c 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 = 1.0d0 / ((x * s) * (c * (s * (x * c))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c 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[(1.0 / N[(N[(x * s), $MachinePrecision] * N[(c * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

2. Taylor expanded in x around 0 56.9%

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

   1. associate-/r* 56.4%

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

   2. *-commutative 56.4%

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

   3. unpow2 56.4%

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

   4. unpow2 56.4%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.7%

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

   8. rem-square-sqrt 67.7%

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

   9. unpow2 67.7%

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

   10. swap-sqr 74.6%

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

   11. unpow2 74.6%

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

   12. unpow2 74.6%

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

   13. rem-sqrt-square 79.8%

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

   14. *-commutative 79.8%

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

4. Simplified 79.8%

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

   1. unpow2 79.8%

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

   2. add-sqr-sqrt 42.3%

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

   3. fabs-sqr 42.3%

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

   4. add-sqr-sqrt 57.6%

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

   5. associate-*r* 56.8%

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

   6. add-sqr-sqrt 37.3%

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

   7. fabs-sqr 37.3%

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

   8. add-sqr-sqrt 77.6%

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

   9. associate-*r* 77.2%

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

6. Applied egg-rr 77.2%

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

   1. expm1-log1p-u 53.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot c\right)\right)} \cdot \left(s \cdot x\right)} \]

   2. expm1-udef 33.5%

      \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot c\right)} - 1\right)} \cdot \left(s \cdot x\right)} \]

   3. associate-*l* 33.4%

      \[\leadsto \frac{1}{\left(e^{\mathsf{log1p}\left(\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot c\right)}\right)} - 1\right) \cdot \left(s \cdot x\right)} \]

8. Applied egg-rr 33.4%

   \[\leadsto \frac{1}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(c \cdot s\right) \cdot \left(x \cdot c\right)\right)} - 1\right)} \cdot \left(s \cdot x\right)} \]
9. Step-by-step derivation

   1. expm1-def 52.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(c \cdot s\right) \cdot \left(x \cdot c\right)\right)\right)} \cdot \left(s \cdot x\right)} \]

   2. expm1-log1p 76.6%

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

   3. associate-*l* 77.0%

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

   4. *-commutative 77.0%

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

10. Simplified 77.0%

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

11. Final simplification 77.0%

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

Alternative 7: 78.5% accurate, 24.1× speedup

Math

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

FPCore

NOTE: c 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 (* x (* c s)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

NOTE: c 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 = x * (c * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: c 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[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.9%

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

2. Taylor expanded in x around 0 56.9%

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

   1. associate-/r* 56.4%

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

   2. *-commutative 56.4%

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

   3. unpow2 56.4%

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

   4. unpow2 56.4%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.7%

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

   8. rem-square-sqrt 67.7%

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

   9. unpow2 67.7%

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

   10. swap-sqr 74.6%

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

   11. unpow2 74.6%

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

   12. unpow2 74.6%

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

   13. rem-sqrt-square 79.8%

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

   14. *-commutative 79.8%

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

4. Simplified 79.8%

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

   1. unpow-prod-down 67.7%

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

   2. unpow2 67.7%

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

   3. pow2 67.7%

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

   4. sqr-abs 67.7%

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

   5. swap-sqr 79.8%

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

   6. associate-*r* 78.4%

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

   7. associate-*r* 80.9%

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

6. Applied egg-rr 80.9%

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

7. Final simplification 80.9%

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

Alternative 8: 79.4% accurate, 24.1× speedup

Math

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

FPCore

NOTE: c 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

c = abs(c);
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: c 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

c = Math.abs(c);
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

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

Julia

c = abs(c)
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

c = abs(c)
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: c 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 64.9%

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

2. Taylor expanded in x around 0 56.9%

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

   1. associate-/r* 56.4%

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

   2. *-commutative 56.4%

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

   3. unpow2 56.4%

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

   4. unpow2 56.4%

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

   5. swap-sqr 67.2%

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

   6. unpow2 67.2%

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

   7. associate-/r* 67.7%

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

   8. rem-square-sqrt 67.7%

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

   9. unpow2 67.7%

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

   10. swap-sqr 74.6%

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

   11. unpow2 74.6%

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

   12. unpow2 74.6%

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

   13. rem-sqrt-square 79.8%

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

   14. *-commutative 79.8%

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

4. Simplified 79.8%

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

   1. pow-flip 80.0%

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

   2. add-sqr-sqrt 42.3%

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

   3. fabs-sqr 42.3%

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

   4. add-sqr-sqrt 80.0%

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

   5. *-commutative 80.0%

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

   6. *-commutative 80.0%

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

   7. associate-*r* 81.1%

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

   8. metadata-eval 81.1%

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

6. Applied egg-rr 81.1%

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

   1. metadata-eval 81.1%

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

   2. pow-div 81.1%

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

   3. inv-pow 81.1%

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

   4. associate-*l* 78.6%

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

   5. pow1 78.6%

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

   6. associate-*l* 80.0%

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

8. Applied egg-rr 80.0%

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

9. Final simplification 80.0%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))