mixedcos

Percentage Accurate: 67.0% → 98.0%

Time: 12.8s

Alternatives: 8

Speedup: 24.1×

• 31.0% 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: 67.0% 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: 98.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{-\frac{t_0}{c}}{x \cdot s}}{c \cdot \left(x \cdot \left(-s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(x \cdot c\right)} \cdot \left(\frac{\frac{1}{c}}{x} \cdot \frac{1}{s}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
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 1.5e+118)
     (/ (/ (- (/ t_0 c)) (* x s)) (* c (* x (- s))))
     (* (/ t_0 (* s (* x c))) (* (/ (/ 1.0 c) x) (/ 1.0 s))))))

C

x = abs(x);
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 <= 1.5e+118) {
		tmp = (-(t_0 / c) / (x * s)) / (c * (x * -s));
	} else {
		tmp = (t_0 / (s * (x * c))) * (((1.0 / c) / x) * (1.0 / s));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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 <= 1.5d+118) then
        tmp = (-(t_0 / c) / (x * s)) / (c * (x * -s))
    else
        tmp = (t_0 / (s * (x * c))) * (((1.0d0 / c) / x) * (1.0d0 / s))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
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 <= 1.5e+118) {
		tmp = (-(t_0 / c) / (x * s)) / (c * (x * -s));
	} else {
		tmp = (t_0 / (s * (x * c))) * (((1.0 / c) / x) * (1.0 / s));
	}
	return tmp;
}

Python

x = abs(x)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if x <= 1.5e+118:
		tmp = (-(t_0 / c) / (x * s)) / (c * (x * -s))
	else:
		tmp = (t_0 / (s * (x * c))) * (((1.0 / c) / x) * (1.0 / s))
	return tmp

Julia

x = abs(x)
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 <= 1.5e+118)
		tmp = Float64(Float64(Float64(-Float64(t_0 / c)) / Float64(x * s)) / Float64(c * Float64(x * Float64(-s))));
	else
		tmp = Float64(Float64(t_0 / Float64(s * Float64(x * c))) * Float64(Float64(Float64(1.0 / c) / x) * Float64(1.0 / s)));
	end
	return tmp
end

MATLAB

x = abs(x)
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 <= 1.5e+118)
		tmp = (-(t_0 / c) / (x * s)) / (c * (x * -s));
	else
		tmp = (t_0 / (s * (x * c))) * (((1.0 / c) / x) * (1.0 / s));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
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, 1.5e+118], N[(N[((-N[(t$95$0 / c), $MachinePrecision]) / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / c), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e118

   1. Initial program 69.3%

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

      1. *-un-lft-identity 69.3%

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

      2. times-frac 68.9%

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

      3. pow-flip 68.9%

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

      4. metadata-eval 68.9%

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

      5. *-commutative 68.9%

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

      6. associate-*r* 64.4%

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

      7. unpow2 64.4%

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

      8. pow-prod-down 75.9%

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

   3. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. div-inv 75.9%

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

      3. associate-*l* 75.9%

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

      4. *-commutative 75.9%

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

      5. pow-flip 75.9%

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

      6. add-sqr-sqrt 43.7%

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

      7. fabs-sqr 43.7%

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

      8. add-sqr-sqrt 75.9%

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

      9. metadata-eval 75.9%

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

      10. unpow-prod-down 96.8%

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

      11. *-commutative 96.8%

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

      12. metadata-eval 96.8%

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

      13. pow-flip 96.8%

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

      14. div-inv 96.8%

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

      15. unpow2 96.8%

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

      16. associate-/l/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   if 1.5e118 < x

   1. Initial program 50.5%

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

      1. *-un-lft-identity 50.5%

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

      2. times-frac 50.5%

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

      3. pow-flip 50.4%

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

      4. metadata-eval 50.4%

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

      5. *-commutative 50.4%

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

      6. associate-*r* 38.9%

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

      7. unpow2 38.9%

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

      8. pow-prod-down 67.1%

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

   3. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

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

      2. div-inv 67.1%

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

      3. associate-*l* 67.1%

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

      4. *-commutative 67.1%

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

      5. pow-flip 67.0%

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

      6. add-sqr-sqrt 39.6%

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

      7. fabs-sqr 39.6%

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

      8. add-sqr-sqrt 67.0%

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

      9. metadata-eval 67.0%

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

      10. unpow-prod-down 91.5%

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

      11. *-commutative 91.5%

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

      12. metadata-eval 91.5%

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

      13. pow-flip 89.7%

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

      14. div-inv 89.7%

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

      15. unpow2 89.7%

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

      16. associate-/l/ 91.5%

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

   5. Applied egg-rr 91.6%

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

      1. associate-/r* 86.2%

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

      2. distribute-frac-neg 86.2%

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

      3. associate-/r* 86.2%

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

      4. distribute-frac-neg 86.2%

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

      5. distribute-rgt-neg-out 86.2%

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

      6. frac-2neg 86.2%

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

      7. associate-/r* 91.5%

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

      8. div-inv 91.6%

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

      9. *-commutative 91.6%

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

      10. associate-*r* 89.1%

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

      11. *-commutative 89.1%

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

      12. associate-*r* 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. associate-/r* 97.1%

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

      2. div-inv 97.1%

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

      3. associate-/r* 97.3%

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

   9. Applied egg-rr 97.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{-\frac{\cos \left(x \cdot 2\right)}{c}}{x \cdot s}}{c \cdot \left(x \cdot \left(-s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \left(\frac{\frac{1}{c}}{x} \cdot \frac{1}{s}\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
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 (* s (* x c))))
   (if (<= x 4e-26)
     (/ (/ (/ 1.0 (* x s)) c) (* c (* x s)))
     (* (/ (cos (* x 2.0)) t_0) (/ 1.0 t_0)))))

C

x = abs(x);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 4e-26) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = (cos((x * 2.0)) / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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 = s * (x * c)
    if (x <= 4d-26) then
        tmp = ((1.0d0 / (x * s)) / c) / (c * (x * s))
    else
        tmp = (cos((x * 2.0d0)) / t_0) * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if (x <= 4e-26) {
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	} else {
		tmp = (Math.cos((x * 2.0)) / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Python

x = abs(x)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 4e-26:
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s))
	else:
		tmp = (math.cos((x * 2.0)) / t_0) * (1.0 / t_0)
	return tmp

Julia

x = abs(x)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (x <= 4e-26)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / Float64(c * Float64(x * s)));
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 4e-26)
		tmp = ((1.0 / (x * s)) / c) / (c * (x * s));
	else
		tmp = (cos((x * 2.0)) / t_0) * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
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[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e-26], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000002e-26

   1. Initial program 68.4%

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

   2. Taylor expanded in x around 0 59.1%

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

      1. associate-/r* 58.5%

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

      2. *-commutative 58.5%

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

      3. unpow2 58.5%

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

      4. unpow2 58.5%

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

      5. swap-sqr 69.2%

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

      6. unpow2 69.2%

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

      7. associate-/r* 69.7%

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

      8. rem-square-sqrt 69.7%

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

      9. unpow2 69.7%

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

      10. swap-sqr 74.9%

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

      11. unpow2 74.9%

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

      12. unpow2 74.9%

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

      13. rem-sqrt-square 83.1%

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

      14. *-commutative 83.1%

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

   4. Simplified 83.1%

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

      1. clear-num 83.1%

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

      2. add-sqr-sqrt 83.1%

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

      3. clear-num 83.1%

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

      4. sqrt-div 83.1%

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

      5. metadata-eval 83.1%

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

      6. unpow2 83.1%

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

      7. sqrt-prod 57.0%

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

      8. add-sqr-sqrt 55.9%

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

      9. add-sqr-sqrt 33.7%

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

      10. fabs-sqr 33.7%

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

      11. add-sqr-sqrt 54.5%

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

      12. clear-num 54.5%

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

      13. sqrt-div 54.5%

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

      14. metadata-eval 54.5%

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

      15. unpow2 54.5%

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

   6. Applied egg-rr 83.2%

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

      1. un-div-inv 83.2%

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

      2. *-commutative 83.2%

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

      3. *-commutative 83.2%

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

      4. associate-/r* 82.8%

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

      5. *-commutative 82.8%

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

   8. Applied egg-rr 82.8%

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

   if 4.0000000000000002e-26 < x

   1. Initial program 61.7%

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

      1. *-un-lft-identity 61.7%

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

      2. times-frac 61.6%

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

      3. pow-flip 61.6%

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

      4. metadata-eval 61.6%

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

      5. *-commutative 61.6%

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

      6. associate-*r* 55.1%

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

      7. unpow2 55.1%

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

      8. pow-prod-down 72.8%

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

   3. Applied egg-rr 72.8%

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

      1. *-commutative 72.8%

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

      2. div-inv 72.8%

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

      3. associate-*l* 72.8%

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

      4. *-commutative 72.8%

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

      5. pow-flip 72.8%

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

      6. add-sqr-sqrt 37.4%

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

      7. fabs-sqr 37.4%

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

      8. add-sqr-sqrt 72.8%

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

      9. metadata-eval 72.8%

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

      10. unpow-prod-down 94.9%

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

      11. *-commutative 94.9%

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

      12. metadata-eval 94.9%

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

      13. pow-flip 93.9%

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

      14. div-inv 93.9%

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

      15. unpow2 93.9%

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

      16. associate-/l/ 94.9%

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

   5. Applied egg-rr 95.0%

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

      1. associate-/r* 91.9%

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

      2. distribute-frac-neg 91.9%

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

      3. associate-/r* 91.9%

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

      4. distribute-frac-neg 91.9%

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

      5. distribute-rgt-neg-out 91.9%

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

      6. frac-2neg 91.9%

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

      7. associate-/r* 94.9%

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

      8. div-inv 94.9%

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

      9. *-commutative 94.9%

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

      10. associate-*r* 93.5%

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

      11. *-commutative 93.5%

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

      12. associate-*r* 98.1%

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

   7. Applied egg-rr 98.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 3: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 3.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{-\frac{t_1}{c}}{x \cdot s}}{c \cdot \left(x \cdot \left(-s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
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 (* s (* x c))) (t_1 (cos (* x 2.0))))
   (if (<= x 3.1e+103)
     (/ (/ (- (/ t_1 c)) (* x s)) (* c (* x (- s))))
     (* (/ t_1 t_0) (/ 1.0 t_0)))))

C

x = abs(x);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (x <= 3.1e+103) {
		tmp = (-(t_1 / c) / (x * s)) / (c * (x * -s));
	} else {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = cos((x * 2.0d0))
    if (x <= 3.1d+103) then
        tmp = (-(t_1 / c) / (x * s)) / (c * (x * -s))
    else
        tmp = (t_1 / t_0) * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 3.1e+103) {
		tmp = (-(t_1 / c) / (x * s)) / (c * (x * -s));
	} else {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Python

x = abs(x)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if x <= 3.1e+103:
		tmp = (-(t_1 / c) / (x * s)) / (c * (x * -s))
	else:
		tmp = (t_1 / t_0) * (1.0 / t_0)
	return tmp

Julia

x = abs(x)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 3.1e+103)
		tmp = Float64(Float64(Float64(-Float64(t_1 / c)) / Float64(x * s)) / Float64(c * Float64(x * Float64(-s))));
	else
		tmp = Float64(Float64(t_1 / t_0) * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 3.1e+103)
		tmp = (-(t_1 / c) / (x * s)) / (c * (x * -s));
	else
		tmp = (t_1 / t_0) * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
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[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.1e+103], N[(N[((-N[(t$95$1 / c), $MachinePrecision]) / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * (-s)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.1000000000000002e103

   1. Initial program 69.4%

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

      1. *-un-lft-identity 69.4%

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

      2. times-frac 68.9%

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

      3. pow-flip 68.9%

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

      4. metadata-eval 68.9%

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

      5. *-commutative 68.9%

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

      6. associate-*r* 64.3%

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

      7. unpow2 64.3%

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

      8. pow-prod-down 76.0%

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

   3. Applied egg-rr 76.0%

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

      1. *-commutative 76.0%

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

      2. div-inv 76.0%

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

      3. associate-*l* 76.0%

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

      4. *-commutative 76.0%

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

      5. pow-flip 76.0%

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

      6. add-sqr-sqrt 43.4%

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

      7. fabs-sqr 43.4%

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

      8. add-sqr-sqrt 76.0%

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

      9. metadata-eval 76.0%

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

      10. unpow-prod-down 96.8%

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

      11. *-commutative 96.8%

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

      12. metadata-eval 96.8%

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

      13. pow-flip 96.7%

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

      14. div-inv 96.7%

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

      15. unpow2 96.7%

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

      16. associate-/l/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   if 3.1000000000000002e103 < x

   1. Initial program 51.7%

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

      1. *-un-lft-identity 51.7%

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

      2. times-frac 51.7%

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

      3. pow-flip 51.7%

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

      4. metadata-eval 51.7%

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

      5. *-commutative 51.7%

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

      6. associate-*r* 41.1%

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

      7. unpow2 41.1%

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

      8. pow-prod-down 67.1%

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

   3. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

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

      2. div-inv 67.1%

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

      3. associate-*l* 67.1%

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

      4. *-commutative 67.1%

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

      5. pow-flip 67.1%

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

      6. add-sqr-sqrt 41.7%

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

      7. fabs-sqr 41.7%

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

      8. add-sqr-sqrt 67.1%

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

      9. metadata-eval 67.1%

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

      10. unpow-prod-down 92.1%

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

      11. *-commutative 92.1%

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

      12. metadata-eval 92.1%

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

      13. pow-flip 90.5%

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

      14. div-inv 90.5%

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

      15. unpow2 90.5%

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

      16. associate-/l/ 92.1%

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

   5. Applied egg-rr 92.3%

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

      1. associate-/r* 87.3%

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

      2. distribute-frac-neg 87.3%

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

      3. associate-/r* 87.3%

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

      4. distribute-frac-neg 87.3%

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

      5. distribute-rgt-neg-out 87.3%

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

      6. frac-2neg 87.3%

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

      7. associate-/r* 92.1%

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

      8. div-inv 92.2%

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

      9. *-commutative 92.2%

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

      10. associate-*r* 89.9%

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

      11. *-commutative 89.9%

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

      12. associate-*r* 97.3%

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{-\frac{\cos \left(x \cdot 2\right)}{c}}{x \cdot s}}{c \cdot \left(x \cdot \left(-s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 4: 94.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 4.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot c}}{s \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
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 4.5e-46)
     (/ (/ (/ 1.0 (* x s)) c) t_0)
     (/ (/ (cos (* x 2.0)) (* x c)) (* s t_0)))))

C

x = abs(x);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (x <= 4.5e-46) {
		tmp = ((1.0 / (x * s)) / c) / t_0;
	} else {
		tmp = (cos((x * 2.0)) / (x * c)) / (s * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
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 <= 4.5d-46) then
        tmp = ((1.0d0 / (x * s)) / c) / t_0
    else
        tmp = (cos((x * 2.0d0)) / (x * c)) / (s * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
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 <= 4.5e-46) {
		tmp = ((1.0 / (x * s)) / c) / t_0;
	} else {
		tmp = (Math.cos((x * 2.0)) / (x * c)) / (s * t_0);
	}
	return tmp;
}

Python

x = abs(x)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if x <= 4.5e-46:
		tmp = ((1.0 / (x * s)) / c) / t_0
	else:
		tmp = (math.cos((x * 2.0)) / (x * c)) / (s * t_0)
	return tmp

Julia

x = abs(x)
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 <= 4.5e-46)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s)) / c) / t_0);
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / Float64(x * c)) / Float64(s * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
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 <= 4.5e-46)
		tmp = ((1.0 / (x * s)) / c) / t_0;
	else
		tmp = (cos((x * 2.0)) / (x * c)) / (s * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
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, 4.5e-46], N[(N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] / N[(s * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.50000000000000001e-46

   1. Initial program 68.8%

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

   2. Taylor expanded in x around 0 59.1%

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

      1. associate-/r* 58.6%

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

      2. *-commutative 58.6%

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

      3. unpow2 58.6%

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

      4. unpow2 58.6%

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

      5. swap-sqr 69.6%

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

      6. unpow2 69.6%

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

      7. associate-/r* 70.2%

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

      8. rem-square-sqrt 70.2%

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

      9. unpow2 70.2%

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

      10. swap-sqr 75.4%

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

      11. unpow2 75.4%

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

      12. unpow2 75.4%

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

      13. rem-sqrt-square 82.5%

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

      14. *-commutative 82.5%

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

   4. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. add-sqr-sqrt 82.5%

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

      3. clear-num 82.5%

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

      4. sqrt-div 82.5%

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

      5. metadata-eval 82.5%

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

      6. unpow2 82.5%

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

      7. sqrt-prod 57.0%

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

      8. add-sqr-sqrt 56.4%

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

      9. add-sqr-sqrt 33.9%

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

      10. fabs-sqr 33.9%

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

      11. add-sqr-sqrt 54.4%

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

      12. clear-num 54.4%

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

      13. sqrt-div 54.4%

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

      14. metadata-eval 54.4%

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

      15. unpow2 54.4%

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

   6. Applied egg-rr 82.6%

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

      1. un-div-inv 82.6%

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

      2. *-commutative 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-/r* 82.2%

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

      5. *-commutative 82.2%

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

   8. Applied egg-rr 82.2%

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

   if 4.50000000000000001e-46 < x

   1. Initial program 61.2%

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

      1. *-un-lft-identity 61.2%

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

      2. times-frac 61.1%

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

      3. pow-flip 61.1%

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

      4. metadata-eval 61.1%

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

      5. *-commutative 61.1%

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

      6. associate-*r* 55.3%

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

      7. unpow2 55.3%

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

      8. pow-prod-down 71.2%

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

   3. Applied egg-rr 71.2%

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

      1. *-commutative 71.2%

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

      2. div-inv 71.2%

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

      3. associate-*l* 71.2%

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

      4. *-commutative 71.2%

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

      5. pow-flip 71.2%

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

      6. add-sqr-sqrt 36.5%

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

      7. fabs-sqr 36.5%

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

      8. add-sqr-sqrt 71.2%

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

      9. metadata-eval 71.2%

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

      10. unpow-prod-down 95.4%

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

      11. *-commutative 95.4%

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

      12. metadata-eval 95.4%

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

      13. pow-flip 94.5%

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

      14. div-inv 94.5%

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

      15. unpow2 94.5%

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

      16. associate-/l/ 95.4%

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

   5. Applied egg-rr 95.5%

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

      1. associate-/r* 92.7%

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

      2. distribute-frac-neg 92.7%

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

      3. associate-/r* 92.7%

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

      4. distribute-frac-neg 92.7%

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

      5. distribute-rgt-neg-out 92.7%

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

      6. frac-2neg 92.7%

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

      7. associate-/r* 95.4%

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

      8. div-inv 95.3%

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

      9. *-commutative 95.3%

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

      10. associate-*r* 94.1%

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

      11. *-commutative 94.1%

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

      12. associate-*r* 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. associate-/r* 98.2%

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

      2. frac-times 90.7%

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

      3. div-inv 90.7%

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

      4. associate-*l* 86.6%

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

   9. Applied egg-rr 86.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot s}}{c}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot c}}{s \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
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

x = abs(x);
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: x should be positive before calling this function
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

x = Math.abs(x);
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

x = abs(x)
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

x = abs(x)
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

x = abs(x)
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: x should be positive before calling this function
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 66.8%

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

   1. *-un-lft-identity 66.8%

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

   2. times-frac 66.3%

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

   3. pow-flip 66.4%

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

   4. metadata-eval 66.4%

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

   5. *-commutative 66.4%

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

   6. associate-*r* 60.9%

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

   7. unpow2 60.9%

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

   8. pow-prod-down 74.7%

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

3. Applied egg-rr 74.7%

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

   1. *-commutative 74.7%

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

   2. div-inv 74.7%

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

   3. associate-*l* 74.7%

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

   4. *-commutative 74.7%

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

   5. pow-flip 74.7%

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

   6. add-sqr-sqrt 43.1%

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

   7. fabs-sqr 43.1%

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

   8. add-sqr-sqrt 74.7%

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

   9. metadata-eval 74.7%

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

   10. unpow-prod-down 96.1%

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

   11. *-commutative 96.1%

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

   12. metadata-eval 96.1%

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

   13. pow-flip 95.8%

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

   14. div-inv 95.8%

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

   15. unpow2 95.8%

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

   16. associate-/l/ 96.1%

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

5. Applied egg-rr 96.1%

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

6. Final simplification 96.1%

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

Alternative 6: 79.4% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
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

x = abs(x);
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: x should be positive before calling this function
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

x = Math.abs(x);
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

x = abs(x)
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

x = abs(x)
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

x = abs(x)
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: x should be positive before calling this function
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 66.8%

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

2. Taylor expanded in x around 0 55.3%

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

   1. associate-/r* 54.9%

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

   2. *-commutative 54.9%

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

   3. unpow2 54.9%

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

   4. unpow2 54.9%

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

   5. swap-sqr 64.8%

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

   6. unpow2 64.8%

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

   7. associate-/r* 65.2%

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

   8. rem-square-sqrt 65.2%

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

   9. unpow2 65.2%

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

   10. swap-sqr 69.7%

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

   11. unpow2 69.7%

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

   12. unpow2 69.7%

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

   13. rem-sqrt-square 76.7%

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

   14. *-commutative 76.7%

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

4. Simplified 76.7%

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

   1. unpow-prod-down 65.2%

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

   2. pow2 65.2%

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

   3. pow2 65.2%

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

   4. sqr-abs 65.2%

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

   5. swap-sqr 76.7%

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

6. Applied egg-rr 76.7%

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

7. Final simplification 76.7%

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

Alternative 7: 79.5% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
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

x = abs(x);
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: x should be positive before calling this function
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

x = Math.abs(x);
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

x = abs(x)
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

x = abs(x)
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

x = abs(x)
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: x should be positive before calling this function
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 66.8%

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

   1. *-un-lft-identity 66.8%

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

   2. times-frac 66.3%

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

   3. pow-flip 66.4%

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

   4. metadata-eval 66.4%

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

   5. *-commutative 66.4%

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

   6. associate-*r* 60.9%

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

   7. unpow2 60.9%

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

   8. pow-prod-down 74.7%

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

3. Applied egg-rr 74.7%

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

   1. *-commutative 74.7%

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

   2. div-inv 74.7%

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

   3. associate-*l* 74.7%

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

   4. *-commutative 74.7%

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

   5. pow-flip 74.7%

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

   6. add-sqr-sqrt 43.1%

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

   7. fabs-sqr 43.1%

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

   8. add-sqr-sqrt 74.7%

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

   9. metadata-eval 74.7%

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

   10. unpow-prod-down 96.1%

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

   11. *-commutative 96.1%

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

   12. metadata-eval 96.1%

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

   13. pow-flip 95.8%

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

   14. div-inv 95.8%

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

   15. unpow2 95.8%

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

   16. associate-/l/ 96.1%

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

5. Applied egg-rr 96.1%

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

6. Taylor expanded in x around 0 76.7%

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

7. Final simplification 76.7%

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

Alternative 8: 79.5% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
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 (/ (/ (/ 1.0 (* x s)) c) (* c (* x s))))

C

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

Fortran

NOTE: x should be positive before calling this function
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 = ((1.0d0 / (x * s)) / c) / (c * (x * s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
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[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

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

2. Taylor expanded in x around 0 55.3%

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

   1. associate-/r* 54.9%

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

   2. *-commutative 54.9%

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

   3. unpow2 54.9%

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

   4. unpow2 54.9%

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

   5. swap-sqr 64.8%

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

   6. unpow2 64.8%

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

   7. associate-/r* 65.2%

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

   8. rem-square-sqrt 65.2%

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

   9. unpow2 65.2%

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

   10. swap-sqr 69.7%

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

   11. unpow2 69.7%

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

   12. unpow2 69.7%

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

   13. rem-sqrt-square 76.7%

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

   14. *-commutative 76.7%

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

4. Simplified 76.7%

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

   1. clear-num 76.7%

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

   2. add-sqr-sqrt 76.7%

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

   3. clear-num 76.7%

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

   4. sqrt-div 76.7%

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

   5. metadata-eval 76.7%

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

   6. unpow2 76.7%

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

   7. sqrt-prod 51.2%

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

   8. add-sqr-sqrt 56.5%

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

   9. add-sqr-sqrt 32.6%

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

   10. fabs-sqr 32.6%

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

   11. add-sqr-sqrt 54.3%

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

   12. clear-num 54.3%

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

   13. sqrt-div 54.3%

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

   14. metadata-eval 54.3%

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

   15. unpow2 54.3%

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

6. Applied egg-rr 76.7%

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

   1. un-div-inv 76.7%

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

   2. *-commutative 76.7%

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

   3. *-commutative 76.7%

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

   4. associate-/r* 76.5%

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

   5. *-commutative 76.5%

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

8. Applied egg-rr 76.5%

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

9. Final simplification 76.5%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))