mixedcos

Percentage Accurate: 66.1% → 99.5%

Time: 12.5s

Alternatives: 6

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{\frac{x}{\frac{1}{s}}}{\frac{1}{c}} \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c 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
 (if (<= x 1.04e-10)
   (/ 1.0 (* (/ (/ x (/ 1.0 s)) (/ 1.0 c)) (* c (* x s))))
   (* (pow (* s (* x c)) -2.0) (cos (* x 2.0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.04e-10) {
		tmp = 1.0 / (((x / (1.0 / s)) / (1.0 / c)) * (c * (x * s)));
	} else {
		tmp = pow((s * (x * c)), -2.0) * cos((x * 2.0));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c 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) :: tmp
    if (x <= 1.04d-10) then
        tmp = 1.0d0 / (((x / (1.0d0 / s)) / (1.0d0 / c)) * (c * (x * s)))
    else
        tmp = ((s * (x * c)) ** (-2.0d0)) * cos((x * 2.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.04e-10) {
		tmp = 1.0 / (((x / (1.0 / s)) / (1.0 / c)) * (c * (x * s)));
	} else {
		tmp = Math.pow((s * (x * c)), -2.0) * Math.cos((x * 2.0));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	tmp = 0
	if x <= 1.04e-10:
		tmp = 1.0 / (((x / (1.0 / s)) / (1.0 / c)) * (c * (x * s)))
	else:
		tmp = math.pow((s * (x * c)), -2.0) * math.cos((x * 2.0))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 1.04e-10)
		tmp = Float64(1.0 / Float64(Float64(Float64(x / Float64(1.0 / s)) / Float64(1.0 / c)) * Float64(c * Float64(x * s))));
	else
		tmp = Float64((Float64(s * Float64(x * c)) ^ -2.0) * cos(Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.04e-10)
		tmp = 1.0 / (((x / (1.0 / s)) / (1.0 / c)) * (c * (x * s)));
	else
		tmp = ((s * (x * c)) ^ -2.0) * cos((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c 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_] := If[LessEqual[x, 1.04e-10], N[(1.0 / N[(N[(N[(x / N[(1.0 / s), $MachinePrecision]), $MachinePrecision] / N[(1.0 / c), $MachinePrecision]), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04e-10

   1. Initial program 61.8%

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

      1. associate-/r* 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*r* 57.0%

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

      4. unpow2 57.0%

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

      5. associate-/l/ 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in x around 0 54.7%

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

      1. associate-/r* 54.7%

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

      2. *-commutative 54.7%

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

      3. unpow2 54.7%

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

      4. unpow2 54.7%

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

      5. swap-sqr 68.6%

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

      6. unpow2 68.6%

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

      7. associate-/r* 68.6%

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

      8. unpow2 68.6%

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

      9. rem-square-sqrt 68.6%

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

      10. swap-sqr 78.2%

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

      11. unpow2 78.2%

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

      12. unpow2 78.2%

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

      13. rem-sqrt-square 85.9%

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

      14. *-commutative 85.9%

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

   6. Simplified 85.9%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-prod 85.9%

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

      3. unpow2 85.9%

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

      4. pow2 85.9%

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

      5. add-sqr-sqrt 85.9%

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

      6. unpow2 85.9%

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

      7. add-sqr-sqrt 45.5%

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

      8. fabs-sqr 45.5%

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

      9. add-sqr-sqrt 54.3%

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

      10. add-sqr-sqrt 38.8%

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

      11. fabs-sqr 38.8%

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

      12. add-sqr-sqrt 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. associate-*r* 84.1%

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

      2. *-commutative 84.1%

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

      3. /-rgt-identity 84.1%

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

      4. clear-num 84.1%

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

      5. div-inv 84.1%

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

      6. clear-num 84.1%

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

      7. div-inv 84.2%

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

      8. div-inv 84.2%

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

      9. associate-/r* 86.0%

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

   10. Applied egg-rr 86.0%

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

   if 1.04e-10 < x

   1. Initial program 69.9%

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

   2. Taylor expanded in c around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. unpow2 67.3%

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

      3. unpow2 67.3%

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

      4. swap-sqr 88.0%

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

      5. unpow2 88.0%

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

      6. unpow2 88.0%

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

      7. rem-square-sqrt 88.0%

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

      8. swap-sqr 91.6%

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

      9. unpow2 91.6%

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

      10. unpow2 91.6%

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

      11. rem-sqrt-square 93.7%

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

      12. *-commutative 93.7%

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

   4. Simplified 93.7%

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

   5. Taylor expanded in x around inf 88.0%

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

      1. associate-/r* 88.0%

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

      2. *-commutative 88.0%

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

      3. unpow2 88.0%

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

      4. sqr-abs 88.0%

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

      5. unpow2 88.0%

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

      6. associate-/l/ 88.0%

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

      7. *-commutative 88.0%

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

      8. unpow2 88.0%

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

      9. unpow2 88.0%

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

      10. swap-sqr 93.7%

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

      11. associate-/l/ 93.7%

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

      12. *-rgt-identity 93.7%

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

      13. associate-*r/ 93.7%

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

      14. *-rgt-identity 93.7%

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

      15. associate-*r/ 93.6%

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

      16. associate-*l* 93.6%

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

   7. Simplified 93.8%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

      3. /-rgt-identity 96.2%

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

      4. clear-num 96.1%

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

      5. div-inv 96.0%

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

      6. clear-num 94.9%

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

      7. div-inv 95.0%

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

      8. associate-/l/ 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. associate-/r/ 96.2%

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

       2. /-rgt-identity 96.2%

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

       3. associate-*r* 95.1%

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

   11. Applied egg-rr 95.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.04 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{\frac{x}{\frac{1}{s}}}{\frac{1}{c}} \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x \cdot 2\right)\\ \end{array} \]

Alternative 2: 97.0% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c 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
 (* (cos (* x 2.0)) (pow (* c (* x s)) -2.0)))

C

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

Fortran

NOTE: x should be positive before calling this function
NOTE: c 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 = cos((x * 2.0d0)) * ((c * (x * s)) ** (-2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: c 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[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

2. Taylor expanded in c around 0 59.5%

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

   1. *-commutative 59.5%

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

   2. unpow2 59.5%

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

   3. unpow2 59.5%

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

   4. swap-sqr 78.7%

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

   5. unpow2 78.7%

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

   6. unpow2 78.7%

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

   7. rem-square-sqrt 78.7%

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

   8. swap-sqr 88.5%

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

   9. unpow2 88.5%

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

   10. unpow2 88.5%

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

   11. rem-sqrt-square 96.0%

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

   12. *-commutative 96.0%

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

4. Simplified 96.0%

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

5. Taylor expanded in x around inf 78.7%

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

   1. associate-/r* 78.7%

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

   2. *-commutative 78.7%

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

   3. unpow2 78.7%

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

   4. sqr-abs 78.7%

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

   5. unpow2 78.7%

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

   6. associate-/l/ 78.7%

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

   7. *-commutative 78.7%

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

   8. unpow2 78.7%

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

   9. unpow2 78.7%

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

   10. swap-sqr 96.0%

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

   11. associate-/l/ 96.0%

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

   12. *-rgt-identity 96.0%

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

   13. associate-*r/ 96.0%

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

   14. *-rgt-identity 96.0%

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

   15. associate-*r/ 96.0%

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

   16. associate-*l* 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 3: 96.7% accurate, 2.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c 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);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return cos((x * 2.0)) / (t_0 * t_0);
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c 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);
c = Math.abs(c);
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)
c = abs(c)
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)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0))
end

MATLAB

x = abs(x)
c = abs(c)
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: c 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[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.7%

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

2. Taylor expanded in c around 0 59.5%

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

   1. *-commutative 59.5%

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

   2. unpow2 59.5%

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

   3. unpow2 59.5%

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

   4. swap-sqr 78.7%

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

   5. unpow2 78.7%

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

   6. unpow2 78.7%

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

   7. rem-square-sqrt 78.7%

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

   8. swap-sqr 88.5%

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

   9. unpow2 88.5%

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

   10. unpow2 88.5%

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

   11. rem-sqrt-square 96.0%

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

   12. *-commutative 96.0%

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

4. Simplified 96.0%

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

   1. add-sqr-sqrt 50.6%

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

   2. sqrt-prod 81.8%

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

   3. unpow2 81.8%

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

   4. pow2 81.8%

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

   5. add-sqr-sqrt 81.8%

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

   6. unpow2 81.8%

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

   7. add-sqr-sqrt 42.5%

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

   8. fabs-sqr 42.5%

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

   9. add-sqr-sqrt 56.7%

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

   10. add-sqr-sqrt 37.3%

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

   11. fabs-sqr 37.3%

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

   12. add-sqr-sqrt 81.8%

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

6. Applied egg-rr 96.0%

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

7. Final simplification 96.0%

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

Alternative 4: 79.3% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. associate-/r* 63.7%

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

   2. *-commutative 63.7%

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

   3. associate-*r* 59.5%

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

   4. unpow2 59.5%

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

   5. associate-/l/ 58.5%

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

3. Simplified 58.5%

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

4. Taylor expanded in x around 0 54.8%

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

   1. associate-/r* 54.8%

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

   2. *-commutative 54.8%

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

   3. unpow2 54.8%

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

   4. unpow2 54.8%

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

   5. swap-sqr 67.7%

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

   6. unpow2 67.7%

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

   7. associate-/r* 67.7%

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

   8. unpow2 67.7%

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

   9. rem-square-sqrt 67.7%

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

   10. swap-sqr 75.9%

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

   11. unpow2 75.9%

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

   12. unpow2 75.9%

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

   13. rem-sqrt-square 81.8%

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

   14. *-commutative 81.8%

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

6. Simplified 81.8%

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

   1. add-sqr-sqrt 50.6%

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

   2. sqrt-prod 81.8%

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

   3. unpow2 81.8%

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

   4. pow2 81.8%

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

   5. add-sqr-sqrt 81.8%

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

   6. unpow2 81.8%

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

   7. add-sqr-sqrt 42.5%

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

   8. fabs-sqr 42.5%

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

   9. add-sqr-sqrt 56.7%

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

   10. add-sqr-sqrt 37.3%

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

   11. fabs-sqr 37.3%

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

   12. add-sqr-sqrt 81.8%

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

8. Applied egg-rr 81.8%

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

   1. associate-*r* 80.4%

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

   2. *-commutative 80.4%

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

   3. /-rgt-identity 80.4%

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

   4. clear-num 80.4%

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

   5. div-inv 80.4%

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

   6. clear-num 80.3%

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

   7. div-inv 80.4%

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

   8. div-inv 80.4%

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

   9. associate-/r* 81.9%

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

10. Applied egg-rr 81.9%

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

11. Final simplification 81.9%

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

Alternative 5: 79.3% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. associate-/r* 63.7%

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

   2. *-commutative 63.7%

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

   3. associate-*r* 59.5%

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

   4. unpow2 59.5%

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

   5. associate-/l/ 58.5%

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

3. Simplified 58.5%

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

4. Taylor expanded in x around 0 54.8%

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

   1. associate-/r* 54.8%

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

   2. *-commutative 54.8%

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

   3. unpow2 54.8%

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

   4. unpow2 54.8%

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

   5. swap-sqr 67.7%

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

   6. unpow2 67.7%

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

   7. associate-/r* 67.7%

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

   8. unpow2 67.7%

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

   9. rem-square-sqrt 67.7%

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

   10. swap-sqr 75.9%

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

   11. unpow2 75.9%

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

   12. unpow2 75.9%

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

   13. rem-sqrt-square 81.8%

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

   14. *-commutative 81.8%

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

6. Simplified 81.8%

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

   1. add-sqr-sqrt 50.6%

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

   2. sqrt-prod 81.8%

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

   3. unpow2 81.8%

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

   4. pow2 81.8%

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

   5. add-sqr-sqrt 81.8%

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

   6. unpow2 81.8%

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

   7. add-sqr-sqrt 42.5%

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

   8. fabs-sqr 42.5%

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

   9. add-sqr-sqrt 56.7%

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

   10. add-sqr-sqrt 37.3%

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

   11. fabs-sqr 37.3%

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

   12. add-sqr-sqrt 81.8%

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

8. Applied egg-rr 81.8%

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

   1. /-rgt-identity 81.8%

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

   2. associate-/l* 81.8%

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

   3. associate-/r* 81.8%

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

10. Applied egg-rr 81.8%

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

11. Final simplification 81.8%

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

Alternative 6: 79.3% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c 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);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c 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);
c = Math.abs(c);
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)
c = abs(c)
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)
c = abs(c)
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)
c = abs(c)
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: c 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 63.7%

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

   1. associate-/r* 63.7%

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

   2. *-commutative 63.7%

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

   3. associate-*r* 59.5%

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

   4. unpow2 59.5%

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

   5. associate-/l/ 58.5%

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

3. Simplified 58.5%

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

4. Taylor expanded in x around 0 54.8%

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

   1. associate-/r* 54.8%

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

   2. *-commutative 54.8%

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

   3. unpow2 54.8%

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

   4. unpow2 54.8%

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

   5. swap-sqr 67.7%

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

   6. unpow2 67.7%

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

   7. associate-/r* 67.7%

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

   8. unpow2 67.7%

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

   9. rem-square-sqrt 67.7%

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

   10. swap-sqr 75.9%

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

   11. unpow2 75.9%

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

   12. unpow2 75.9%

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

   13. rem-sqrt-square 81.8%

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

   14. *-commutative 81.8%

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

6. Simplified 81.8%

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

   1. add-sqr-sqrt 50.6%

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

   2. sqrt-prod 81.8%

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

   3. unpow2 81.8%

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

   4. pow2 81.8%

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

   5. add-sqr-sqrt 81.8%

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

   6. unpow2 81.8%

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

   7. add-sqr-sqrt 42.5%

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

   8. fabs-sqr 42.5%

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

   9. add-sqr-sqrt 56.7%

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

   10. add-sqr-sqrt 37.3%

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

   11. fabs-sqr 37.3%

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

   12. add-sqr-sqrt 81.8%

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

8. Applied egg-rr 81.8%

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

9. Final simplification 81.8%

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

Reproduce

herbie shell --seed 2023308 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))