mixedcos

Percentage Accurate: 65.6% → 98.6%

Time: 22.7s

Alternatives: 11

Speedup: 24.1×

• 31.5% 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: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

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 2.4e+98)
   (/ (cos (* x 2.0)) (pow (* c (* x s)) 2.0))
   (/ (cos (* x -2.0)) (* s (* (* x c) (* s (* x c)))))))

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 <= 2.4e+98) {
		tmp = cos((x * 2.0)) / pow((c * (x * s)), 2.0);
	} else {
		tmp = cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))));
	}
	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 <= 2.4d+98) then
        tmp = cos((x * 2.0d0)) / ((c * (x * s)) ** 2.0d0)
    else
        tmp = cos((x * (-2.0d0))) / (s * ((x * c) * (s * (x * c))))
    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 <= 2.4e+98) {
		tmp = Math.cos((x * 2.0)) / Math.pow((c * (x * s)), 2.0);
	} else {
		tmp = Math.cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))));
	}
	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 <= 2.4e+98:
		tmp = math.cos((x * 2.0)) / math.pow((c * (x * s)), 2.0)
	else:
		tmp = math.cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))))
	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 <= 2.4e+98)
		tmp = Float64(cos(Float64(x * 2.0)) / (Float64(c * Float64(x * s)) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x * -2.0)) / Float64(s * Float64(Float64(x * c) * Float64(s * Float64(x * c)))));
	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 <= 2.4e+98)
		tmp = cos((x * 2.0)) / ((c * (x * s)) ^ 2.0);
	else
		tmp = cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))));
	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, 2.4e+98], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3999999999999999e98

   1. Initial program 74.0%

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

      1. add-sqr-sqrt 74.0%

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

      2. pow2 74.0%

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

      3. sqrt-prod 74.0%

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

      4. unpow2 74.0%

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

      5. sqrt-prod 37.0%

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

      6. add-sqr-sqrt 78.6%

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

      7. *-commutative 78.6%

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

      8. associate-*r* 71.4%

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

      9. unpow2 71.4%

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

      10. pow-prod-down 88.9%

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

      11. sqrt-pow1 97.6%

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

      12. metadata-eval 97.6%

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

      13. pow1 97.6%

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

      14. *-commutative 97.6%

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

   3. Applied egg-rr 97.6%

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

   if 2.3999999999999999e98 < x

   1. Initial program 54.6%

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

      1. *-commutative 54.6%

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

      2. /-rgt-identity 54.6%

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

      3. metadata-eval 54.6%

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

      4. associate-/l/ 54.6%

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

      5. metadata-eval 54.6%

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

      6. /-rgt-identity 54.6%

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

      7. cos-neg 54.6%

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

      8. *-commutative 54.6%

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

      9. distribute-rgt-neg-in 54.6%

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

      10. metadata-eval 54.6%

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

      11. *-commutative 54.6%

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

      12. associate-*l* 39.9%

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

      13. unpow2 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in x around inf 40.0%

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

      1. *-commutative 40.0%

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

      2. unpow2 40.0%

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

      3. unpow2 40.0%

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

      4. associate-*r* 54.6%

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

      5. *-commutative 54.6%

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

      6. unpow2 54.6%

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

      7. associate-*r* 59.1%

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

      8. associate-*r* 64.0%

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

      9. *-commutative 64.0%

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

      10. unpow2 64.0%

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

      11. associate-*l* 78.6%

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

      12. *-commutative 78.6%

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

      13. unpow2 78.6%

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

      14. associate-*l* 87.8%

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

      15. *-commutative 87.8%

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

      16. associate-*l* 92.5%

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

      17. unpow2 92.5%

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

      18. associate-*r* 96.7%

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

      19. *-commutative 96.7%

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

   6. Simplified 96.7%

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

      1. unpow2 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*r* 90.2%

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

   8. Applied egg-rr 90.2%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+98}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 2: 95.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.8e-98)
   (/ 1.0 (* c (* (* x s) (* c (* x s)))))
   (/ (cos (* x -2.0)) (* (* c s) (* x (* s (* x c)))))))

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 <= 1.8e-98) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = cos((x * -2.0)) / ((c * s) * (x * (s * (x * c))));
	}
	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 <= 1.8d-98) then
        tmp = 1.0d0 / (c * ((x * s) * (c * (x * s))))
    else
        tmp = cos((x * (-2.0d0))) / ((c * s) * (x * (s * (x * c))))
    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 <= 1.8e-98) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = Math.cos((x * -2.0)) / ((c * s) * (x * (s * (x * c))));
	}
	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 <= 1.8e-98:
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))))
	else:
		tmp = math.cos((x * -2.0)) / ((c * s) * (x * (s * (x * c))))
	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 <= 1.8e-98)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * Float64(c * Float64(x * s)))));
	else
		tmp = Float64(cos(Float64(x * -2.0)) / Float64(Float64(c * s) * Float64(x * Float64(s * Float64(x * c)))));
	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 <= 1.8e-98)
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	else
		tmp = cos((x * -2.0)) / ((c * s) * (x * (s * (x * c))));
	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, 1.8e-98], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(c * s), $MachinePrecision] * N[(x * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8000000000000001e-98

   1. Initial program 72.5%

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

   2. Taylor expanded in x around 0 59.4%

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

      1. unpow2 59.4%

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

      2. unpow2 59.4%

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

      3. associate-*r* 65.1%

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

      4. *-commutative 65.1%

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

      5. unpow2 65.1%

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

      6. associate-*r* 70.8%

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

      7. associate-*r* 72.0%

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

      8. *-commutative 72.0%

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

      9. unpow2 72.0%

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

      10. associate-*l* 78.1%

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

      11. *-commutative 78.1%

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

      12. unpow2 78.1%

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

      13. associate-*l* 82.0%

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

      14. *-commutative 82.0%

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

      15. associate-*l* 84.2%

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

      16. unpow2 84.2%

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

      17. associate-*r* 84.6%

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

      18. *-commutative 84.6%

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

   4. Simplified 84.6%

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

      1. unpow2 84.6%

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

      2. associate-*r* 82.9%

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

      3. associate-*l* 81.3%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in s around 0 80.4%

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

      1. /-rgt-identity 80.4%

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

      2. associate-*r* 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-*r* 84.2%

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

      5. associate-/l* 84.2%

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

      6. frac-2neg 84.2%

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

      7. metadata-eval 84.2%

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

      8. distribute-lft-neg-in 84.2%

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

      9. associate-/l/ 84.1%

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

      10. div-inv 84.0%

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

      11. div-inv 84.1%

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

      12. associate-/l* 82.0%

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

   9. Applied egg-rr 80.4%

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

       1. associate-/l/ 80.3%

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

       2. associate-/r/ 80.4%

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

       3. /-rgt-identity 80.4%

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

       4. *-commutative 80.4%

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

       5. *-commutative 80.4%

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

       6. *-commutative 80.4%

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

       7. associate-*r* 82.0%

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

   11. Simplified 82.0%

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

   if 1.8000000000000001e-98 < x

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

      2. /-rgt-identity 67.0%

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

      3. metadata-eval 67.0%

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

      4. associate-/l/ 66.9%

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

      5. metadata-eval 66.9%

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

      6. /-rgt-identity 66.9%

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

      7. cos-neg 66.9%

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

      8. *-commutative 66.9%

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

      9. distribute-rgt-neg-in 66.9%

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

      10. metadata-eval 66.9%

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

      11. *-commutative 66.9%

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

      12. associate-*l* 58.8%

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

      13. unpow2 58.8%

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

   3. Simplified 58.8%

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

   4. Taylor expanded in x around inf 58.9%

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

      1. *-commutative 58.9%

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

      2. unpow2 58.9%

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

      3. unpow2 58.9%

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

      4. associate-*r* 67.0%

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

      5. *-commutative 67.0%

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

      6. unpow2 67.0%

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

      7. associate-*r* 70.7%

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

      8. associate-*r* 74.6%

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

      9. *-commutative 74.6%

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

      10. unpow2 74.6%

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

      11. associate-*l* 84.1%

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

      12. *-commutative 84.1%

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

      13. unpow2 84.1%

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

      14. associate-*l* 93.2%

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

      15. *-commutative 93.2%

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

      16. associate-*l* 95.7%

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

      17. unpow2 95.7%

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

      18. associate-*r* 98.0%

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

      19. *-commutative 98.0%

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

   6. Simplified 98.0%

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

      1. unpow2 61.3%

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

      2. associate-*r* 60.5%

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

      3. associate-*l* 60.3%

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

   8. Applied egg-rr 83.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 3: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.1e-6)
   (pow (* c (* x s)) -2.0)
   (/ (cos (* x -2.0)) (* (* x c) (* s (* s (* x c)))))))

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 <= 1.1e-6) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = cos((x * -2.0)) / ((x * c) * (s * (s * (x * c))));
	}
	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 <= 1.1d-6) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = cos((x * (-2.0d0))) / ((x * c) * (s * (s * (x * c))))
    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 <= 1.1e-6) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = Math.cos((x * -2.0)) / ((x * c) * (s * (s * (x * c))));
	}
	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 <= 1.1e-6:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = math.cos((x * -2.0)) / ((x * c) * (s * (s * (x * c))))
	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 <= 1.1e-6)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(cos(Float64(x * -2.0)) / Float64(Float64(x * c) * Float64(s * Float64(s * Float64(x * c)))));
	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 <= 1.1e-6)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = cos((x * -2.0)) / ((x * c) * (s * (s * (x * c))));
	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, 1.1e-6], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x * c), $MachinePrecision] * N[(s * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-6

   1. Initial program 73.2%

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

   2. Taylor expanded in x around 0 61.3%

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

      1. unpow2 61.3%

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

      2. unpow2 61.3%

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

      3. associate-*r* 66.5%

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

      4. *-commutative 66.5%

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

      5. unpow2 66.5%

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

      6. associate-*r* 72.1%

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

      7. associate-*r* 73.7%

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

      8. *-commutative 73.7%

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

      9. unpow2 73.7%

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

      10. associate-*l* 79.7%

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

      11. *-commutative 79.7%

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

      12. unpow2 79.7%

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

      13. associate-*l* 83.7%

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

      14. *-commutative 83.7%

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

      15. associate-*l* 85.7%

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

      16. unpow2 85.7%

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

      17. associate-*r* 86.0%

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

      18. *-commutative 86.0%

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

   4. Simplified 86.0%

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

      1. expm1-log1p-u 84.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 75.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\right)} - 1} \]

      3. pow-flip 75.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(s \cdot \left(c \cdot x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      4. metadata-eval 75.7%

         \[\leadsto e^{\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   6. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 84.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 86.0%

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

      3. associate-*r* 85.5%

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

      4. *-commutative 85.5%

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

      5. associate-*r* 85.6%

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

   8. Simplified 85.6%

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

   if 1.1000000000000001e-6 < x

   1. Initial program 62.5%

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

      1. *-commutative 62.5%

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

      2. /-rgt-identity 62.5%

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

      3. metadata-eval 62.5%

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

      4. associate-/l/ 62.5%

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

      5. metadata-eval 62.5%

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

      6. /-rgt-identity 62.5%

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

      7. cos-neg 62.5%

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

      8. *-commutative 62.5%

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

      9. distribute-rgt-neg-in 62.5%

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

      10. metadata-eval 62.5%

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

      11. *-commutative 62.5%

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

      12. associate-*l* 51.6%

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

      13. unpow2 51.6%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in x around inf 51.6%

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

      1. *-commutative 51.6%

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

      2. unpow2 51.6%

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

      3. unpow2 51.6%

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

      4. associate-*r* 62.5%

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

      5. *-commutative 62.5%

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

      6. unpow2 62.5%

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

      7. associate-*r* 65.8%

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

      8. associate-*r* 69.4%

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

      9. *-commutative 69.4%

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

      10. unpow2 69.4%

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

      11. associate-*l* 80.5%

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

      12. *-commutative 80.5%

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

      13. unpow2 80.5%

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

      14. associate-*l* 90.8%

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

      15. *-commutative 90.8%

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

      16. associate-*l* 94.3%

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

      17. unpow2 94.3%

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

      18. associate-*r* 97.5%

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

      19. *-commutative 97.5%

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

   6. Simplified 97.5%

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

      1. unpow2 97.5%

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

      2. associate-*r* 89.2%

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

   8. Applied egg-rr 89.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-6}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 4: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

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 1.05e-21)
   (/ 1.0 (* c (* (* x s) (* c (* x s)))))
   (/ (cos (* x -2.0)) (* s (* (* x c) (* s (* x c)))))))

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 <= 1.05e-21) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))));
	}
	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 <= 1.05d-21) then
        tmp = 1.0d0 / (c * ((x * s) * (c * (x * s))))
    else
        tmp = cos((x * (-2.0d0))) / (s * ((x * c) * (s * (x * c))))
    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 <= 1.05e-21) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = Math.cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))));
	}
	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 <= 1.05e-21:
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))))
	else:
		tmp = math.cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))))
	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 <= 1.05e-21)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * Float64(c * Float64(x * s)))));
	else
		tmp = Float64(cos(Float64(x * -2.0)) / Float64(s * Float64(Float64(x * c) * Float64(s * Float64(x * c)))));
	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 <= 1.05e-21)
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	else
		tmp = cos((x * -2.0)) / (s * ((x * c) * (s * (x * c))));
	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, 1.05e-21], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(s * N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000006e-21

   1. Initial program 73.1%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

      3. associate-*r* 66.3%

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

      4. *-commutative 66.3%

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

      5. unpow2 66.3%

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

      6. associate-*r* 72.0%

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

      7. associate-*r* 73.6%

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

      8. *-commutative 73.6%

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

      9. unpow2 73.6%

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

      10. associate-*l* 79.6%

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

      11. *-commutative 79.6%

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

      12. unpow2 79.6%

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

      13. associate-*l* 83.6%

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

      14. *-commutative 83.6%

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

      15. associate-*l* 85.6%

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

      16. unpow2 85.6%

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

      17. associate-*r* 86.0%

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

      18. *-commutative 86.0%

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

   4. Simplified 86.0%

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

      1. unpow2 86.0%

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

      2. associate-*r* 84.4%

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

      3. associate-*l* 83.0%

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

   6. Applied egg-rr 83.0%

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

   7. Taylor expanded in s around 0 82.2%

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

      1. /-rgt-identity 82.2%

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

      2. associate-*r* 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*r* 85.6%

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

      5. associate-/l* 85.6%

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

      6. frac-2neg 85.6%

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

      7. metadata-eval 85.6%

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

      8. distribute-lft-neg-in 85.6%

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

      9. associate-/l/ 85.5%

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

      10. div-inv 85.5%

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

      11. div-inv 85.5%

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

      12. associate-/l* 83.5%

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

   9. Applied egg-rr 82.1%

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

       1. associate-/l/ 82.1%

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

       2. associate-/r/ 82.1%

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

       3. /-rgt-identity 82.1%

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

       4. *-commutative 82.1%

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

       5. *-commutative 82.1%

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

       6. *-commutative 82.1%

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

       7. associate-*r* 83.6%

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

   11. Simplified 83.6%

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

   if 1.05000000000000006e-21 < x

   1. Initial program 63.2%

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

      1. *-commutative 63.2%

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

      2. /-rgt-identity 63.2%

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

      3. metadata-eval 63.2%

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

      4. associate-/l/ 63.2%

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

      5. metadata-eval 63.2%

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

      6. /-rgt-identity 63.2%

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

      7. cos-neg 63.2%

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

      8. *-commutative 63.2%

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

      9. distribute-rgt-neg-in 63.2%

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

      10. metadata-eval 63.2%

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

      11. *-commutative 63.2%

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

      12. associate-*l* 52.4%

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

      13. unpow2 52.4%

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

   3. Simplified 52.4%

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

   4. Taylor expanded in x around inf 52.5%

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

      1. *-commutative 52.5%

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

      2. unpow2 52.5%

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

      3. unpow2 52.5%

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

      4. associate-*r* 63.2%

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

      5. *-commutative 63.2%

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

      6. unpow2 63.2%

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

      7. associate-*r* 66.4%

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

      8. associate-*r* 70.0%

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

      9. *-commutative 70.0%

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

      10. unpow2 70.0%

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

      11. associate-*l* 80.8%

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

      12. *-commutative 80.8%

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

      13. unpow2 80.8%

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

      14. associate-*l* 91.0%

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

      15. *-commutative 91.0%

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

      16. associate-*l* 94.4%

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

      17. unpow2 94.4%

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

      18. associate-*r* 97.5%

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

      19. *-commutative 97.5%

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

   6. Simplified 97.5%

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

      1. unpow2 97.5%

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

      2. *-commutative 97.5%

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

      3. associate-*r* 92.7%

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

   8. Applied egg-rr 92.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: 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 (* s (* x c))))
   (if (<= x 2.05e-22)
     (/ 1.0 (* c (* (* x s) (* 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 = s * (x * c);
	double tmp;
	if (x <= 2.05e-22) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = (cos((x * 2.0)) / t_0) / t_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) :: t_0
    real(8) :: tmp
    t_0 = s * (x * c)
    if (x <= 2.05d-22) then
        tmp = 1.0d0 / (c * ((x * s) * (c * (x * s))))
    else
        tmp = (cos((x * 2.0d0)) / t_0) / t_0
    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 t_0 = s * (x * c);
	double tmp;
	if (x <= 2.05e-22) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = (Math.cos((x * 2.0)) / t_0) / t_0;
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if x <= 2.05e-22:
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))))
	else:
		tmp = (math.cos((x * 2.0)) / t_0) / t_0
	return tmp

Julia

x = abs(x)
c = abs(c)
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 <= 2.05e-22)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * Float64(c * Float64(x * s)))));
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / t_0) / t_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)
	t_0 = s * (x * c);
	tmp = 0.0;
	if (x <= 2.05e-22)
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	else
		tmp = (cos((x * 2.0)) / t_0) / t_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_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.05e-22], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.05e-22

   1. Initial program 73.1%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

      3. associate-*r* 66.3%

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

      4. *-commutative 66.3%

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

      5. unpow2 66.3%

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

      6. associate-*r* 72.0%

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

      7. associate-*r* 73.6%

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

      8. *-commutative 73.6%

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

      9. unpow2 73.6%

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

      10. associate-*l* 79.6%

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

      11. *-commutative 79.6%

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

      12. unpow2 79.6%

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

      13. associate-*l* 83.6%

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

      14. *-commutative 83.6%

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

      15. associate-*l* 85.6%

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

      16. unpow2 85.6%

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

      17. associate-*r* 86.0%

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

      18. *-commutative 86.0%

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

   4. Simplified 86.0%

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

      1. unpow2 86.0%

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

      2. associate-*r* 84.4%

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

      3. associate-*l* 83.0%

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

   6. Applied egg-rr 83.0%

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

   7. Taylor expanded in s around 0 82.2%

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

      1. /-rgt-identity 82.2%

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

      2. associate-*r* 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*r* 85.6%

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

      5. associate-/l* 85.6%

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

      6. frac-2neg 85.6%

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

      7. metadata-eval 85.6%

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

      8. distribute-lft-neg-in 85.6%

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

      9. associate-/l/ 85.5%

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

      10. div-inv 85.5%

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

      11. div-inv 85.5%

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

      12. associate-/l* 83.5%

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

   9. Applied egg-rr 82.1%

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

       1. associate-/l/ 82.1%

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

       2. associate-/r/ 82.1%

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

       3. /-rgt-identity 82.1%

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

       4. *-commutative 82.1%

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

       5. *-commutative 82.1%

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

       6. *-commutative 82.1%

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

       7. associate-*r* 83.6%

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

   11. Simplified 83.6%

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

   if 2.05e-22 < x

   1. Initial program 63.2%

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

   3. Applied egg-rr 80.5%

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

      1. frac-times 73.8%

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

      2. *-un-lft-identity 73.8%

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

      3. unpow2 73.8%

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

      4. associate-/l* 73.7%

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

      5. add-sqr-sqrt 58.9%

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

      6. sqrt-unprod 32.4%

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

      7. *-commutative 32.4%

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

      8. *-commutative 32.4%

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

      9. swap-sqr 32.4%

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

      10. metadata-eval 32.4%

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

      11. metadata-eval 32.4%

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

      12. swap-sqr 32.4%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 73.7%

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

      15. metadata-eval 73.7%

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

      16. pow-flip 73.7%

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

      17. associate-/l* 73.8%

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

      18. div-inv 73.7%

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

   5. Applied egg-rr 97.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 6: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \end{array} \]

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 5.1e-92)
   (/ 1.0 (* c (* (* x s) (* c (* x s)))))
   (/ (/ (/ (cos (* x 2.0)) s) (* x c)) (* s (* x c)))))

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 <= 5.1e-92) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = ((cos((x * 2.0)) / s) / (x * c)) / (s * (x * c));
	}
	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 <= 5.1d-92) then
        tmp = 1.0d0 / (c * ((x * s) * (c * (x * s))))
    else
        tmp = ((cos((x * 2.0d0)) / s) / (x * c)) / (s * (x * c))
    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 <= 5.1e-92) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = ((Math.cos((x * 2.0)) / s) / (x * c)) / (s * (x * c));
	}
	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 <= 5.1e-92:
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))))
	else:
		tmp = ((math.cos((x * 2.0)) / s) / (x * c)) / (s * (x * c))
	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 <= 5.1e-92)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * Float64(c * Float64(x * s)))));
	else
		tmp = Float64(Float64(Float64(cos(Float64(x * 2.0)) / s) / Float64(x * c)) / Float64(s * Float64(x * c)));
	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 <= 5.1e-92)
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	else
		tmp = ((cos((x * 2.0)) / s) / (x * c)) / (s * (x * c));
	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, 5.1e-92], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.09999999999999972e-92

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 59.6%

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

      1. unpow2 59.6%

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

      2. unpow2 59.6%

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

      3. associate-*r* 65.3%

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

      4. *-commutative 65.3%

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

      5. unpow2 65.3%

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

      6. associate-*r* 71.0%

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

      7. associate-*r* 72.2%

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

      8. *-commutative 72.2%

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

      9. unpow2 72.2%

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

      10. associate-*l* 78.2%

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

      11. *-commutative 78.2%

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

      12. unpow2 78.2%

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

      13. associate-*l* 82.1%

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

      14. *-commutative 82.1%

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

      15. associate-*l* 84.3%

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

      16. unpow2 84.3%

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

      17. associate-*r* 84.7%

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

      18. *-commutative 84.7%

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

   4. Simplified 84.7%

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

      1. unpow2 84.7%

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

      2. associate-*r* 83.0%

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

      3. associate-*l* 81.4%

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

   6. Applied egg-rr 81.4%

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

   7. Taylor expanded in s around 0 80.5%

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

      1. /-rgt-identity 80.5%

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

      2. associate-*r* 82.6%

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

      3. *-commutative 82.6%

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

      4. associate-*r* 84.3%

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

      5. associate-/l* 84.3%

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

      6. frac-2neg 84.3%

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

      7. metadata-eval 84.3%

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

      8. distribute-lft-neg-in 84.3%

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

      9. associate-/l/ 84.2%

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

      10. div-inv 84.1%

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

      11. div-inv 84.2%

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

      12. associate-/l* 82.0%

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

   9. Applied egg-rr 80.5%

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

       1. associate-/l/ 80.5%

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

       2. associate-/r/ 80.5%

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

       3. /-rgt-identity 80.5%

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

       4. *-commutative 80.5%

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

       5. *-commutative 80.5%

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

       6. *-commutative 80.5%

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

       7. associate-*r* 82.1%

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

   11. Simplified 82.1%

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

   if 5.09999999999999972e-92 < x

   1. Initial program 66.5%

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

   3. Applied egg-rr 82.3%

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

      1. frac-times 74.6%

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

      2. *-un-lft-identity 74.6%

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

      3. unpow2 74.6%

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

      4. associate-/l* 74.6%

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

      5. add-sqr-sqrt 63.2%

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

      6. sqrt-unprod 42.8%

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

      7. *-commutative 42.8%

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

      8. *-commutative 42.8%

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

      9. swap-sqr 42.8%

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

      10. metadata-eval 42.8%

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

      11. metadata-eval 42.8%

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

      12. swap-sqr 42.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 74.6%

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

      15. metadata-eval 74.6%

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

      16. pow-flip 74.6%

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

      17. associate-/l* 74.6%

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

      18. div-inv 74.6%

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

   5. Applied egg-rr 98.0%

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

      1. *-un-lft-identity 98.0%

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

      2. *-commutative 98.0%

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

      3. times-frac 98.1%

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

      4. *-commutative 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. associate-*l/ 98.1%

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

      2. *-commutative 98.1%

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

      3. *-rgt-identity 98.1%

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

      4. *-commutative 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 7: 79.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{+15}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}\\ \end{array} \end{array} \]

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 2.25e+15)
   (pow (* c (* x s)) -2.0)
   (/ -1.0 (pow (* x (* c s)) 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 <= 2.25e+15) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = -1.0 / pow((x * (c * s)), 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 <= 2.25d+15) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (-1.0d0) / ((x * (c * s)) ** 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 <= 2.25e+15) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = -1.0 / Math.pow((x * (c * s)), 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 <= 2.25e+15:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = -1.0 / math.pow((x * (c * s)), 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 <= 2.25e+15)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(-1.0 / (Float64(x * Float64(c * s)) ^ 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 <= 2.25e+15)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = -1.0 / ((x * (c * s)) ^ 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, 2.25e+15], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(-1.0 / N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25e15

   1. Initial program 73.3%

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

   2. Taylor expanded in x around 0 60.7%

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

      1. unpow2 60.7%

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

      2. unpow2 60.7%

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

      3. associate-*r* 65.8%

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

      4. *-commutative 65.8%

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

      5. unpow2 65.8%

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

      6. associate-*r* 71.3%

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

      7. associate-*r* 72.9%

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

      8. *-commutative 72.9%

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

      9. unpow2 72.9%

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

      10. associate-*l* 78.7%

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

      11. *-commutative 78.7%

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

      12. unpow2 78.7%

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

      13. associate-*l* 82.8%

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

      14. *-commutative 82.8%

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

      15. associate-*l* 84.7%

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

      16. unpow2 84.7%

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

      17. associate-*r* 85.0%

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

      18. *-commutative 85.0%

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

   4. Simplified 85.0%

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

      1. expm1-log1p-u 83.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\right)\right)} \]

      2. expm1-udef 74.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\right)} - 1} \]

      3. pow-flip 74.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(s \cdot \left(c \cdot x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      4. metadata-eval 74.8%

         \[\leadsto e^{\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   6. Applied egg-rr 74.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 83.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 85.0%

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

      3. associate-*r* 84.5%

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

      4. *-commutative 84.5%

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

      5. associate-*r* 84.7%

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

   8. Simplified 84.7%

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

   if 2.25e15 < x

   1. Initial program 61.5%

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

   2. Taylor expanded in x around 0 42.8%

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

      1. unpow2 42.8%

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

      2. unpow2 42.8%

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

      3. associate-*r* 46.1%

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

      4. *-commutative 46.1%

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

      5. unpow2 46.1%

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

      6. associate-*r* 46.3%

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

      7. associate-*r* 46.7%

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

      8. *-commutative 46.7%

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

      9. unpow2 46.7%

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

      10. associate-*l* 48.3%

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

      11. *-commutative 48.3%

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

      12. unpow2 48.3%

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

      13. associate-*l* 49.5%

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

      14. *-commutative 49.5%

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

      15. associate-*l* 49.4%

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

      16. unpow2 49.4%

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

      17. associate-*r* 49.1%

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

      18. *-commutative 49.1%

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

   4. Simplified 49.1%

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

      1. add-sqr-sqrt 49.1%

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

      2. sqrt-div 49.1%

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

      3. metadata-eval 49.1%

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

      4. unpow2 49.1%

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

      5. sqrt-prod 30.7%

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

      6. add-sqr-sqrt 45.3%

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

      7. sqrt-div 45.3%

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

      8. metadata-eval 45.3%

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

      9. unpow2 45.3%

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

      10. sqrt-prod 30.7%

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

      11. add-sqr-sqrt 49.1%

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

   6. Applied egg-rr 49.1%

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

      1. unpow2 49.1%

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

      2. associate-/r* 49.1%

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

      3. associate-/l/ 49.4%

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

      4. associate-/r* 49.4%

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

   8. Simplified 49.4%

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

      1. associate-/r* 49.4%

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

      2. associate-/l/ 49.1%

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

      3. associate-/r* 49.1%

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

      4. inv-pow 49.1%

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

      5. metadata-eval 49.1%

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

      6. *-commutative 49.1%

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

      7. unpow-prod-down 49.1%

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

      8. metadata-eval 49.1%

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

      9. inv-pow 49.1%

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

      10. metadata-eval 49.1%

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

      11. metadata-eval 49.1%

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

      12. metadata-eval 49.1%

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

      13. sqrt-pow1 47.8%

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

      14. pow-flip 47.6%

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

      15. *-commutative 47.6%

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

      16. sqrt-div 47.6%

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

      17. metadata-eval 47.6%

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

      18. unpow2 47.6%

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

      19. sqrt-prod 21.5%

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

      20. add-sqr-sqrt 49.1%

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

   10. Applied egg-rr 49.1%

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

       1. *-commutative 49.1%

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

       2. associate-*r/ 49.1%

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

       3. *-rgt-identity 49.1%

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

       4. *-commutative 49.1%

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

   12. Simplified 49.1%

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

       1. unpow2 49.1%

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

       2. associate-/r* 49.1%

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

       3. associate-*r* 47.9%

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

       4. *-commutative 47.9%

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

       5. associate-*r* 49.2%

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

       6. frac-2neg 49.2%

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

       7. metadata-eval 49.2%

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

       8. distribute-lft-neg-in 49.2%

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

       9. associate-/l/ 49.2%

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

       10. associate-/r* 49.2%

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

       11. associate-*r* 48.3%

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

       12. *-commutative 48.3%

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

       13. associate-*r* 49.4%

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

       14. times-frac 48.9%

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

       15. *-un-lft-identity 48.9%

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

       16. associate-/l/ 48.9%

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

   14. Applied egg-rr 45.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{+15}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}\\ \end{array} \]

Alternative 8: 79.7% accurate, 3.0× speedup

Math

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

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 (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 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 = (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.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.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(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 = (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[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 70.9%

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

2. Taylor expanded in x around 0 57.1%

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

   1. unpow2 57.1%

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

   2. unpow2 57.1%

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

   3. associate-*r* 61.8%

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

   4. *-commutative 61.8%

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

   5. unpow2 61.8%

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

   6. associate-*r* 66.4%

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

   7. associate-*r* 67.7%

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

   8. *-commutative 67.7%

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

   9. unpow2 67.7%

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

   10. associate-*l* 72.7%

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

   11. *-commutative 72.7%

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

   12. unpow2 72.7%

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

   13. associate-*l* 76.1%

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

   14. *-commutative 76.1%

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

   15. associate-*l* 77.7%

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

   16. unpow2 77.7%

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

   17. associate-*r* 77.9%

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

   18. *-commutative 77.9%

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

4. Simplified 77.9%

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

   1. expm1-log1p-u 77.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\right)\right)} \]

   2. expm1-udef 69.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\right)} - 1} \]

   3. pow-flip 69.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(s \cdot \left(c \cdot x\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

   4. metadata-eval 69.5%

      \[\leadsto e^{\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

6. Applied egg-rr 69.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 77.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(s \cdot \left(c \cdot x\right)\right)}^{-2}\right)\right)} \]

   2. expm1-log1p 77.9%

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

   3. associate-*r* 77.3%

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

   4. *-commutative 77.3%

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

   5. associate-*r* 77.7%

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

8. Simplified 77.7%

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

9. Final simplification 77.7%

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

Alternative 9: 79.7% accurate, 20.9× speedup

Math

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

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 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(Float64(Float64(Float64(-1.0 / c) / Float64(x * s)) / -1.0) / 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 / 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[(N[(N[(N[(-1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.9%

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

2. Taylor expanded in x around 0 57.1%

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

   1. unpow2 57.1%

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

   2. unpow2 57.1%

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

   3. associate-*r* 61.8%

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

   4. *-commutative 61.8%

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

   5. unpow2 61.8%

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

   6. associate-*r* 66.4%

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

   7. associate-*r* 67.7%

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

   8. *-commutative 67.7%

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

   9. unpow2 67.7%

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

   10. associate-*l* 72.7%

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

   11. *-commutative 72.7%

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

   12. unpow2 72.7%

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

   13. associate-*l* 76.1%

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

   14. *-commutative 76.1%

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

   15. associate-*l* 77.7%

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

   16. unpow2 77.7%

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

   17. associate-*r* 77.9%

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

   18. *-commutative 77.9%

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

4. Simplified 77.9%

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

   1. add-sqr-sqrt 77.9%

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

   2. sqrt-div 77.9%

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

   3. metadata-eval 77.9%

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

   4. unpow2 77.9%

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

   5. sqrt-prod 43.4%

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

   6. add-sqr-sqrt 54.3%

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

   7. sqrt-div 54.3%

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

   8. metadata-eval 54.3%

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

   9. unpow2 54.3%

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

   10. sqrt-prod 36.3%

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

   11. add-sqr-sqrt 77.9%

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

6. Applied egg-rr 77.9%

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

   1. unpow2 77.9%

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

   2. associate-/r* 77.9%

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

   3. associate-/l/ 77.5%

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

   4. associate-/r* 77.5%

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

8. Simplified 77.5%

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

   1. unpow2 77.5%

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

   2. frac-2neg 77.5%

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

   3. associate-/r* 77.6%

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

   4. associate-/l/ 76.8%

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

   5. associate-/r* 76.8%

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

   6. frac-times 73.3%

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

   7. *-un-lft-identity 73.3%

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

   8. distribute-neg-frac 73.3%

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

   9. metadata-eval 73.3%

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

   10. *-commutative 73.3%

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

   11. associate-*r* 73.7%

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

   12. *-commutative 73.7%

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

10. Applied egg-rr 73.7%

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

    1. *-commutative 73.7%

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

    2. associate-/r* 76.2%

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

    3. *-rgt-identity 76.2%

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

    4. *-rgt-identity 76.2%

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

    5. metadata-eval 76.2%

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

    6. distribute-neg-frac 76.2%

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

    7. associate-/l/ 76.2%

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

    8. associate-/l/ 76.2%

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

    9. distribute-neg-frac 76.2%

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

    10. metadata-eval 76.2%

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

    11. *-commutative 76.2%

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

    12. *-commutative 76.2%

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

    13. *-commutative 76.2%

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

    14. associate-*r* 77.6%

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

12. Simplified 77.6%

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

    1. add-sqr-sqrt 40.3%

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

    2. neg-mul-1 40.3%

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

    3. times-frac 40.3%

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

    4. associate-/r* 39.5%

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

    5. associate-/r* 39.5%

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

14. Applied egg-rr 39.5%

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

    1. associate-*l/ 39.5%

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

16. Simplified 77.7%

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

17. Final simplification 77.7%

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

Alternative 10: 79.7% accurate, 22.4× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: 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 s)) (- 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 * s)) / -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 * s)) / -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 * s)) / -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 * s)) / -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(Float64(Float64(-1.0 / Float64(x * s)) / Float64(-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 * s)) / -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[(N[(N[(-1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / (-c)), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.9%

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

2. Taylor expanded in x around 0 57.1%

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

   1. unpow2 57.1%

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

   2. unpow2 57.1%

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

   3. associate-*r* 61.8%

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

   4. *-commutative 61.8%

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

   5. unpow2 61.8%

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

   6. associate-*r* 66.4%

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

   7. associate-*r* 67.7%

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

   8. *-commutative 67.7%

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

   9. unpow2 67.7%

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

   10. associate-*l* 72.7%

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

   11. *-commutative 72.7%

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

   12. unpow2 72.7%

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

   13. associate-*l* 76.1%

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

   14. *-commutative 76.1%

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

   15. associate-*l* 77.7%

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

   16. unpow2 77.7%

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

   17. associate-*r* 77.9%

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

   18. *-commutative 77.9%

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

4. Simplified 77.9%

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

   1. add-sqr-sqrt 77.9%

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

   2. sqrt-div 77.9%

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

   3. metadata-eval 77.9%

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

   4. unpow2 77.9%

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

   5. sqrt-prod 43.4%

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

   6. add-sqr-sqrt 54.3%

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

   7. sqrt-div 54.3%

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

   8. metadata-eval 54.3%

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

   9. unpow2 54.3%

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

   10. sqrt-prod 36.3%

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

   11. add-sqr-sqrt 77.9%

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

6. Applied egg-rr 77.9%

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

   1. unpow2 77.9%

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

   2. associate-/r* 77.9%

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

   3. associate-/l/ 77.5%

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

   4. associate-/r* 77.5%

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

8. Simplified 77.5%

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

   1. unpow2 77.5%

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

   2. frac-2neg 77.5%

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

   3. associate-/r* 77.6%

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

   4. associate-/l/ 76.8%

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

   5. associate-/r* 76.8%

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

   6. frac-times 73.3%

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

   7. *-un-lft-identity 73.3%

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

   8. distribute-neg-frac 73.3%

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

   9. metadata-eval 73.3%

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

   10. *-commutative 73.3%

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

   11. associate-*r* 73.7%

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

   12. *-commutative 73.7%

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

10. Applied egg-rr 73.7%

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

    1. *-commutative 73.7%

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

    2. associate-/r* 76.2%

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

    3. *-rgt-identity 76.2%

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

    4. *-rgt-identity 76.2%

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

    5. metadata-eval 76.2%

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

    6. distribute-neg-frac 76.2%

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

    7. associate-/l/ 76.2%

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

    8. associate-/l/ 76.2%

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

    9. distribute-neg-frac 76.2%

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

    10. metadata-eval 76.2%

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

    11. *-commutative 76.2%

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

    12. *-commutative 76.2%

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

    13. *-commutative 76.2%

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

    14. associate-*r* 77.6%

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

12. Simplified 77.6%

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

13. Final simplification 77.6%

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

Alternative 11: 78.9% accurate, 24.1× speedup

Math

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

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 (* c (* (* x s) (* 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) * (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) * (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) * (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) * (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(c * Float64(Float64(x * s) * 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 / (c * ((x * s) * (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[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.9%

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

2. Taylor expanded in x around 0 57.1%

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

   1. unpow2 57.1%

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

   2. unpow2 57.1%

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

   3. associate-*r* 61.8%

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

   4. *-commutative 61.8%

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

   5. unpow2 61.8%

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

   6. associate-*r* 66.4%

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

   7. associate-*r* 67.7%

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

   8. *-commutative 67.7%

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

   9. unpow2 67.7%

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

   10. associate-*l* 72.7%

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

   11. *-commutative 72.7%

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

   12. unpow2 72.7%

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

   13. associate-*l* 76.1%

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

   14. *-commutative 76.1%

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

   15. associate-*l* 77.7%

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

   16. unpow2 77.7%

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

   17. associate-*r* 77.9%

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

   18. *-commutative 77.9%

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

4. Simplified 77.9%

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

   1. unpow2 77.9%

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

   2. associate-*r* 76.4%

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

   3. associate-*l* 75.2%

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

6. Applied egg-rr 75.2%

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

7. Taylor expanded in s around 0 74.7%

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

   1. /-rgt-identity 74.7%

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

   2. associate-*r* 76.3%

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

   3. *-commutative 76.3%

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

   4. associate-*r* 77.7%

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

   5. associate-/l* 77.7%

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

   6. frac-2neg 77.7%

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

   7. metadata-eval 77.7%

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

   8. distribute-lft-neg-in 77.7%

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

   9. associate-/l/ 77.6%

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

   10. div-inv 77.6%

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

   11. div-inv 77.6%

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

   12. associate-/l* 76.1%

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

9. Applied egg-rr 74.7%

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

    1. associate-/l/ 74.7%

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

    2. associate-/r/ 74.7%

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

    3. /-rgt-identity 74.7%

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

    4. *-commutative 74.7%

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

    5. *-commutative 74.7%

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

    6. *-commutative 74.7%

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

    7. associate-*r* 76.1%

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

11. Simplified 76.1%

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

12. Final simplification 76.1%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))