mixedcos

Percentage Accurate: 66.8% → 99.0%

Time: 13.5s

Alternatives: 6

Speedup: 24.1×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s\_m} \cdot \frac{\frac{\cos \left(x\_m \cdot 2\right)}{s\_m \cdot \left(c\_m \cdot x\_m\right)}}{c\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<= x_m 2.7e-14)
     (/ (/ 1.0 t_0) t_0)
     (*
      (/ 1.0 s_m)
      (/ (/ (cos (* x_m 2.0)) (* s_m (* c_m x_m))) (* c_m x_m))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 2.7e-14) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (1.0 / s_m) * ((cos((x_m * 2.0)) / (s_m * (c_m * x_m))) / (c_m * x_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if (x_m <= 2.7d-14) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = (1.0d0 / s_m) * ((cos((x_m * 2.0d0)) / (s_m * (c_m * x_m))) / (c_m * x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 2.7e-14) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (1.0 / s_m) * ((Math.cos((x_m * 2.0)) / (s_m * (c_m * x_m))) / (c_m * x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 2.7e-14:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = (1.0 / s_m) * ((math.cos((x_m * 2.0)) / (s_m * (c_m * x_m))) / (c_m * x_m))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 2.7e-14)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(1.0 / s_m) * Float64(Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(c_m * x_m))) / Float64(c_m * x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 2.7e-14)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = (1.0 / s_m) * ((cos((x_m * 2.0)) / (s_m * (c_m * x_m))) / (c_m * x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.7e-14], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / s$95$m), $MachinePrecision] * N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6999999999999999e-14

   1. Initial program 63.7%

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

      1. associate-/r* 63.7%

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

      2. cos-neg 63.7%

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

      3. distribute-rgt-neg-out 63.7%

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

      4. distribute-rgt-neg-out 63.7%

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

      5. *-commutative 63.7%

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

      6. distribute-rgt-neg-in 63.7%

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

      7. metadata-eval 63.7%

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

      8. *-commutative 63.7%

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

      9. associate-*l* 57.9%

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

      10. unpow2 57.9%

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

   3. Simplified 57.9%

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

   6. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.1%

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

      2. *-un-lft-identity 97.1%

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

      3. *-commutative 97.1%

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

   8. Applied egg-rr 97.1%

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

   9. Taylor expanded in x around 0 89.6%

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

   if 2.6999999999999999e-14 < x

   1. Initial program 67.4%

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

      1. associate-/r* 68.3%

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

      2. cos-neg 68.3%

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

      3. distribute-rgt-neg-out 68.3%

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

      4. distribute-rgt-neg-out 68.3%

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

      5. *-commutative 68.3%

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

      6. distribute-rgt-neg-in 68.3%

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

      7. metadata-eval 68.3%

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

      8. *-commutative 68.3%

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

      9. associate-*l* 64.5%

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

      10. unpow2 64.5%

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

   3. Simplified 64.5%

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

      1. associate-/l/ 64.4%

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

      2. add-sqr-sqrt 39.5%

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

      3. add-sqr-sqrt 64.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 36.6%

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

      6. swap-sqr 36.6%

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

      7. metadata-eval 36.6%

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

      8. metadata-eval 36.6%

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

      9. swap-sqr 36.6%

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

      10. *-commutative 36.6%

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

      11. *-commutative 36.6%

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

      12. sqrt-unprod 61.7%

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

      13. add-sqr-sqrt 64.4%

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

      14. *-un-lft-identity 64.4%

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

      15. *-commutative 64.4%

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

      16. unpow2 64.4%

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

      17. associate-*r* 67.4%

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

      18. associate-*l* 70.9%

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

      19. *-commutative 70.9%

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

      20. *-commutative 70.9%

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

   6. Applied egg-rr 90.4%

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

   7. Taylor expanded in c around 0 64.4%

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

      1. associate-/r* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. unpow2 64.5%

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

      5. unpow2 64.5%

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

      6. swap-sqr 79.8%

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

      7. unpow2 79.8%

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

      8. associate-/l/ 78.8%

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

      9. *-commutative 78.8%

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

      10. unpow2 78.8%

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

      11. unpow2 78.8%

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

      12. swap-sqr 92.5%

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

      13. associate-/r* 94.2%

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

      14. *-lft-identity 94.2%

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

      15. associate-*r/ 94.2%

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

      16. associate-*l/ 94.2%

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

      17. associate-/r/ 94.2%

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

      18. associate-/l/ 92.4%

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

   9. Simplified 94.9%

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

       1. *-commutative 94.9%

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

       2. *-commutative 94.9%

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

       3. associate-*r* 98.3%

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

       4. *-commutative 98.3%

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

       5. unpow-prod-down 84.1%

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

       6. metadata-eval 84.1%

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

       7. pow-flip 83.4%

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

       8. pow2 83.4%

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

       9. metadata-eval 83.4%

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

       10. pow-prod-up 83.4%

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

       11. inv-pow 83.4%

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

       12. inv-pow 83.4%

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

       13. un-div-inv 83.4%

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

       14. times-frac 90.4%

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

       15. *-un-lft-identity 90.4%

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

       16. associate-/l/ 88.5%

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

       17. div-inv 88.6%

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

       18. *-commutative 88.6%

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

   11. Applied egg-rr 94.0%

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

       1. unpow2 94.0%

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

       2. *-commutative 94.0%

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

       3. associate-*r* 88.6%

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

       4. associate-*r* 87.4%

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

       5. associate-/r* 88.3%

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

       6. *-commutative 88.3%

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

       7. associate-*r* 90.9%

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

       8. *-commutative 90.9%

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

       9. associate-/l/ 91.6%

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

       10. associate-/r* 94.2%

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

       11. *-un-lft-identity 94.2%

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

       12. associate-*r* 89.5%

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

       13. *-commutative 89.5%

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

       14. times-frac 86.0%

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

   13. Applied egg-rr 91.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(c \cdot x\right)}}{c \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (* (pow (* s_m (* c_m x_m)) -2.0) (cos (* x_m 2.0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return pow((s_m * (c_m * x_m)), -2.0) * cos((x_m * 2.0));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[Power[N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.7%

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

   1. associate-/r* 65.0%

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

   2. cos-neg 65.0%

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

   3. distribute-rgt-neg-out 65.0%

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

   4. distribute-rgt-neg-out 65.0%

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

   5. *-commutative 65.0%

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

   6. distribute-rgt-neg-in 65.0%

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

   7. metadata-eval 65.0%

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

   8. *-commutative 65.0%

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

   9. associate-*l* 59.7%

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

   10. unpow2 59.7%

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

3. Simplified 59.7%

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

   1. associate-/l/ 59.7%

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

   2. add-sqr-sqrt 44.6%

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

   3. add-sqr-sqrt 59.7%

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

   4. add-sqr-sqrt 24.7%

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

   5. sqrt-unprod 43.8%

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

   6. swap-sqr 43.8%

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

   7. metadata-eval 43.8%

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

   8. metadata-eval 43.8%

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

   9. swap-sqr 43.8%

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

   10. *-commutative 43.8%

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

   11. *-commutative 43.8%

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

   12. sqrt-unprod 33.7%

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

   13. add-sqr-sqrt 59.7%

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

   14. *-un-lft-identity 59.7%

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

   15. *-commutative 59.7%

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

   16. unpow2 59.7%

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

   17. associate-*r* 64.7%

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

   18. associate-*l* 67.7%

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

   19. *-commutative 67.7%

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

   20. *-commutative 67.7%

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

6. Applied egg-rr 87.6%

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

7. Taylor expanded in c around 0 59.7%

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

   1. associate-/r* 59.7%

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

   2. *-commutative 59.7%

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

   3. *-commutative 59.7%

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

   4. unpow2 59.7%

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

   5. unpow2 59.7%

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

   6. swap-sqr 78.1%

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

   7. unpow2 78.1%

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

   8. associate-/l/ 77.8%

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

   9. *-commutative 77.8%

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

   10. unpow2 77.8%

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

   11. unpow2 77.8%

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

   12. swap-sqr 95.6%

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

   13. associate-/r* 96.3%

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

   14. *-lft-identity 96.3%

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

   15. associate-*r/ 96.3%

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

   16. associate-*l/ 96.3%

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

   17. associate-/r/ 96.3%

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

   18. associate-/l/ 95.6%

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

9. Simplified 98.3%

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

Alternative 3: 99.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 2.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{t\_1}{c\_m}}{\left(s\_m \cdot x\_m\right) \cdot \left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 2.7e-34)
     (/ (/ t_1 c_m) (* (* s_m x_m) (* c_m (* s_m x_m))))
     (/ t_1 (* t_0 t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 2.7e-34) {
		tmp = (t_1 / c_m) / ((s_m * x_m) * (c_m * (s_m * x_m)));
	} else {
		tmp = t_1 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s_m * (c_m * x_m)
    t_1 = cos((x_m * 2.0d0))
    if (x_m <= 2.7d-34) then
        tmp = (t_1 / c_m) / ((s_m * x_m) * (c_m * (s_m * x_m)))
    else
        tmp = t_1 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 2.7e-34) {
		tmp = (t_1 / c_m) / ((s_m * x_m) * (c_m * (s_m * x_m)));
	} else {
		tmp = t_1 / (t_0 * t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (c_m * x_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 2.7e-34:
		tmp = (t_1 / c_m) / ((s_m * x_m) * (c_m * (s_m * x_m)))
	else:
		tmp = t_1 / (t_0 * t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 2.7e-34)
		tmp = Float64(Float64(t_1 / c_m) / Float64(Float64(s_m * x_m) * Float64(c_m * Float64(s_m * x_m))));
	else
		tmp = Float64(t_1 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (c_m * x_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 2.7e-34)
		tmp = (t_1 / c_m) / ((s_m * x_m) * (c_m * (s_m * x_m)));
	else
		tmp = t_1 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 2.7e-34], N[(N[(t$95$1 / c$95$m), $MachinePrecision] / N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.70000000000000017e-34

   1. Initial program 63.7%

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

      1. associate-/r* 63.7%

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

      2. cos-neg 63.7%

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

      3. distribute-rgt-neg-out 63.7%

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

      4. distribute-rgt-neg-out 63.7%

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

      5. *-commutative 63.7%

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

      6. distribute-rgt-neg-in 63.7%

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

      7. metadata-eval 63.7%

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

      8. *-commutative 63.7%

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

      9. associate-*l* 57.8%

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

      10. unpow2 57.8%

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

   3. Simplified 57.8%

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

   6. Applied egg-rr 97.0%

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

      1. associate-/r* 97.1%

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

      2. frac-times 93.3%

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

      3. metadata-eval 93.3%

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

      4. times-frac 93.3%

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

      5. *-un-lft-identity 93.3%

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

      6. *-un-lft-identity 93.3%

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

      7. *-commutative 93.3%

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

   8. Applied egg-rr 93.3%

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

   if 2.70000000000000017e-34 < x

   1. Initial program 67.3%

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

      1. associate-/r* 68.3%

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

      2. cos-neg 68.3%

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

      3. distribute-rgt-neg-out 68.3%

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

      4. distribute-rgt-neg-out 68.3%

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

      5. *-commutative 68.3%

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

      6. distribute-rgt-neg-in 68.3%

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

      7. metadata-eval 68.3%

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

      8. *-commutative 68.3%

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

      9. associate-*l* 64.5%

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

      10. unpow2 64.5%

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

   3. Simplified 64.5%

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

      1. associate-/l/ 64.5%

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

      2. add-sqr-sqrt 40.7%

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

      3. add-sqr-sqrt 64.5%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 37.8%

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

      6. swap-sqr 37.8%

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

      7. metadata-eval 37.8%

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

      8. metadata-eval 37.8%

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

      9. swap-sqr 37.8%

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

      10. *-commutative 37.8%

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

      11. *-commutative 37.8%

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

      12. sqrt-unprod 61.9%

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

      13. add-sqr-sqrt 64.5%

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

      14. *-un-lft-identity 64.5%

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

      15. *-commutative 64.5%

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

      16. unpow2 64.5%

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

      17. associate-*r* 67.3%

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

      18. associate-*l* 70.7%

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

      19. *-commutative 70.7%

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

      20. *-commutative 70.7%

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

   6. Applied egg-rr 90.8%

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

   7. Taylor expanded in c around 0 64.5%

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

      1. associate-/r* 64.5%

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

      2. *-commutative 64.5%

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

      3. *-commutative 64.5%

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

      4. unpow2 64.5%

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

      5. unpow2 64.5%

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

      6. swap-sqr 79.2%

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

      7. unpow2 79.2%

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

      8. associate-/l/ 78.3%

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

      9. *-commutative 78.3%

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

      10. unpow2 78.3%

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

      11. unpow2 78.3%

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

      12. swap-sqr 92.8%

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

      13. associate-/r* 94.4%

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

      14. *-lft-identity 94.4%

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

      15. associate-*r/ 94.4%

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

      16. associate-*l/ 94.4%

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

      17. associate-/r/ 94.4%

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

      18. associate-/l/ 92.7%

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

   9. Simplified 95.1%

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

       1. *-commutative 95.1%

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

       2. *-commutative 95.1%

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

       3. associate-*r* 98.4%

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

       4. *-commutative 98.4%

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

       5. unpow-prod-down 84.7%

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

       6. metadata-eval 84.7%

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

       7. pow-flip 84.1%

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

       8. pow2 84.1%

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

       9. metadata-eval 84.1%

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

       10. pow-prod-up 84.0%

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

       11. inv-pow 84.0%

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

       12. inv-pow 84.0%

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

       13. un-div-inv 84.1%

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

       14. times-frac 90.8%

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

       15. *-un-lft-identity 90.8%

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

       16. associate-/l/ 89.0%

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

       17. div-inv 89.1%

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

       18. *-commutative 89.1%

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

   11. Applied egg-rr 94.2%

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

       1. unpow2 94.2%

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

       2. *-commutative 94.2%

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

       3. *-commutative 94.2%

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

   13. Applied egg-rr 94.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{\left(s \cdot x\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ t_1 := \frac{\frac{\frac{1}{x\_m}}{s\_m}}{c\_m}\\ \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))) (t_1 (/ (/ (/ 1.0 x_m) s_m) c_m)))
   (if (<= x_m 3.6e-13) (* t_1 t_1) (/ (cos (* x_m 2.0)) (* t_0 t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double t_1 = ((1.0 / x_m) / s_m) / c_m;
	double tmp;
	if (x_m <= 3.6e-13) {
		tmp = t_1 * t_1;
	} else {
		tmp = cos((x_m * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s_m * (c_m * x_m)
    t_1 = ((1.0d0 / x_m) / s_m) / c_m
    if (x_m <= 3.6d-13) then
        tmp = t_1 * t_1
    else
        tmp = cos((x_m * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double t_1 = ((1.0 / x_m) / s_m) / c_m;
	double tmp;
	if (x_m <= 3.6e-13) {
		tmp = t_1 * t_1;
	} else {
		tmp = Math.cos((x_m * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (c_m * x_m)
	t_1 = ((1.0 / x_m) / s_m) / c_m
	tmp = 0
	if x_m <= 3.6e-13:
		tmp = t_1 * t_1
	else:
		tmp = math.cos((x_m * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	t_1 = Float64(Float64(Float64(1.0 / x_m) / s_m) / c_m)
	tmp = 0.0
	if (x_m <= 3.6e-13)
		tmp = Float64(t_1 * t_1);
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (c_m * x_m);
	t_1 = ((1.0 / x_m) / s_m) / c_m;
	tmp = 0.0;
	if (x_m <= 3.6e-13)
		tmp = t_1 * t_1;
	else
		tmp = cos((x_m * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 3.6e-13], N[(t$95$1 * t$95$1), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5999999999999998e-13

   1. Initial program 63.9%

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

      1. associate-/r* 63.9%

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

      2. cos-neg 63.9%

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

      3. distribute-rgt-neg-out 63.9%

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

      4. distribute-rgt-neg-out 63.9%

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

      5. *-commutative 63.9%

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

      6. distribute-rgt-neg-in 63.9%

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

      7. metadata-eval 63.9%

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

      8. *-commutative 63.9%

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

      9. associate-*l* 58.1%

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

      10. unpow2 58.1%

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

   3. Simplified 58.1%

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

      1. associate-/l/ 58.1%

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

      2. add-sqr-sqrt 46.7%

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

      3. add-sqr-sqrt 58.1%

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

      4. add-sqr-sqrt 33.8%

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

      5. sqrt-unprod 46.8%

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

      6. swap-sqr 46.8%

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

      7. metadata-eval 46.8%

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

      8. metadata-eval 46.8%

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

      9. swap-sqr 46.8%

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

      10. *-commutative 46.8%

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

      11. *-commutative 46.8%

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

      12. sqrt-unprod 23.5%

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

      13. add-sqr-sqrt 58.1%

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

      14. *-un-lft-identity 58.1%

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

      15. *-commutative 58.1%

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

      16. unpow2 58.1%

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

      17. associate-*r* 63.9%

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

      18. associate-*l* 66.6%

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

      19. *-commutative 66.6%

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

      20. *-commutative 66.6%

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

   6. Applied egg-rr 86.7%

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

   7. Taylor expanded in c around 0 58.1%

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

      1. associate-/r* 58.1%

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

      2. *-commutative 58.1%

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

      3. *-commutative 58.1%

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

      4. unpow2 58.1%

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

      5. unpow2 58.1%

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

      6. swap-sqr 77.5%

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

      7. unpow2 77.5%

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

      8. associate-/l/ 77.5%

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

      9. *-commutative 77.5%

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

      10. unpow2 77.5%

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

      11. unpow2 77.5%

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

      12. swap-sqr 96.8%

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

      13. associate-/r* 97.1%

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

      14. *-lft-identity 97.1%

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

      15. associate-*r/ 97.1%

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

      16. associate-*l/ 97.1%

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

      17. associate-/r/ 97.0%

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

      18. associate-/l/ 96.7%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in x around 0 55.7%

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

       1. associate-*r* 55.6%

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

       2. *-commutative 55.6%

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

       3. associate-*l* 55.7%

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

       4. unpow2 55.7%

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

       5. unpow2 55.7%

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

       6. unpow2 55.7%

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

       7. swap-sqr 67.8%

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

       8. swap-sqr 91.4%

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

       9. unpow2 91.4%

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

       10. exp-to-pow 54.6%

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

       11. exp-to-pow 91.4%

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

       12. *-commutative 91.4%

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

       13. *-commutative 91.4%

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

       14. associate-*r* 88.3%

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

   12. Simplified 88.3%

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

       1. pow-flip 88.6%

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

       2. *-commutative 88.6%

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

       3. *-commutative 88.6%

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

       4. associate-*r* 91.7%

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

       5. *-commutative 91.7%

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

       6. associate-*r* 89.6%

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

       7. sqr-pow 89.6%

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

       8. metadata-eval 89.6%

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

       9. metadata-eval 89.6%

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

       10. metadata-eval 89.6%

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

       11. metadata-eval 89.6%

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

       12. inv-pow 89.6%

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

       13. inv-pow 89.6%

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

       14. *-commutative 89.6%

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

       15. associate-/r* 89.6%

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

       16. associate-/r* 89.7%

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

       17. *-commutative 89.7%

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

       18. associate-/r* 89.7%

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

       19. associate-/r* 89.6%

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

   14. Applied egg-rr 89.6%

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

   if 3.5999999999999998e-13 < x

   1. Initial program 66.9%

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

      1. associate-/r* 67.9%

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

      2. cos-neg 67.9%

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

      3. distribute-rgt-neg-out 67.9%

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

      4. distribute-rgt-neg-out 67.9%

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

      5. *-commutative 67.9%

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

      6. distribute-rgt-neg-in 67.9%

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

      7. metadata-eval 67.9%

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

      8. *-commutative 67.9%

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

      9. associate-*l* 64.0%

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

      10. unpow2 64.0%

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

   3. Simplified 64.0%

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

      1. associate-/l/ 63.9%

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

      2. add-sqr-sqrt 38.7%

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

      3. add-sqr-sqrt 63.9%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 35.7%

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

      6. swap-sqr 35.7%

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

      7. metadata-eval 35.7%

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

      8. metadata-eval 35.7%

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

      9. swap-sqr 35.7%

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

      10. *-commutative 35.7%

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

      11. *-commutative 35.7%

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

      12. sqrt-unprod 61.2%

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

      13. add-sqr-sqrt 63.9%

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

      14. *-un-lft-identity 63.9%

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

      15. *-commutative 63.9%

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

      16. unpow2 63.9%

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

      17. associate-*r* 66.9%

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

      18. associate-*l* 70.5%

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

      19. *-commutative 70.5%

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

      20. *-commutative 70.5%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in c around 0 63.9%

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

      1. associate-/r* 64.0%

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

      2. *-commutative 64.0%

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

      3. *-commutative 64.0%

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

      4. unpow2 64.0%

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

      5. unpow2 64.0%

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

      6. swap-sqr 79.5%

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

      7. unpow2 79.5%

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

      8. associate-/l/ 78.5%

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

      9. *-commutative 78.5%

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

      10. unpow2 78.5%

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

      11. unpow2 78.5%

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

      12. swap-sqr 92.4%

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

      13. associate-/r* 94.1%

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

      14. *-lft-identity 94.1%

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

      15. associate-*r/ 94.1%

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

      16. associate-*l/ 94.2%

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

      17. associate-/r/ 94.2%

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

      18. associate-/l/ 92.4%

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

   9. Simplified 94.8%

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

       1. *-commutative 94.8%

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

       2. *-commutative 94.8%

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

       3. associate-*r* 98.3%

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

       4. *-commutative 98.3%

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

       5. unpow-prod-down 83.8%

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

       6. metadata-eval 83.8%

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

       7. pow-flip 83.2%

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

       8. pow2 83.2%

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

       9. metadata-eval 83.2%

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

       10. pow-prod-up 83.2%

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

       11. inv-pow 83.2%

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

       12. inv-pow 83.2%

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

       13. un-div-inv 83.2%

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

       14. times-frac 90.2%

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

       15. *-un-lft-identity 90.2%

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

       16. associate-/l/ 88.4%

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

       17. div-inv 88.4%

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

       18. *-commutative 88.4%

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

   11. Applied egg-rr 93.9%

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

       1. unpow2 93.9%

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

       2. *-commutative 93.9%

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

       3. *-commutative 93.9%

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

   13. Applied egg-rr 93.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s}}{c} \cdot \frac{\frac{\frac{1}{x}}{s}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m)))) (/ (/ 1.0 t_0) t_0)))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	return (1.0 / t_0) / t_0;
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	return (1.0 / t_0) / t_0

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 64.7%

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

   1. associate-/r* 65.0%

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

   2. cos-neg 65.0%

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

   3. distribute-rgt-neg-out 65.0%

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

   4. distribute-rgt-neg-out 65.0%

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

   5. *-commutative 65.0%

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

   6. distribute-rgt-neg-in 65.0%

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

   7. metadata-eval 65.0%

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

   8. *-commutative 65.0%

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

   9. associate-*l* 59.7%

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

   10. unpow2 59.7%

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

3. Simplified 59.7%

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

6. Applied egg-rr 96.3%

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

   1. associate-*l/ 96.3%

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

   2. *-un-lft-identity 96.3%

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

   3. *-commutative 96.3%

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

8. Applied egg-rr 96.3%

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

9. Taylor expanded in x around 0 83.1%

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

10. Final simplification 83.1%

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

Alternative 6: 79.1% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m)))) (/ 1.0 (* t_0 t_0))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	return 1.0 / (t_0 * t_0);
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 64.7%

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

   1. associate-/r* 65.0%

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

   2. cos-neg 65.0%

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

   3. distribute-rgt-neg-out 65.0%

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

   4. distribute-rgt-neg-out 65.0%

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

   5. *-commutative 65.0%

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

   6. distribute-rgt-neg-in 65.0%

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

   7. metadata-eval 65.0%

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

   8. *-commutative 65.0%

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

   9. associate-*l* 59.7%

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

   10. unpow2 59.7%

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

3. Simplified 59.7%

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

5. Taylor expanded in x around 0 55.9%

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

   1. associate-/r* 55.9%

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

   2. *-commutative 55.9%

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

   3. unpow2 55.9%

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

   4. unpow2 55.9%

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

   5. swap-sqr 70.6%

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

   6. unpow2 70.6%

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

   7. associate-/r* 70.6%

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

   8. unpow2 70.6%

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

   9. unpow2 70.6%

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

   10. swap-sqr 82.9%

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

   11. unpow2 82.9%

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

7. Simplified 82.9%

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

   1. unpow2 82.9%

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

9. Applied egg-rr 82.9%

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

10. Final simplification 82.9%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))