mixedcos

Percentage Accurate: 65.7% → 99.3%

Time: 15.5s

Alternatives: 12

Speedup: 24.1×

• 31.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \frac{c\_m}{\frac{1}{x\_m}}\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ t_2 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 9 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (/ 1.0 x_m))))
        (t_1 (cos (* x_m 2.0)))
        (t_2 (* c_m (* x_m s_m))))
   (if (<= x_m 9e+30) (/ (/ t_1 t_2) t_2) (/ 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 / (1.0 / x_m));
	double t_1 = cos((x_m * 2.0));
	double t_2 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 9e+30) {
		tmp = (t_1 / t_2) / t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = s_m * (c_m / (1.0d0 / x_m))
    t_1 = cos((x_m * 2.0d0))
    t_2 = c_m * (x_m * s_m)
    if (x_m <= 9d+30) then
        tmp = (t_1 / t_2) / t_2
    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 / (1.0 / x_m));
	double t_1 = Math.cos((x_m * 2.0));
	double t_2 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 9e+30) {
		tmp = (t_1 / t_2) / t_2;
	} 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 / (1.0 / x_m))
	t_1 = math.cos((x_m * 2.0))
	t_2 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 9e+30:
		tmp = (t_1 / t_2) / t_2
	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 / Float64(1.0 / x_m)))
	t_1 = cos(Float64(x_m * 2.0))
	t_2 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 9e+30)
		tmp = Float64(Float64(t_1 / t_2) / t_2);
	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 / (1.0 / x_m));
	t_1 = cos((x_m * 2.0));
	t_2 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 9e+30)
		tmp = (t_1 / t_2) / t_2;
	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 / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 9e+30], N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.9999999999999999e30

   1. Initial program 65.2%

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

   3. Taylor expanded in x around inf 57.3%

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

      1. associate-/r* 57.3%

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

      2. *-commutative 57.3%

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

      3. unpow2 57.3%

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

      4. unpow2 57.3%

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

      5. swap-sqr 74.7%

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

      6. unpow2 74.7%

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

      7. associate-/r* 74.7%

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

      8. unpow2 74.7%

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

      9. unpow2 74.7%

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

      10. swap-sqr 94.3%

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

      11. unpow2 94.3%

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

      12. *-commutative 94.3%

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

   5. Simplified 94.3%

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

   7. Applied egg-rr 93.9%

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

      1. associate-/r* 93.9%

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

      2. div-inv 93.9%

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

      3. div-inv 93.9%

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

      4. associate-/l/ 94.0%

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

      5. remove-double-div 94.0%

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

      6. *-commutative 94.0%

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

      7. associate-/r/ 91.7%

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

      8. associate-/r/ 94.0%

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

      9. div-inv 94.0%

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

      10. associate-/l/ 93.9%

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

      11. remove-double-div 94.4%

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

      12. *-commutative 94.4%

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

   9. Applied egg-rr 94.4%

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

       1. un-div-inv 94.4%

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

   11. Applied egg-rr 94.4%

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

   if 8.9999999999999999e30 < x

   1. Initial program 64.5%

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

   3. Taylor expanded in x around inf 60.8%

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

      1. associate-/r* 57.6%

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

      2. *-commutative 57.6%

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

      3. unpow2 57.6%

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

      4. unpow2 57.6%

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

      5. swap-sqr 75.5%

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

      6. unpow2 75.5%

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

      7. associate-/r* 78.8%

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

      8. unpow2 78.8%

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

      9. unpow2 78.8%

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

      10. swap-sqr 98.1%

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

      11. unpow2 98.1%

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

      12. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   7. Applied egg-rr 98.0%

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

      1. associate-/r/ 95.0%

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

      2. associate-/r/ 94.9%

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

   9. Simplified 94.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \frac{c}{\frac{1}{x}}\right) \cdot \left(s \cdot \frac{c}{\frac{1}{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \left(c\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;{s\_m}^{2} \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{t\_1}{s\_m \cdot \left(\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right) \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 (* x_m (* c_m s_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= (pow s_m 2.0) 5e+27)
     (/ t_1 (* s_m (* (* c_m (* x_m s_m)) (* x_m c_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 = x_m * (c_m * s_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (pow(s_m, 2.0) <= 5e+27) {
		tmp = t_1 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_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 = x_m * (c_m * s_m)
    t_1 = cos((x_m * 2.0d0))
    if ((s_m ** 2.0d0) <= 5d+27) then
        tmp = t_1 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_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 = x_m * (c_m * s_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (Math.pow(s_m, 2.0) <= 5e+27) {
		tmp = t_1 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_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 = x_m * (c_m * s_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if math.pow(s_m, 2.0) <= 5e+27:
		tmp = t_1 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_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(x_m * Float64(c_m * s_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if ((s_m ^ 2.0) <= 5e+27)
		tmp = Float64(t_1 / Float64(s_m * Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(x_m * c_m))));
	else
		tmp = Float64(Float64(t_1 / 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 = x_m * (c_m * s_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 5e+27)
		tmp = t_1 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_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[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 5e+27], N[(t$95$1 / N[(s$95$m * N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 4.99999999999999979e27

   1. Initial program 58.7%

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

   3. Taylor expanded in x around inf 54.7%

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

      1. associate-/r* 54.7%

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

      2. *-commutative 54.7%

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

      3. unpow2 54.7%

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

      4. unpow2 54.7%

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

      5. swap-sqr 71.3%

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

      6. unpow2 71.3%

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

      7. associate-/r* 71.3%

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

      8. unpow2 71.3%

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

      9. unpow2 71.3%

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

      10. swap-sqr 94.7%

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

      11. unpow2 94.7%

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

      12. *-commutative 94.7%

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

   5. Simplified 94.7%

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

      1. unpow2 94.7%

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

      2. associate-*r* 92.2%

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

      3. associate-*r* 90.8%

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

      4. *-commutative 90.8%

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

      5. associate-*l* 89.8%

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

      6. *-commutative 89.8%

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

   7. Applied egg-rr 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-*r* 90.8%

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

      3. *-commutative 90.8%

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

      4. *-commutative 90.8%

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

      5. *-commutative 90.8%

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

   9. Simplified 90.8%

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

   if 4.99999999999999979e27 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 72.4%

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

      1. associate-/r* 70.7%

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

      2. *-commutative 70.7%

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

      3. unpow2 70.7%

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

      4. sqr-neg 70.7%

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

      5. unpow2 70.7%

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

      6. cos-neg 70.7%

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

      7. *-commutative 70.7%

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

      8. distribute-rgt-neg-in 70.7%

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

      9. metadata-eval 70.7%

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

      10. unpow2 70.7%

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

      11. sqr-neg 70.7%

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

      12. unpow2 70.7%

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

      13. associate-*r* 60.5%

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

      14. unpow2 60.5%

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

      15. *-commutative 60.5%

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

   3. Simplified 60.5%

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

   6. Applied egg-rr 78.6%

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

      1. rem-cbrt-cube 95.8%

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

      2. unpow2 95.8%

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

      3. associate-/r* 95.9%

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

      4. *-commutative 95.9%

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

      5. *-commutative 95.9%

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

      6. associate-*l* 94.4%

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

      7. *-commutative 94.4%

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

      8. associate-*l* 98.1%

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

   8. Applied egg-rr 98.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 0.00012:\\ \;\;\;\;{t\_1}^{-2}\\ \mathbf{elif}\;x\_m \leq 3.6 \cdot 10^{+221}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_1 \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

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 (cos (* x_m 2.0))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 0.00012)
     (pow t_1 -2.0)
     (if (<= x_m 3.6e+221)
       (/ t_0 (* s_m (* (* x_m c_m) (* x_m (* c_m s_m)))))
       (/ t_0 (* t_1 (* s_m (* x_m c_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 = cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 0.00012) {
		tmp = pow(t_1, -2.0);
	} else if (x_m <= 3.6e+221) {
		tmp = t_0 / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	} else {
		tmp = t_0 / (t_1 * (s_m * (x_m * c_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) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 0.00012d0) then
        tmp = t_1 ** (-2.0d0)
    else if (x_m <= 3.6d+221) then
        tmp = t_0 / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))))
    else
        tmp = t_0 / (t_1 * (s_m * (x_m * c_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 = Math.cos((x_m * 2.0));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 0.00012) {
		tmp = Math.pow(t_1, -2.0);
	} else if (x_m <= 3.6e+221) {
		tmp = t_0 / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	} else {
		tmp = t_0 / (t_1 * (s_m * (x_m * c_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 = math.cos((x_m * 2.0))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 0.00012:
		tmp = math.pow(t_1, -2.0)
	elif x_m <= 3.6e+221:
		tmp = t_0 / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))))
	else:
		tmp = t_0 / (t_1 * (s_m * (x_m * c_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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 0.00012)
		tmp = t_1 ^ -2.0;
	elseif (x_m <= 3.6e+221)
		tmp = Float64(t_0 / Float64(s_m * Float64(Float64(x_m * c_m) * Float64(x_m * Float64(c_m * s_m)))));
	else
		tmp = Float64(t_0 / Float64(t_1 * Float64(s_m * Float64(x_m * c_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 = cos((x_m * 2.0));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 0.00012)
		tmp = t_1 ^ -2.0;
	elseif (x_m <= 3.6e+221)
		tmp = t_0 / (s_m * ((x_m * c_m) * (x_m * (c_m * s_m))));
	else
		tmp = t_0 / (t_1 * (s_m * (x_m * c_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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00012], N[Power[t$95$1, -2.0], $MachinePrecision], If[LessEqual[x$95$m, 3.6e+221], N[(t$95$0 / N[(s$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$1 * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.20000000000000003e-4

   1. Initial program 64.8%

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

      1. associate-/r* 64.8%

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

      2. *-commutative 64.8%

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

      3. unpow2 64.8%

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

      4. sqr-neg 64.8%

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

      5. unpow2 64.8%

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

      6. cos-neg 64.8%

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

      7. *-commutative 64.8%

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

      8. distribute-rgt-neg-in 64.8%

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

      9. metadata-eval 64.8%

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

      10. unpow2 64.8%

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

      11. sqr-neg 64.8%

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

      12. unpow2 64.8%

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

      13. associate-*r* 56.6%

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

      14. unpow2 56.6%

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

      15. *-commutative 56.6%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. associate-/r* 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

      4. unpow2 53.2%

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

      5. swap-sqr 67.6%

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

      6. unpow2 67.6%

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

      7. associate-/r* 67.6%

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

      8. unpow2 67.6%

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

      9. unpow2 67.6%

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

      10. swap-sqr 81.5%

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

      11. unpow2 81.5%

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

   7. Simplified 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow-flip 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-*l* 83.6%

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

      5. metadata-eval 83.6%

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

   9. Applied egg-rr 83.6%

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

       1. *-lft-identity 83.6%

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

       2. associate-*r* 81.5%

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

       3. *-commutative 81.5%

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

       4. *-commutative 81.5%

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

   11. Simplified 81.5%

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

   if 1.20000000000000003e-4 < x < 3.60000000000000009e221

   1. Initial program 71.9%

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

   3. Taylor expanded in x around inf 69.5%

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

      1. associate-/r* 67.1%

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

      2. *-commutative 67.1%

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

      3. unpow2 67.1%

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

      4. unpow2 67.1%

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

      5. swap-sqr 76.5%

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

      6. unpow2 76.5%

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

      7. associate-/r* 78.9%

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

      8. unpow2 78.9%

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

      9. unpow2 78.9%

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

      10. swap-sqr 99.8%

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

      11. unpow2 99.8%

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

      12. *-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*r* 97.7%

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

      3. associate-*r* 97.6%

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

      4. *-commutative 97.6%

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

      5. associate-*l* 97.6%

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

      6. *-commutative 97.6%

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

   7. Applied egg-rr 97.6%

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

   if 3.60000000000000009e221 < x

   1. Initial program 55.1%

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

   3. Taylor expanded in x around inf 50.0%

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

      1. associate-/r* 45.8%

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

      2. *-commutative 45.8%

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

      3. unpow2 45.8%

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

      4. unpow2 45.8%

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

      5. swap-sqr 75.0%

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

      6. unpow2 75.0%

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

      7. associate-/r* 79.1%

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

      8. unpow2 79.1%

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

      9. unpow2 79.1%

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

      10. swap-sqr 95.3%

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

      11. unpow2 95.3%

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

      12. *-commutative 95.3%

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

   5. Simplified 95.3%

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

      1. unpow2 95.3%

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

      2. associate-*r* 91.5%

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

      3. associate-*r* 91.5%

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

      4. *-commutative 91.5%

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

      5. associate-*l* 79.6%

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

      6. *-commutative 79.6%

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

   7. Applied egg-rr 79.6%

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

      1. associate-*l* 79.6%

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

      2. associate-*r* 91.5%

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

      3. *-commutative 91.5%

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

      4. *-commutative 91.5%

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

      5. *-commutative 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00012:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+221}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;{t\_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{t\_0 \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

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 (* x_m s_m))))
   (if (<= x_m 7.5e-9)
     (pow t_0 -2.0)
     (/ (cos (* x_m 2.0)) (* t_0 (* s_m (* x_m c_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 * (x_m * s_m);
	double tmp;
	if (x_m <= 7.5e-9) {
		tmp = pow(t_0, -2.0);
	} else {
		tmp = cos((x_m * 2.0)) / (t_0 * (s_m * (x_m * c_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 * (x_m * s_m)
    if (x_m <= 7.5d-9) then
        tmp = t_0 ** (-2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / (t_0 * (s_m * (x_m * c_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 * (x_m * s_m);
	double tmp;
	if (x_m <= 7.5e-9) {
		tmp = Math.pow(t_0, -2.0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (t_0 * (s_m * (x_m * c_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 * (x_m * s_m)
	tmp = 0
	if x_m <= 7.5e-9:
		tmp = math.pow(t_0, -2.0)
	else:
		tmp = math.cos((x_m * 2.0)) / (t_0 * (s_m * (x_m * c_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(x_m * s_m))
	tmp = 0.0
	if (x_m <= 7.5e-9)
		tmp = t_0 ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(t_0 * Float64(s_m * Float64(x_m * c_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 * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 7.5e-9)
		tmp = t_0 ^ -2.0;
	else
		tmp = cos((x_m * 2.0)) / (t_0 * (s_m * (x_m * c_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[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 7.5e-9], N[Power[t$95$0, -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.49999999999999933e-9

   1. Initial program 65.5%

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

      1. associate-/r* 65.4%

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

      2. *-commutative 65.4%

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

      3. unpow2 65.4%

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

      4. sqr-neg 65.4%

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

      5. unpow2 65.4%

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

      6. cos-neg 65.4%

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

      7. *-commutative 65.4%

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

      8. distribute-rgt-neg-in 65.4%

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

      9. metadata-eval 65.4%

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

      10. unpow2 65.4%

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

      11. sqr-neg 65.4%

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

      12. unpow2 65.4%

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

      13. associate-*r* 57.2%

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

      14. unpow2 57.2%

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

      15. *-commutative 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 53.7%

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

      1. associate-/r* 53.7%

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

      2. *-commutative 53.7%

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

      3. unpow2 53.7%

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

      4. unpow2 53.7%

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

      5. swap-sqr 68.2%

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

      6. unpow2 68.2%

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

      7. associate-/r* 68.3%

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

      8. unpow2 68.3%

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

      9. unpow2 68.3%

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

      10. swap-sqr 81.6%

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

      11. unpow2 81.6%

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

   7. Simplified 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. pow-flip 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-*l* 83.7%

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

      5. metadata-eval 83.7%

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

   9. Applied egg-rr 83.7%

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

       1. *-lft-identity 83.7%

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

       2. associate-*r* 81.6%

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

       3. *-commutative 81.6%

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

       4. *-commutative 81.6%

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

   11. Simplified 81.6%

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

   if 7.49999999999999933e-9 < x

   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. Add Preprocessing

   3. Taylor expanded in x around inf 60.7%

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

      1. associate-/r* 57.7%

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

      2. *-commutative 57.7%

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

      3. unpow2 57.7%

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

      4. unpow2 57.7%

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

      5. swap-sqr 73.8%

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

      6. unpow2 73.8%

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

      7. associate-/r* 76.8%

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

      8. unpow2 76.8%

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

      9. unpow2 76.8%

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

      10. swap-sqr 98.2%

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

      11. unpow2 98.2%

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

      12. *-commutative 98.2%

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

   5. Simplified 98.2%

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

      1. unpow2 98.2%

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

      2. associate-*r* 95.6%

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

      3. associate-*r* 95.5%

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

      4. *-commutative 95.5%

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

      5. associate-*l* 91.4%

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

      6. *-commutative 91.4%

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

   7. Applied egg-rr 91.4%

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

      1. associate-*l* 91.3%

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

      2. associate-*r* 95.6%

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

      3. *-commutative 95.6%

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

      4. *-commutative 95.6%

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

      5. *-commutative 95.6%

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

   9. Simplified 95.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 0.00012:\\ \;\;\;\;{t\_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{s\_m \cdot \left(t\_0 \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

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 (* x_m s_m))))
   (if (<= x_m 0.00012)
     (pow t_0 -2.0)
     (/ (cos (* x_m 2.0)) (* s_m (* t_0 (* x_m c_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 * (x_m * s_m);
	double tmp;
	if (x_m <= 0.00012) {
		tmp = pow(t_0, -2.0);
	} else {
		tmp = cos((x_m * 2.0)) / (s_m * (t_0 * (x_m * c_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 * (x_m * s_m)
    if (x_m <= 0.00012d0) then
        tmp = t_0 ** (-2.0d0)
    else
        tmp = cos((x_m * 2.0d0)) / (s_m * (t_0 * (x_m * c_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 * (x_m * s_m);
	double tmp;
	if (x_m <= 0.00012) {
		tmp = Math.pow(t_0, -2.0);
	} else {
		tmp = Math.cos((x_m * 2.0)) / (s_m * (t_0 * (x_m * c_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 * (x_m * s_m)
	tmp = 0
	if x_m <= 0.00012:
		tmp = math.pow(t_0, -2.0)
	else:
		tmp = math.cos((x_m * 2.0)) / (s_m * (t_0 * (x_m * c_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(x_m * s_m))
	tmp = 0.0
	if (x_m <= 0.00012)
		tmp = t_0 ^ -2.0;
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(s_m * Float64(t_0 * Float64(x_m * c_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 * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 0.00012)
		tmp = t_0 ^ -2.0;
	else
		tmp = cos((x_m * 2.0)) / (s_m * (t_0 * (x_m * c_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[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00012], N[Power[t$95$0, -2.0], $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(t$95$0 * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.20000000000000003e-4

   1. Initial program 64.8%

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

      1. associate-/r* 64.8%

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

      2. *-commutative 64.8%

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

      3. unpow2 64.8%

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

      4. sqr-neg 64.8%

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

      5. unpow2 64.8%

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

      6. cos-neg 64.8%

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

      7. *-commutative 64.8%

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

      8. distribute-rgt-neg-in 64.8%

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

      9. metadata-eval 64.8%

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

      10. unpow2 64.8%

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

      11. sqr-neg 64.8%

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

      12. unpow2 64.8%

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

      13. associate-*r* 56.6%

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

      14. unpow2 56.6%

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

      15. *-commutative 56.6%

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

   3. Simplified 56.6%

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

   5. Taylor expanded in x around 0 53.2%

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

      1. associate-/r* 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

      4. unpow2 53.2%

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

      5. swap-sqr 67.6%

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

      6. unpow2 67.6%

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

      7. associate-/r* 67.6%

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

      8. unpow2 67.6%

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

      9. unpow2 67.6%

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

      10. swap-sqr 81.5%

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

      11. unpow2 81.5%

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

   7. Simplified 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow-flip 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-*l* 83.6%

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

      5. metadata-eval 83.6%

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

   9. Applied egg-rr 83.6%

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

       1. *-lft-identity 83.6%

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

       2. associate-*r* 81.5%

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

       3. *-commutative 81.5%

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

       4. *-commutative 81.5%

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

   11. Simplified 81.5%

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

   if 1.20000000000000003e-4 < x

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 62.4%

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

      1. associate-/r* 59.4%

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

      2. *-commutative 59.4%

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

      3. unpow2 59.4%

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

      4. unpow2 59.4%

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

      5. swap-sqr 75.9%

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

      6. unpow2 75.9%

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

      7. associate-/r* 79.0%

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

      8. unpow2 79.0%

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

      9. unpow2 79.0%

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

      10. swap-sqr 98.2%

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

      11. unpow2 98.2%

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

      12. *-commutative 98.2%

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

   5. Simplified 98.2%

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

      1. unpow2 98.2%

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

      2. associate-*r* 95.4%

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

      3. associate-*r* 95.4%

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

      4. *-commutative 95.4%

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

      5. associate-*l* 91.1%

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

      6. *-commutative 91.1%

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

   7. Applied egg-rr 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-*r* 95.4%

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

      3. *-commutative 95.4%

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

      4. *-commutative 95.4%

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

      5. *-commutative 95.4%

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

   9. Simplified 95.4%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00012:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

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 (* x_m s_m)))) (/ (/ (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 = c_m * (x_m * s_m);
	return (cos((x_m * 2.0)) / 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 * (x_m * s_m)
    code = (cos((x_m * 2.0d0)) / 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 * (x_m * s_m);
	return (Math.cos((x_m * 2.0)) / 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 * (x_m * s_m)
	return (math.cos((x_m * 2.0)) / 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(x_m * s_m))
	return Float64(Float64(cos(Float64(x_m * 2.0)) / 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 * (x_m * s_m);
	tmp = (cos((x_m * 2.0)) / 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[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.0%

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

3. Taylor expanded in x around inf 58.1%

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

   1. associate-/r* 57.3%

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

   2. *-commutative 57.3%

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

   3. unpow2 57.3%

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

   4. unpow2 57.3%

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

   5. swap-sqr 74.9%

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

   6. unpow2 74.9%

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

   7. associate-/r* 75.7%

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

   8. unpow2 75.7%

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

   9. unpow2 75.7%

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

   10. swap-sqr 95.2%

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

   11. unpow2 95.2%

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

   12. *-commutative 95.2%

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

5. Simplified 95.2%

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

7. Applied egg-rr 94.9%

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

   1. associate-/r* 94.9%

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

   2. div-inv 94.9%

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

   3. div-inv 94.9%

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

   4. associate-/l/ 95.0%

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

   5. remove-double-div 95.0%

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

   6. *-commutative 95.0%

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

   7. associate-/r/ 92.5%

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

   8. associate-/r/ 95.0%

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

   9. div-inv 94.9%

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

   10. associate-/l/ 94.9%

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

   11. remove-double-div 95.2%

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

   12. *-commutative 95.2%

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

9. Applied egg-rr 95.2%

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

    1. un-div-inv 95.3%

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

11. Applied egg-rr 95.3%

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

Alternative 7: 79.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\frac{1}{x\_m}}{s\_m}}{c\_m}\\ \mathbf{if}\;x\_m \leq 4 \cdot 10^{+236}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;-{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{-2}\\ \end{array} \end{array} \]

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 (/ (/ (/ 1.0 x_m) s_m) c_m)))
   (if (<= x_m 4e+236) (* t_0 t_0) (- (pow (* c_m (* x_m s_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) {
	double t_0 = ((1.0 / x_m) / s_m) / c_m;
	double tmp;
	if (x_m <= 4e+236) {
		tmp = t_0 * t_0;
	} else {
		tmp = -pow((c_m * (x_m * s_m)), -2.0);
	}
	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) :: tmp
    t_0 = ((1.0d0 / x_m) / s_m) / c_m
    if (x_m <= 4d+236) then
        tmp = t_0 * t_0
    else
        tmp = -((c_m * (x_m * s_m)) ** (-2.0d0))
    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 = ((1.0 / x_m) / s_m) / c_m;
	double tmp;
	if (x_m <= 4e+236) {
		tmp = t_0 * t_0;
	} else {
		tmp = -Math.pow((c_m * (x_m * s_m)), -2.0);
	}
	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 = ((1.0 / x_m) / s_m) / c_m
	tmp = 0
	if x_m <= 4e+236:
		tmp = t_0 * t_0
	else:
		tmp = -math.pow((c_m * (x_m * s_m)), -2.0)
	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(Float64(Float64(1.0 / x_m) / s_m) / c_m)
	tmp = 0.0
	if (x_m <= 4e+236)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(-(Float64(c_m * Float64(x_m * s_m)) ^ -2.0));
	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 = ((1.0 / x_m) / s_m) / c_m;
	tmp = 0.0;
	if (x_m <= 4e+236)
		tmp = t_0 * t_0;
	else
		tmp = -((c_m * (x_m * s_m)) ^ -2.0);
	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[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 4e+236], N[(t$95$0 * t$95$0), $MachinePrecision], (-N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000021e236

   1. Initial program 66.3%

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

      1. associate-/r* 65.8%

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

      2. *-commutative 65.8%

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

      3. unpow2 65.8%

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

      4. sqr-neg 65.8%

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

      5. unpow2 65.8%

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

      6. cos-neg 65.8%

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

      7. *-commutative 65.8%

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

      8. distribute-rgt-neg-in 65.8%

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

      9. metadata-eval 65.8%

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

      10. unpow2 65.8%

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

      11. sqr-neg 65.8%

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

      12. unpow2 65.8%

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

      13. associate-*r* 58.8%

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

      14. unpow2 58.8%

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

      15. *-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around 0 53.7%

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

      1. associate-/r* 53.3%

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

      2. *-commutative 53.3%

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

      3. unpow2 53.3%

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

      4. unpow2 53.3%

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

      5. swap-sqr 65.5%

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

      6. unpow2 65.5%

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

      7. associate-/r* 65.9%

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

      8. unpow2 65.9%

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

      9. unpow2 65.9%

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

      10. swap-sqr 78.6%

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

      11. unpow2 78.6%

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

   7. Simplified 78.6%

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

      1. unpow2 78.6%

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

      2. unpow2 78.6%

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

      3. unpow-prod-down 65.9%

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

      4. *-commutative 65.9%

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

      5. unpow-prod-down 53.7%

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

      6. associate-/l/ 53.7%

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

      7. associate-/l/ 53.7%

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

      8. add-sqr-sqrt 53.7%

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

   9. Applied egg-rr 78.1%

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

   if 4.00000000000000021e236 < x

   1. Initial program 50.9%

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

      1. associate-/r* 45.9%

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

      2. *-commutative 45.9%

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

      3. unpow2 45.9%

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

      4. sqr-neg 45.9%

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

      5. unpow2 45.9%

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

      6. cos-neg 45.9%

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

      7. *-commutative 45.9%

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

      8. distribute-rgt-neg-in 45.9%

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

      9. metadata-eval 45.9%

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

      10. unpow2 45.9%

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

      11. sqr-neg 45.9%

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

      12. unpow2 45.9%

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

      13. associate-*r* 40.0%

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

      14. unpow2 40.0%

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

      15. *-commutative 40.0%

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

   3. Simplified 40.0%

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

      1. associate-/l/ 45.0%

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

      2. associate-/r* 45.0%

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

      3. associate-/l/ 45.0%

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

      4. unpow2 45.0%

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

      5. *-un-lft-identity 45.0%

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

      6. times-frac 50.4%

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

   6. Applied egg-rr 85.1%

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

   7. Taylor expanded in x around 0 50.4%

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

      1. associate-/r* 50.4%

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

      2. *-commutative 50.4%

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

      3. unpow2 50.4%

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

      4. unpow2 50.4%

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

      5. swap-sqr 62.9%

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

      6. unpow2 62.9%

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

      7. *-commutative 62.9%

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

   9. Simplified 62.9%

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

       1. unpow2 62.9%

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

       2. *-commutative 62.9%

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

       3. associate-*r* 61.7%

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

       4. *-commutative 61.7%

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

   11. Applied egg-rr 61.7%

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

       1. frac-2neg 61.7%

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

       2. metadata-eval 61.7%

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

       3. clear-num 61.7%

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

       4. frac-times 61.7%

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

       5. metadata-eval 61.7%

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

       6. add-sqr-sqrt 41.5%

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

       7. sqrt-unprod 62.2%

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

       8. sqr-neg 62.2%

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

       9. sqrt-unprod 26.1%

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

       10. add-sqr-sqrt 67.9%

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

       11. associate-*l* 72.7%

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

       12. pow2 72.7%

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

   13. Applied egg-rr 72.7%

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

       1. associate-/r* 72.7%

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

       2. metadata-eval 72.7%

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

       3. associate-*r/ 72.7%

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

       4. associate-/r/ 72.7%

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

       5. associate-*l/ 72.7%

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

       6. *-commutative 72.7%

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

       7. /-rgt-identity 72.7%

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

       8. associate-/l* 72.7%

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

       9. unpow2 72.7%

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

       10. associate-*l* 73.4%

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

       11. associate-/l/ 73.4%

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

       12. *-lft-identity 73.4%

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

       13. associate-*l/ 73.4%

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

       14. unpow-1 73.4%

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

       15. associate-/l/ 73.4%

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

       16. *-commutative 73.4%

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

       17. unpow-1 73.4%

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

       18. pow-sqr 73.4%

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

       19. *-commutative 73.4%

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

       20. metadata-eval 73.4%

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

   15. Simplified 73.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s}}{c} \cdot \frac{\frac{\frac{1}{x}}{s}}{c}\\ \mathbf{else}:\\ \;\;\;\;-{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 20.9× speedup

Math

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

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 (/ (/ (/ 1.0 x_m) s_m) c_m))) (* 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 = ((1.0 / x_m) / s_m) / c_m;
	return 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 = ((1.0d0 / x_m) / s_m) / c_m
    code = 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 = ((1.0 / x_m) / s_m) / c_m;
	return 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 = ((1.0 / x_m) / s_m) / c_m
	return 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(Float64(Float64(1.0 / x_m) / s_m) / c_m)
	return 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 = ((1.0 / x_m) / s_m) / c_m;
	tmp = 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[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. unpow2 64.3%

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

   4. sqr-neg 64.3%

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

   5. unpow2 64.3%

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

   6. cos-neg 64.3%

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

   7. *-commutative 64.3%

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

   8. distribute-rgt-neg-in 64.3%

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

   9. metadata-eval 64.3%

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

   10. unpow2 64.3%

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

   11. sqr-neg 64.3%

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

   12. unpow2 64.3%

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

   13. associate-*r* 57.3%

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

   14. unpow2 57.3%

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

   15. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in x around 0 53.0%

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

   1. associate-/r* 52.2%

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

   2. *-commutative 52.2%

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

   3. unpow2 52.2%

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

   4. unpow2 52.2%

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

   5. swap-sqr 64.4%

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

   6. unpow2 64.4%

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

   7. associate-/r* 65.2%

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

   8. unpow2 65.2%

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

   9. unpow2 65.2%

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

   10. swap-sqr 77.4%

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

   11. unpow2 77.4%

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

7. Simplified 77.4%

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

   1. unpow2 77.4%

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

   2. unpow2 77.4%

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

   3. unpow-prod-down 65.2%

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

   4. *-commutative 65.2%

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

   5. unpow-prod-down 53.0%

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

   6. associate-/l/ 53.0%

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

   7. associate-/l/ 53.0%

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

   8. add-sqr-sqrt 53.0%

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

9. Applied egg-rr 77.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{s}}{c} \cdot \frac{\frac{\frac{1}{x}}{s}}{c}} \]
10. Add Preprocessing

Alternative 9: 79.3% accurate, 22.4× speedup

Math

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

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
 (/ -1.0 (* (* c_m (* x_m s_m)) (* c_m (* x_m (- s_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) {
	return -1.0 / ((c_m * (x_m * s_m)) * (c_m * (x_m * -s_m)));
}

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 = (-1.0d0) / ((c_m * (x_m * s_m)) * (c_m * (x_m * -s_m)))
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 -1.0 / ((c_m * (x_m * s_m)) * (c_m * (x_m * -s_m)));
}

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 -1.0 / ((c_m * (x_m * s_m)) * (c_m * (x_m * -s_m)))

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(-1.0 / Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(c_m * Float64(x_m * Float64(-s_m)))))
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 = -1.0 / ((c_m * (x_m * s_m)) * (c_m * (x_m * -s_m)));
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[(-1.0 / N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * N[(x$95$m * (-s$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. unpow2 64.3%

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

   4. sqr-neg 64.3%

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

   5. unpow2 64.3%

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

   6. cos-neg 64.3%

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

   7. *-commutative 64.3%

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

   8. distribute-rgt-neg-in 64.3%

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

   9. metadata-eval 64.3%

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

   10. unpow2 64.3%

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

   11. sqr-neg 64.3%

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

   12. unpow2 64.3%

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

   13. associate-*r* 57.3%

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

   14. unpow2 57.3%

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

   15. *-commutative 57.3%

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

3. Simplified 57.3%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
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. associate-/r* 58.1%

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

   3. associate-/l/ 58.1%

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

   4. unpow2 58.1%

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

   5. *-un-lft-identity 58.1%

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

   6. times-frac 65.7%

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

6. Applied egg-rr 86.4%

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

7. Taylor expanded in x around 0 60.2%

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

   1. associate-/r* 60.2%

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

   2. *-commutative 60.2%

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

   3. unpow2 60.2%

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

   4. unpow2 60.2%

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

   5. swap-sqr 73.5%

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

   6. unpow2 73.5%

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

   7. *-commutative 73.5%

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

9. Simplified 73.5%

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

    1. unpow2 73.5%

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

    2. *-commutative 73.5%

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

    3. associate-*r* 72.2%

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

    4. *-commutative 72.2%

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

11. Applied egg-rr 72.2%

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

    1. associate-*r/ 63.6%

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

    2. associate-*l* 64.8%

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

    3. frac-times 77.3%

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

    4. associate-/r* 77.4%

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

    5. associate-/r* 77.4%

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

    6. frac-2neg 77.4%

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

    7. metadata-eval 77.4%

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

    8. frac-times 77.4%

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

    9. metadata-eval 77.4%

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

13. Applied egg-rr 77.4%

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

14. Final simplification 77.4%

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

Alternative 10: 78.1% accurate, 24.1× speedup

Math

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

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 (* x_m (* c_m s_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 = x_m * (c_m * s_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 = x_m * (c_m * s_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 = x_m * (c_m * s_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 = x_m * (c_m * s_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(x_m * Float64(c_m * s_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 = x_m * (c_m * s_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[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. unpow2 64.3%

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

   4. sqr-neg 64.3%

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

   5. unpow2 64.3%

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

   6. cos-neg 64.3%

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

   7. *-commutative 64.3%

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

   8. distribute-rgt-neg-in 64.3%

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

   9. metadata-eval 64.3%

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

   10. unpow2 64.3%

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

   11. sqr-neg 64.3%

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

   12. unpow2 64.3%

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

   13. associate-*r* 57.3%

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

   14. unpow2 57.3%

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

   15. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in x around 0 53.0%

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

   1. associate-/r* 52.2%

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

   2. *-commutative 52.2%

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

   3. unpow2 52.2%

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

   4. unpow2 52.2%

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

   5. swap-sqr 64.4%

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

   6. unpow2 64.4%

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

   7. associate-/r* 65.2%

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

   8. unpow2 65.2%

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

   9. unpow2 65.2%

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

   10. swap-sqr 77.4%

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

   11. unpow2 77.4%

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

7. Simplified 77.4%

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

   1. unpow2 77.4%

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

   2. *-commutative 77.4%

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

   3. associate-*l* 76.6%

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

   4. *-commutative 76.6%

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

   5. associate-*l* 78.8%

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

9. Applied egg-rr 78.8%

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

10. Final simplification 78.8%

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

Alternative 11: 75.3% accurate, 24.1× speedup

Math

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

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
 (/ 1.0 (* x_m (* c_m (* s_m (* c_m (* x_m s_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) {
	return 1.0 / (x_m * (c_m * (s_m * (c_m * (x_m * s_m)))));
}

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 = 1.0d0 / (x_m * (c_m * (s_m * (c_m * (x_m * s_m)))))
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 1.0 / (x_m * (c_m * (s_m * (c_m * (x_m * s_m)))));
}

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 1.0 / (x_m * (c_m * (s_m * (c_m * (x_m * s_m)))))

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(1.0 / Float64(x_m * Float64(c_m * Float64(s_m * Float64(c_m * Float64(x_m * s_m))))))
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 = 1.0 / (x_m * (c_m * (s_m * (c_m * (x_m * s_m)))));
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[(1.0 / N[(x$95$m * N[(c$95$m * N[(s$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. unpow2 64.3%

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

   4. sqr-neg 64.3%

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

   5. unpow2 64.3%

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

   6. cos-neg 64.3%

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

   7. *-commutative 64.3%

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

   8. distribute-rgt-neg-in 64.3%

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

   9. metadata-eval 64.3%

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

   10. unpow2 64.3%

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

   11. sqr-neg 64.3%

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

   12. unpow2 64.3%

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

   13. associate-*r* 57.3%

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

   14. unpow2 57.3%

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

   15. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in x around 0 53.0%

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

   1. associate-/r* 52.2%

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

   2. *-commutative 52.2%

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

   3. unpow2 52.2%

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

   4. unpow2 52.2%

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

   5. swap-sqr 64.4%

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

   6. unpow2 64.4%

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

   7. associate-/r* 65.2%

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

   8. unpow2 65.2%

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

   9. unpow2 65.2%

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

   10. swap-sqr 77.4%

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

   11. unpow2 77.4%

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

7. Simplified 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 75.8%

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

   3. associate-*l* 74.4%

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

   4. *-commutative 74.4%

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

   5. *-commutative 74.4%

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

   6. associate-*l* 75.5%

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

9. Applied egg-rr 75.5%

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

    1. associate-*l* 76.7%

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

    2. associate-*r* 75.0%

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

    3. *-commutative 75.0%

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

    4. *-commutative 75.0%

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

11. Simplified 75.0%

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

12. Final simplification 75.0%

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

Alternative 12: 74.0% accurate, 24.1× speedup

Math

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

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
 (/ 1.0 (* s_m (* (* c_m (* x_m s_m)) (* x_m c_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) {
	return 1.0 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_m)));
}

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 = 1.0d0 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_m)))
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 1.0 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_m)));
}

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 1.0 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_m)))

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(1.0 / Float64(s_m * Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(x_m * c_m))))
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 = 1.0 / (s_m * ((c_m * (x_m * s_m)) * (x_m * c_m)));
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[(1.0 / N[(s$95$m * N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 64.3%

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

   2. *-commutative 64.3%

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

   3. unpow2 64.3%

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

   4. sqr-neg 64.3%

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

   5. unpow2 64.3%

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

   6. cos-neg 64.3%

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

   7. *-commutative 64.3%

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

   8. distribute-rgt-neg-in 64.3%

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

   9. metadata-eval 64.3%

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

   10. unpow2 64.3%

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

   11. sqr-neg 64.3%

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

   12. unpow2 64.3%

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

   13. associate-*r* 57.3%

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

   14. unpow2 57.3%

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

   15. *-commutative 57.3%

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

3. Simplified 57.3%

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

5. Taylor expanded in x around 0 53.0%

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

   1. associate-/r* 52.2%

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

   2. *-commutative 52.2%

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

   3. unpow2 52.2%

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

   4. unpow2 52.2%

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

   5. swap-sqr 64.4%

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

   6. unpow2 64.4%

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

   7. associate-/r* 65.2%

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

   8. unpow2 65.2%

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

   9. unpow2 65.2%

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

   10. swap-sqr 77.4%

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

   11. unpow2 77.4%

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

7. Simplified 77.4%

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

   1. unpow2 95.2%

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

   2. associate-*r* 92.5%

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

   3. associate-*r* 90.6%

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

   4. *-commutative 90.6%

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

   5. associate-*l* 90.8%

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

   6. *-commutative 90.8%

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

9. Applied egg-rr 75.4%

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

    1. *-commutative 90.8%

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

    2. associate-*r* 90.6%

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

    3. *-commutative 90.6%

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

    4. *-commutative 90.6%

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

    5. *-commutative 90.6%

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

11. Simplified 74.0%

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

12. Final simplification 74.0%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))