mixedcos

Percentage Accurate: 67.0% → 98.9%

Time: 9.4s

Alternatives: 8

Speedup: 2.4×

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x + x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= ((double) INFINITY)) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / pow((x * (c_m * s_m)), 2.0);
	}
	return tmp;
}

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((x + x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / (t_1 * t_1);
	} else {
		tmp = t_0 / Math.pow((x * (c_m * s_m)), 2.0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((x + x))
	t_1 = c_m * (x * s_m)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= math.inf:
		tmp = t_0 / (t_1 * t_1)
	else:
		tmp = t_0 / math.pow((x * (c_m * s_m)), 2.0)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= Inf)
		tmp = Float64(t_0 / Float64(t_1 * t_1));
	else
		tmp = Float64(t_0 / (Float64(x * Float64(c_m * s_m)) ^ 2.0));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 82.7%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 97.0

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

   4. Applied egg-rr 97.0%

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

      1. count-2 N/A

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

      2. +-lowering-+.f64 97.0

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

   6. Applied egg-rr 97.0%

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 99.4

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

   8. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 97.6

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

   4. Applied egg-rr 97.6%

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

      1. count-2 N/A

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

      2. +-lowering-+.f64 97.6

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

   6. Applied egg-rr 97.6%

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

4. Final simplification 99.0%

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

Alternative 2: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* x (* x (pow s_m 2.0)))))
        -5e-192)
     (/ -2.0 (* c_m (* c_m (* s_m s_m))))
     (/ 1.0 (* t_0 t_0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= -5e-192) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= (-5d-192)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= -5e-192) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= -5e-192:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= -5e-192)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= -5e-192)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-192], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -5.0000000000000001e-192

   1. Initial program 89.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 66.5

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

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x - 0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x + \left(\mathsf{neg}\left(0\right)\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2} + \left(\mathsf{neg}\left(0\right)\right), 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -2 + \color{blue}{0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      10. accelerator-lowering-fma.f64 45.8

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -2, 0\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   7. Simplified 45.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.3

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

   10. Simplified 63.3%

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

   11. Taylor expanded in s around 0

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 63.3

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

   13. Simplified 63.3%

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

   if -5.0000000000000001e-192 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 70.7

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

   5. Simplified 70.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. swap-sqr N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 85.0

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

   7. Applied egg-rr 85.0%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(\left(x \cdot s\_m\right) \cdot \left(c\_m \cdot \left(x \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* x (* x (pow s_m 2.0)))))
      -5e-192)
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))
   (/ 1.0 (* c_m (* (* x s_m) (* c_m (* x s_m)))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= -5e-192) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= (-5d-192)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c_m * ((x * s_m) * (c_m * (x * s_m))))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= -5e-192) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= -5e-192:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= -5e-192)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(x * s_m) * Float64(c_m * Float64(x * s_m)))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= -5e-192)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (c_m * ((x * s_m) * (c_m * (x * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-192], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(N[(x * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -5.0000000000000001e-192

   1. Initial program 89.1%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 66.5

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

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x - 0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x + \left(\mathsf{neg}\left(0\right)\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2} + \left(\mathsf{neg}\left(0\right)\right), 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -2 + \color{blue}{0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      10. accelerator-lowering-fma.f64 45.8

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -2, 0\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   7. Simplified 45.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.3

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

   10. Simplified 63.3%

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

   11. Taylor expanded in s around 0

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 63.3

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

   13. Simplified 63.3%

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

   if -5.0000000000000001e-192 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.4%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 70.7

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

   5. Simplified 70.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. swap-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 82.6

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{-2}{\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{0 - c\_m \cdot \left(s\_m \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* x (* x (pow s_m 2.0)))))
      2e+135)
   (/ -2.0 (* (* s_m s_m) (* c_m c_m)))
   (/ 2.0 (- 0.0 (* c_m (* s_m (* c_m s_m)))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= 2e+135) {
		tmp = -2.0 / ((s_m * s_m) * (c_m * c_m));
	} else {
		tmp = 2.0 / (0.0 - (c_m * (s_m * (c_m * s_m))));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c_m ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= 2d+135) then
        tmp = (-2.0d0) / ((s_m * s_m) * (c_m * c_m))
    else
        tmp = 2.0d0 / (0.0d0 - (c_m * (s_m * (c_m * s_m))))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= 2e+135) {
		tmp = -2.0 / ((s_m * s_m) * (c_m * c_m));
	} else {
		tmp = 2.0 / (0.0 - (c_m * (s_m * (c_m * s_m))));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= 2e+135:
		tmp = -2.0 / ((s_m * s_m) * (c_m * c_m))
	else:
		tmp = 2.0 / (0.0 - (c_m * (s_m * (c_m * s_m))))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= 2e+135)
		tmp = Float64(-2.0 / Float64(Float64(s_m * s_m) * Float64(c_m * c_m)));
	else
		tmp = Float64(2.0 / Float64(0.0 - Float64(c_m * Float64(s_m * Float64(c_m * s_m)))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c_m ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= 2e+135)
		tmp = -2.0 / ((s_m * s_m) * (c_m * c_m));
	else
		tmp = 2.0 / (0.0 - (c_m * (s_m * (c_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+135], N[(-2.0 / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(0.0 - N[(c$95$m * N[(s$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 1.99999999999999992e135

   1. Initial program 78.6%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 71.1

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x - 0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x + \left(\mathsf{neg}\left(0\right)\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2} + \left(\mathsf{neg}\left(0\right)\right), 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -2 + \color{blue}{0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      10. accelerator-lowering-fma.f64 37.7

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -2, 0\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   7. Simplified 37.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 51.8

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

   10. Simplified 51.8%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 53.2

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

   12. Applied egg-rr 53.2%

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

   if 1.99999999999999992e135 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 50.6%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 57.8

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

   4. Applied egg-rr 57.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x - 0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x + \left(\mathsf{neg}\left(0\right)\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2} + \left(\mathsf{neg}\left(0\right)\right), 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -2 + \color{blue}{0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      10. accelerator-lowering-fma.f64 47.1

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -2, 0\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   7. Simplified 47.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 2.0

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

   10. Simplified 2.0%

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

       1. frac-2neg N/A

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

       2. metadata-eval N/A

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

       3. /-lowering-/.f64 N/A

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

       4. neg-sub0 N/A

          \[\leadsto \frac{2}{\color{blue}{0 - s \cdot \left(\left(c \cdot c\right) \cdot s\right)}} \]

       5. +-inverses N/A

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

       6. --lowering--.f64 N/A

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

       7. +-inverses N/A

          \[\leadsto \frac{2}{\color{blue}{0} - s \cdot \left(\left(c \cdot c\right) \cdot s\right)} \]

       8. *-commutative N/A

          \[\leadsto \frac{2}{0 - \color{blue}{\left(\left(c \cdot c\right) \cdot s\right) \cdot s}} \]

       9. associate-*l* N/A

          \[\leadsto \frac{2}{0 - \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)} \cdot s} \]

       10. associate-*l* N/A

           \[\leadsto \frac{2}{0 - \color{blue}{c \cdot \left(\left(c \cdot s\right) \cdot s\right)}} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{0 - \color{blue}{c \cdot \left(\left(c \cdot s\right) \cdot s\right)}} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{2}{0 - c \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot s\right)}} \]

       13. *-lowering-*.f64 35.0

           \[\leadsto \frac{2}{0 - c \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot s\right)} \]

   12. Applied egg-rr 35.0%

       \[\leadsto \color{blue}{\frac{2}{0 - c \cdot \left(\left(c \cdot s\right) \cdot s\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{0 - c \cdot \left(s \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 6 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(c\_m \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{s\_m \cdot \left(s\_m \cdot \left(x \cdot \left(c\_m \cdot \left(x \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (* c_m (* x s_m))))
   (if (<= x 6e-21)
     (/ 1.0 (* t_1 t_1))
     (if (<= x 3.5e+166)
       (/ t_0 (* s_m (* c_m (* s_m (* c_m (* x x))))))
       (/ t_0 (* s_m (* s_m (* x (* c_m (* x c_m))))))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x + x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if (x <= 6e-21) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x <= 3.5e+166) {
		tmp = t_0 / (s_m * (c_m * (s_m * (c_m * (x * x)))));
	} else {
		tmp = t_0 / (s_m * (s_m * (x * (c_m * (x * c_m)))));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    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 + x))
    t_1 = c_m * (x * s_m)
    if (x <= 6d-21) then
        tmp = 1.0d0 / (t_1 * t_1)
    else if (x <= 3.5d+166) then
        tmp = t_0 / (s_m * (c_m * (s_m * (c_m * (x * x)))))
    else
        tmp = t_0 / (s_m * (s_m * (x * (c_m * (x * c_m)))))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((x + x));
	double t_1 = c_m * (x * s_m);
	double tmp;
	if (x <= 6e-21) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x <= 3.5e+166) {
		tmp = t_0 / (s_m * (c_m * (s_m * (c_m * (x * x)))));
	} else {
		tmp = t_0 / (s_m * (s_m * (x * (c_m * (x * c_m)))));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((x + x))
	t_1 = c_m * (x * s_m)
	tmp = 0
	if x <= 6e-21:
		tmp = 1.0 / (t_1 * t_1)
	elif x <= 3.5e+166:
		tmp = t_0 / (s_m * (c_m * (s_m * (c_m * (x * x)))))
	else:
		tmp = t_0 / (s_m * (s_m * (x * (c_m * (x * c_m)))))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 6e-21)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	elseif (x <= 3.5e+166)
		tmp = Float64(t_0 / Float64(s_m * Float64(c_m * Float64(s_m * Float64(c_m * Float64(x * x))))));
	else
		tmp = Float64(t_0 / Float64(s_m * Float64(s_m * Float64(x * Float64(c_m * Float64(x * c_m))))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((x + x));
	t_1 = c_m * (x * s_m);
	tmp = 0.0;
	if (x <= 6e-21)
		tmp = 1.0 / (t_1 * t_1);
	elseif (x <= 3.5e+166)
		tmp = t_0 / (s_m * (c_m * (s_m * (c_m * (x * x)))));
	else
		tmp = t_0 / (s_m * (s_m * (x * (c_m * (x * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6e-21], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+166], N[(t$95$0 / N[(s$95$m * N[(c$95$m * N[(s$95$m * N[(c$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(s$95$m * N[(s$95$m * N[(x * N[(c$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.99999999999999982e-21

   1. Initial program 62.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 66.8

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

   5. Simplified 66.8%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. swap-sqr N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 84.5

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

   7. Applied egg-rr 84.5%

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

   if 5.99999999999999982e-21 < x < 3.4999999999999999e166

   1. Initial program 75.2%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 98.6

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

   4. Applied egg-rr 98.6%

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

      1. /-lowering-/.f64 N/A

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

      2. count-2 N/A

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

      3. cos-lowering-cos.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. swap-sqr N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. *-lowering-*.f64 N/A

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 N/A

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

      22. *-lowering-*.f64 91.3

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

   6. Applied egg-rr 91.3%

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

   if 3.4999999999999999e166 < x

   1. Initial program 67.7%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. /-lowering-/.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 84.3

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

   4. Applied egg-rr 84.3%

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

      1. count-2 N/A

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

      2. associate-/l/ N/A

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

      3. associate-/l/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. count-2 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 84.2

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

   6. Applied egg-rr 84.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 2350:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s\_m \cdot \left(s\_m \cdot \left(x \cdot \left(c\_m \cdot \left(x \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))))
   (if (<= x 2350.0)
     (/ (fma x (fma x -2.0 0.0) 1.0) (* t_0 t_0))
     (/ (cos (+ x x)) (* s_m (* s_m (* x (* c_m (* x c_m)))))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double tmp;
	if (x <= 2350.0) {
		tmp = fma(x, fma(x, -2.0, 0.0), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x + x)) / (s_m * (s_m * (x * (c_m * (x * c_m)))));
	}
	return tmp;
}

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	tmp = 0.0
	if (x <= 2350.0)
		tmp = Float64(fma(x, fma(x, -2.0, 0.0), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(s_m * Float64(s_m * Float64(x * Float64(c_m * Float64(x * c_m))))));
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2350.0], N[(N[(x * N[(x * -2.0 + 0.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(s$95$m * N[(x * N[(c$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2350

   1. Initial program 62.4%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 60.6

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

   4. Applied egg-rr 60.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x - 0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x + \left(\mathsf{neg}\left(0\right)\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2} + \left(\mathsf{neg}\left(0\right)\right), 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -2 + \color{blue}{0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

      10. accelerator-lowering-fma.f64 46.1

          \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -2, 0\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   7. Simplified 46.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(\left(c \cdot c\right) \cdot \left(s \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      3. unswap-sqr N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      4. swap-sqr N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

      11. *-lowering-*.f64 75.5

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

   9. Applied egg-rr 75.5%

      \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]

   if 2350 < x

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. /-lowering-/.f64 N/A

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

      12. count-2 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. +-lowering-+.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 82.9

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

   4. Applied egg-rr 82.9%

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

      1. count-2 N/A

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

      2. associate-/l/ N/A

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

      3. associate-/l/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. count-2 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 83.5

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

   6. Applied egg-rr 83.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2350:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. pow-lowering-pow.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 97.2

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

4. Applied egg-rr 97.2%

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

   1. count-2 N/A

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

   2. +-lowering-+.f64 97.2

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

6. Applied egg-rr 97.2%

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 97.4

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

8. Applied egg-rr 97.4%

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

9. Final simplification 97.4%

   \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)} \]
10. Add Preprocessing

Alternative 8: 31.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 64.8

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

4. Applied egg-rr 64.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. --rgt-identity N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x - 0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{-2 \cdot x + \left(\mathsf{neg}\left(0\right)\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2} + \left(\mathsf{neg}\left(0\right)\right), 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot -2 + \color{blue}{0}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

   10. accelerator-lowering-fma.f64 42.2

       \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -2, 0\right)}, 1\right)}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

7. Simplified 42.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -2, 0\right), 1\right)}}{\left(x \cdot x\right) \cdot \left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 28.1

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

10. Simplified 28.1%

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

11. Taylor expanded in s around 0

    \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. *-lowering-*.f64 N/A

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

    5. *-lowering-*.f64 N/A

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

    6. unpow2 N/A

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

    7. *-lowering-*.f64 28.1

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

13. Simplified 28.1%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))