mixedcos

Percentage Accurate: 66.2% → 96.7%

Time: 9.4s

Alternatives: 9

Speedup: 9.0×

• 30.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.2% 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: 96.7% accurate, 2.4× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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

s_m = fabs(s);
c_m = fabs(c);
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

s_m = abs(s)
c_m = abs(c)
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

s_m = Math.abs(s);
c_m = Math.abs(c);
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

s_m = math.fabs(s)
c_m = math.fabs(c)
[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

s_m = abs(s)
c_m = abs(c)
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

s_m = abs(s);
c_m = abs(c);
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

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $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 66.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow2 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.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 rewrites 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. lift-*.f64 N/A

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

   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. lift-+.f64 97.0

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 97.0

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 94.4

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 96.1

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

6. Applied rewrites 96.1%

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

7. Final simplification 96.1%

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

Alternative 2: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(x \cdot 2\right)}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)}\\ t_1 := x \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -2\right)}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 2.0)) (* (pow c_m 2.0) (* x (* x (pow s_m 2.0))))))
        (t_1 (* x (* c_m s_m))))
   (if (<= t_0 (- INFINITY))
     (/ (* x (* x -2.0)) (* t_1 t_1))
     (if (<= t_0 -2e-154)
       (/ (cos (+ x x)) (* (* c_m c_m) (* s_m (* s_m (* x x)))))
       (/ (/ 1.0 t_1) t_1)))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x * 2.0)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))));
	double t_1 = x * (c_m * s_m);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * (x * -2.0)) / (t_1 * t_1);
	} else if (t_0 <= -2e-154) {
		tmp = cos((x + x)) / ((c_m * c_m) * (s_m * (s_m * (x * x))));
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
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 * 2.0)) / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))));
	double t_1 = x * (c_m * s_m);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * (x * -2.0)) / (t_1 * t_1);
	} else if (t_0 <= -2e-154) {
		tmp = Math.cos((x + x)) / ((c_m * c_m) * (s_m * (s_m * (x * x))));
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((x * 2.0)) / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))
	t_1 = x * (c_m * s_m)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * (x * -2.0)) / (t_1 * t_1)
	elif t_0 <= -2e-154:
		tmp = math.cos((x + x)) / ((c_m * c_m) * (s_m * (s_m * (x * x))))
	else:
		tmp = (1.0 / t_1) / t_1
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(cos(Float64(x * 2.0)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0)))))
	t_1 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * Float64(x * -2.0)) / Float64(t_1 * t_1));
	elseif (t_0 <= -2e-154)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(c_m * c_m) * Float64(s_m * Float64(s_m * Float64(x * x)))));
	else
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
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 * 2.0)) / ((c_m ^ 2.0) * (x * (x * (s_m ^ 2.0))));
	t_1 = x * (c_m * s_m);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * (x * -2.0)) / (t_1 * t_1);
	elseif (t_0 <= -2e-154)
		tmp = cos((x + x)) / ((c_m * c_m) * (s_m * (s_m * (x * x))));
	else
		tmp = (1.0 / t_1) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $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[(N[Cos[N[(x * 2.0), $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]}, Block[{t$95$1 = N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -2e-154], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 63.8

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

   7. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 63.8

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 63.8

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 63.8

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

   9. Applied rewrites 63.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 63.8%

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

   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))) < -1.9999999999999999e-154

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 99.0

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 99.0

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -1.9999999999999999e-154 < (/.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 65.6%

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

   3. 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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 86.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -\infty:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -2\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{elif}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(c\_m \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c\_m}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -2\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 (* x (* c_m s_m))))
   (if (<=
        (/ (cos (* x 2.0)) (* (pow c_m 2.0) (* x (* x (pow s_m 2.0)))))
        -2e-154)
     (/ (* x (* x -2.0)) (* t_0 t_0))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = x * (c_m * s_m);
	double tmp;
	if ((cos((x * 2.0)) / (pow(c_m, 2.0) * (x * (x * pow(s_m, 2.0))))) <= -2e-154) {
		tmp = (x * (x * -2.0)) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
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 = x * (c_m * s_m)
    if ((cos((x * 2.0d0)) / ((c_m ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= (-2d-154)) then
        tmp = (x * (x * (-2.0d0))) / (t_0 * t_0)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = x * (c_m * s_m);
	double tmp;
	if ((Math.cos((x * 2.0)) / (Math.pow(c_m, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= -2e-154) {
		tmp = (x * (x * -2.0)) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = x * (c_m * s_m)
	tmp = 0
	if (math.cos((x * 2.0)) / (math.pow(c_m, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= -2e-154:
		tmp = (x * (x * -2.0)) / (t_0 * t_0)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c_m ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= -2e-154)
		tmp = Float64(Float64(x * Float64(x * -2.0)) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = x * (c_m * s_m);
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((c_m ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= -2e-154)
		tmp = (x * (x * -2.0)) / (t_0 * t_0);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $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[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $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-154], N[(N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $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))) < -1.9999999999999999e-154

   1. Initial program 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 39.9

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 39.9

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*r* N/A

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 39.9

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

   9. Applied rewrites 39.9%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 39.9%

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

   if -1.9999999999999999e-154 < (/.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 65.6%

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

   3. 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. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

   7. Applied rewrites 78.7%

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

   9. Applied rewrites 86.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -2\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 1.28 \cdot 10^{+199}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(\left(s\_m \cdot \left(x \cdot s\_m\right)\right) \cdot \left(x \cdot c\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(s\_m \cdot \left(c\_m \cdot \left(x \cdot \left(x \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 (<= s_m 1.28e+199)
   (/ 1.0 (* c_m (* (* s_m (* x s_m)) (* x c_m))))
   (/ 1.0 (* c_m (* s_m (* c_m (* x (* x s_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if (s_m <= 1.28e+199) {
		tmp = 1.0 / (c_m * ((s_m * (x * s_m)) * (x * c_m)));
	} else {
		tmp = 1.0 / (c_m * (s_m * (c_m * (x * (x * s_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
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 (s_m <= 1.28d+199) then
        tmp = 1.0d0 / (c_m * ((s_m * (x * s_m)) * (x * c_m)))
    else
        tmp = 1.0d0 / (c_m * (s_m * (c_m * (x * (x * s_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if (s_m <= 1.28e+199) {
		tmp = 1.0 / (c_m * ((s_m * (x * s_m)) * (x * c_m)));
	} else {
		tmp = 1.0 / (c_m * (s_m * (c_m * (x * (x * s_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if s_m <= 1.28e+199:
		tmp = 1.0 / (c_m * ((s_m * (x * s_m)) * (x * c_m)))
	else:
		tmp = 1.0 / (c_m * (s_m * (c_m * (x * (x * s_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (s_m <= 1.28e+199)
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(s_m * Float64(x * s_m)) * Float64(x * c_m))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(s_m * Float64(c_m * Float64(x * Float64(x * s_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
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 (s_m <= 1.28e+199)
		tmp = 1.0 / (c_m * ((s_m * (x * s_m)) * (x * c_m)));
	else
		tmp = 1.0 / (c_m * (s_m * (c_m * (x * (x * s_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $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[s$95$m, 1.28e+199], N[(1.0 / N[(c$95$m * N[(N[(s$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(s$95$m * N[(c$95$m * N[(x * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.2799999999999999e199

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   7. Applied rewrites 71.1%

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

   if 1.2799999999999999e199 < s

   1. Initial program 71.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 88.0%

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

   11. Applied rewrites 92.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.28 \cdot 10^{+199}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(s \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(x \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 7.8× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 (* x (* c_m s_m)))) (/ (/ 1.0 t_0) t_0)))

C

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

Fortran

s_m = abs(s)
c_m = abs(c)
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 = x * (c_m * s_m)
    code = (1.0d0 / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.5

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

5. Applied rewrites 68.5%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 79.5%

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

Alternative 6: 78.2% accurate, 9.0× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 (* x (* c_m s_m)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

s_m = abs(s)
c_m = abs(c)
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 = x * (c_m * s_m)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.5

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

5. Applied rewrites 68.5%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 79.4%

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

Alternative 7: 77.9% accurate, 9.0× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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
 (/ 1.0 (* (* x s_m) (* c_m (* c_m (* x s_m))))))

C

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

Fortran

s_m = abs(s)
c_m = abs(c)
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 = 1.0d0 / ((x * s_m) * (c_m * (c_m * (x * s_m))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.5

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

5. Applied rewrites 68.5%

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

7. Applied rewrites 75.5%

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

8. Final simplification 75.5%

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

Alternative 8: 75.6% accurate, 9.0× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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
 (/ 1.0 (* c_m (* s_m (* x (* x (* c_m s_m)))))))

C

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

Fortran

s_m = abs(s)
c_m = abs(c)
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 = 1.0d0 / (c_m * (s_m * (x * (x * (c_m * s_m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.5

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

5. Applied rewrites 68.5%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 74.2%

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

10. Final simplification 74.2%

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

Alternative 9: 75.4% accurate, 9.0× speedup

Math

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

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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
 (/ 1.0 (* c_m (* s_m (* c_m (* x (* x s_m)))))))

C

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

Fortran

s_m = abs(s)
c_m = abs(c)
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 = 1.0d0 / (c_m * (s_m * (c_m * (x * (x * s_m)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in x around 0

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

   1. lower-/.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 68.5

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

5. Applied rewrites 68.5%

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

7. Applied rewrites 72.0%

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

9. Applied rewrites 70.6%

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

11. Applied rewrites 71.8%

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

12. Final simplification 71.8%

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

Reproduce

herbie shell --seed 2024232 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))