mixedcos

Percentage Accurate: 66.7% → 97.0%

Time: 9.7s

Alternatives: 17

Speedup: 9.0×

• 32.1% 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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{t\_0}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* s_m x_m))))
   (if (<= x_m 9e-86)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x_m 1.3e+185)
       (/ t_0 (* (* s_m c_m) (* s_m (* x_m (* x_m c_m)))))
       (/ t_0 (* (* x_m (* c_m (* x_m c_m))) (* s_m s_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 9e-86) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.3e+185) {
		tmp = t_0 / ((s_m * c_m) * (s_m * (x_m * (x_m * c_m))));
	} else {
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (s_m * x_m)
    if (x_m <= 9d-86) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x_m <= 1.3d+185) then
        tmp = t_0 / ((s_m * c_m) * (s_m * (x_m * (x_m * c_m))))
    else
        tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 9e-86) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.3e+185) {
		tmp = t_0 / ((s_m * c_m) * (s_m * (x_m * (x_m * c_m))));
	} else {
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 9e-86:
		tmp = (1.0 / t_1) / t_1
	elif x_m <= 1.3e+185:
		tmp = t_0 / ((s_m * c_m) * (s_m * (x_m * (x_m * c_m))))
	else:
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 9e-86)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x_m <= 1.3e+185)
		tmp = Float64(t_0 / Float64(Float64(s_m * c_m) * Float64(s_m * Float64(x_m * Float64(x_m * c_m)))));
	else
		tmp = Float64(t_0 / Float64(Float64(x_m * Float64(c_m * Float64(x_m * c_m))) * Float64(s_m * s_m)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 9e-86)
		tmp = (1.0 / t_1) / t_1;
	elseif (x_m <= 1.3e+185)
		tmp = t_0 / ((s_m * c_m) * (s_m * (x_m * (x_m * c_m))));
	else
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 9e-86], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 1.3e+185], N[(t$95$0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(x$95$m * N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.9999999999999995e-86

   1. Initial program 64.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 83.1%

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

   if 8.9999999999999995e-86 < x < 1.3e185

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 98.2

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 96.8

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

   8. Applied rewrites 96.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 94.0

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

   10. Applied rewrites 94.0%

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

   if 1.3e185 < x

   1. Initial program 56.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 90.5

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

   4. Applied rewrites 90.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 80.1

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 80.1

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

   8. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   10. Applied rewrites 78.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot c\right) \cdot \left(s \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left({s\_m}^{2} \cdot x\_m\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m \cdot -2, 1\right)}{x\_m \cdot \left(c\_m \cdot \left(s\_m \cdot t\_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* x_m (* (pow s_m 2.0) x_m))))
        -2e-138)
     (/ (fma x_m (* x_m -2.0) 1.0) (* x_m (* c_m (* s_m t_0))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * (x_m * (pow(s_m, 2.0) * x_m)))) <= -2e-138) {
		tmp = fma(x_m, (x_m * -2.0), 1.0) / (x_m * (c_m * (s_m * t_0)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64((s_m ^ 2.0) * x_m)))) <= -2e-138)
		tmp = Float64(fma(x_m, Float64(x_m * -2.0), 1.0) / Float64(x_m * Float64(c_m * Float64(s_m * t_0))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(N[(x$95$m * N[(x$95$m * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x$95$m * N[(c$95$m * N[(s$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

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

      1. lift-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.2

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

   7. Applied rewrites 49.2%

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

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 84.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left({s}^{2} \cdot x\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{x \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left({s\_m}^{2} \cdot x\_m\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x\_m \cdot \left(x\_m \cdot -2\right)}{s\_m \cdot \left(\left(s\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(c\_m \cdot c\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* x_m (* (pow s_m 2.0) x_m))))
        -2e-138)
     (/ (* x_m (* x_m -2.0)) (* s_m (* (* s_m x_m) (* x_m (* c_m c_m)))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * (x_m * (pow(s_m, 2.0) * x_m)))) <= -2e-138) {
		tmp = (x_m * (x_m * -2.0)) / (s_m * ((s_m * x_m) * (x_m * (c_m * c_m))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * (x_m * ((s_m ** 2.0d0) * x_m)))) <= (-2d-138)) then
        tmp = (x_m * (x_m * (-2.0d0))) / (s_m * ((s_m * x_m) * (x_m * (c_m * c_m))))
    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);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * (x_m * (Math.pow(s_m, 2.0) * x_m)))) <= -2e-138) {
		tmp = (x_m * (x_m * -2.0)) / (s_m * ((s_m * x_m) * (x_m * (c_m * c_m))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * (x_m * (math.pow(s_m, 2.0) * x_m)))) <= -2e-138:
		tmp = (x_m * (x_m * -2.0)) / (s_m * ((s_m * x_m) * (x_m * (c_m * c_m))))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64((s_m ^ 2.0) * x_m)))) <= -2e-138)
		tmp = Float64(Float64(x_m * Float64(x_m * -2.0)) / Float64(s_m * Float64(Float64(s_m * x_m) * Float64(x_m * Float64(c_m * c_m)))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * (x_m * ((s_m ^ 2.0) * x_m)))) <= -2e-138)
		tmp = (x_m * (x_m * -2.0)) / (s_m * ((s_m * x_m) * (x_m * (c_m * c_m))));
	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]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(N[(x$95$m * N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

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

      1. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 49.0%

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

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 84.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left({s}^{2} \cdot x\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -2\right)}{s \cdot \left(\left(s \cdot x\right) \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left({s\_m}^{2} \cdot x\_m\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x\_m \cdot \left(x\_m \cdot -2\right)}{s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(s\_m \cdot x\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* x_m (* (pow s_m 2.0) x_m))))
        -2e-138)
     (/ (* x_m (* x_m -2.0)) (* s_m (* c_m (* c_m (* x_m (* s_m x_m))))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * (x_m * (pow(s_m, 2.0) * x_m)))) <= -2e-138) {
		tmp = (x_m * (x_m * -2.0)) / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * (x_m * ((s_m ** 2.0d0) * x_m)))) <= (-2d-138)) then
        tmp = (x_m * (x_m * (-2.0d0))) / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))))
    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);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * (x_m * (Math.pow(s_m, 2.0) * x_m)))) <= -2e-138) {
		tmp = (x_m * (x_m * -2.0)) / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * (x_m * (math.pow(s_m, 2.0) * x_m)))) <= -2e-138:
		tmp = (x_m * (x_m * -2.0)) / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64((s_m ^ 2.0) * x_m)))) <= -2e-138)
		tmp = Float64(Float64(x_m * Float64(x_m * -2.0)) / Float64(s_m * Float64(c_m * Float64(c_m * Float64(x_m * Float64(s_m * x_m))))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * (x_m * ((s_m ^ 2.0) * x_m)))) <= -2e-138)
		tmp = (x_m * (x_m * -2.0)) / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))));
	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]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(N[(x$95$m * N[(x$95$m * -2.0), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * N[(c$95$m * N[(c$95$m * N[(x$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

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

      1. lift-*.f64 N/A

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

      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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.0

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

   7. Applied rewrites 49.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 49.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 44.7

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 44.7

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

   9. Applied rewrites 44.7%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 44.7%

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

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 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 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 84.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left({s}^{2} \cdot x\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot -2\right)}{s \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;{s\_m}^{2} \leq 400:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m + x\_m\right)}{c\_m} \cdot \frac{1}{x\_m}}{c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot x\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{{\left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= (pow s_m 2.0) 400.0)
   (/ (* (/ (cos (+ x_m x_m)) c_m) (/ 1.0 x_m)) (* c_m (* s_m (* s_m x_m))))
   (/ (cos (* 2.0 x_m)) (pow (* x_m (* s_m c_m)) 2.0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (pow(s_m, 2.0) <= 400.0) {
		tmp = ((cos((x_m + x_m)) / c_m) * (1.0 / x_m)) / (c_m * (s_m * (s_m * x_m)));
	} else {
		tmp = cos((2.0 * x_m)) / pow((x_m * (s_m * c_m)), 2.0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((s_m ** 2.0d0) <= 400.0d0) then
        tmp = ((cos((x_m + x_m)) / c_m) * (1.0d0 / x_m)) / (c_m * (s_m * (s_m * x_m)))
    else
        tmp = cos((2.0d0 * x_m)) / ((x_m * (s_m * c_m)) ** 2.0d0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (Math.pow(s_m, 2.0) <= 400.0) {
		tmp = ((Math.cos((x_m + x_m)) / c_m) * (1.0 / x_m)) / (c_m * (s_m * (s_m * x_m)));
	} else {
		tmp = Math.cos((2.0 * x_m)) / Math.pow((x_m * (s_m * c_m)), 2.0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if math.pow(s_m, 2.0) <= 400.0:
		tmp = ((math.cos((x_m + x_m)) / c_m) * (1.0 / x_m)) / (c_m * (s_m * (s_m * x_m)))
	else:
		tmp = math.cos((2.0 * x_m)) / math.pow((x_m * (s_m * c_m)), 2.0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if ((s_m ^ 2.0) <= 400.0)
		tmp = Float64(Float64(Float64(cos(Float64(x_m + x_m)) / c_m) * Float64(1.0 / x_m)) / Float64(c_m * Float64(s_m * Float64(s_m * x_m))));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / (Float64(x_m * Float64(s_m * c_m)) ^ 2.0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 400.0)
		tmp = ((cos((x_m + x_m)) / c_m) * (1.0 / x_m)) / (c_m * (s_m * (s_m * x_m)));
	else
		tmp = cos((2.0 * x_m)) / ((x_m * (s_m * c_m)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 400.0], N[(N[(N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(s$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 400

   1. Initial program 64.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. 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 96.0

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

   4. Applied rewrites 96.0%

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

   6. Applied rewrites 85.9%

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

   if 400 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 66.0%

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

      1. lift-*.f64 N/A

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

      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 95.4

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

   4. Applied rewrites 95.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 400:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{c} \cdot \frac{1}{x}}{c \cdot \left(s \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 10^{-122}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{t\_0}{\left(s\_m \cdot c\_m\right) \cdot \left(x\_m \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* s_m x_m))))
   (if (<= x_m 1e-122)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x_m 1.3e+185)
       (/ t_0 (* (* s_m c_m) (* x_m t_1)))
       (/ t_0 (* (* x_m (* c_m (* x_m c_m))) (* s_m s_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 1e-122) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.3e+185) {
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1));
	} else {
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (s_m * x_m)
    if (x_m <= 1d-122) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x_m <= 1.3d+185) then
        tmp = t_0 / ((s_m * c_m) * (x_m * t_1))
    else
        tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 1e-122) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1.3e+185) {
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1));
	} else {
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 1e-122:
		tmp = (1.0 / t_1) / t_1
	elif x_m <= 1.3e+185:
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1))
	else:
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 1e-122)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x_m <= 1.3e+185)
		tmp = Float64(t_0 / Float64(Float64(s_m * c_m) * Float64(x_m * t_1)));
	else
		tmp = Float64(t_0 / Float64(Float64(x_m * Float64(c_m * Float64(x_m * c_m))) * Float64(s_m * s_m)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 1e-122)
		tmp = (1.0 / t_1) / t_1;
	elseif (x_m <= 1.3e+185)
		tmp = t_0 / ((s_m * c_m) * (x_m * t_1));
	else
		tmp = t_0 / ((x_m * (c_m * (x_m * c_m))) * (s_m * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1e-122], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 1.3e+185], N[(t$95$0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(x$95$m * N[(c$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.00000000000000006e-122

   1. Initial program 64.0%

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

   3. Taylor expanded in x around 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 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 82.1%

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

   if 1.00000000000000006e-122 < x < 1.3e185

   1. Initial program 70.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. 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 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 98.4

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 97.1

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

   6. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 97.1

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

   8. Applied rewrites 97.1%

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

   if 1.3e185 < x

   1. Initial program 56.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 90.5

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

   4. Applied rewrites 90.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 80.1

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 80.1

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

   6. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 80.1

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

   8. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

   10. Applied rewrites 78.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-122}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;s\_m \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{t\_0}{c\_m \cdot \left(\left(s\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot \left(x\_m \cdot c\_m\right)\right)}\\ \mathbf{elif}\;s\_m \leq 4.9 \cdot 10^{+204}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(c\_m \cdot \left(s\_m \cdot t\_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* s_m x_m))))
   (if (<= s_m 2.1e+29)
     (/ t_0 (* c_m (* (* s_m (* s_m x_m)) (* x_m c_m))))
     (if (<= s_m 4.9e+204)
       (/ t_0 (* x_m (* c_m (* s_m t_1))))
       (/ (/ 1.0 t_1) t_1)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (s_m <= 2.1e+29) {
		tmp = t_0 / (c_m * ((s_m * (s_m * x_m)) * (x_m * c_m)));
	} else if (s_m <= 4.9e+204) {
		tmp = t_0 / (x_m * (c_m * (s_m * t_1)));
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (s_m * x_m)
    if (s_m <= 2.1d+29) then
        tmp = t_0 / (c_m * ((s_m * (s_m * x_m)) * (x_m * c_m)))
    else if (s_m <= 4.9d+204) then
        tmp = t_0 / (x_m * (c_m * (s_m * t_1)))
    else
        tmp = (1.0d0 / t_1) / t_1
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (s_m <= 2.1e+29) {
		tmp = t_0 / (c_m * ((s_m * (s_m * x_m)) * (x_m * c_m)));
	} else if (s_m <= 4.9e+204) {
		tmp = t_0 / (x_m * (c_m * (s_m * t_1)));
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (s_m * x_m)
	tmp = 0
	if s_m <= 2.1e+29:
		tmp = t_0 / (c_m * ((s_m * (s_m * x_m)) * (x_m * c_m)))
	elif s_m <= 4.9e+204:
		tmp = t_0 / (x_m * (c_m * (s_m * t_1)))
	else:
		tmp = (1.0 / t_1) / t_1
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (s_m <= 2.1e+29)
		tmp = Float64(t_0 / Float64(c_m * Float64(Float64(s_m * Float64(s_m * x_m)) * Float64(x_m * c_m))));
	elseif (s_m <= 4.9e+204)
		tmp = Float64(t_0 / Float64(x_m * Float64(c_m * Float64(s_m * t_1))));
	else
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (s_m <= 2.1e+29)
		tmp = t_0 / (c_m * ((s_m * (s_m * x_m)) * (x_m * c_m)));
	elseif (s_m <= 4.9e+204)
		tmp = t_0 / (x_m * (c_m * (s_m * t_1)));
	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]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s$95$m, 2.1e+29], N[(t$95$0 / N[(c$95$m * N[(N[(s$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[s$95$m, 4.9e+204], N[(t$95$0 / N[(x$95$m * N[(c$95$m * N[(s$95$m * t$95$1), $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 s < 2.1000000000000002e29

   1. Initial program 67.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 79.7

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

   6. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.0

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

   8. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 86.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 86.2

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

   10. Applied rewrites 86.2%

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

   if 2.1000000000000002e29 < s < 4.8999999999999997e204

   1. Initial program 64.0%

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

      1. lift-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 96.2

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

   6. Applied rewrites 96.2%

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

   if 4.8999999999999997e204 < s

   1. Initial program 47.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(\left(s \cdot \left(s \cdot x\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \mathbf{elif}\;s \leq 4.9 \cdot 10^{+204}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;x\_m \leq 1.2 \cdot 10^{+136}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{c\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* s_m x_m))))
   (if (<= x_m 8e-7)
     (/ 1.0 (* t_1 t_1))
     (if (<= x_m 1.2e+136)
       (/ t_0 (* x_m (* x_m (* c_m (* s_m (* s_m c_m))))))
       (/ t_0 (* c_m (* (* x_m c_m) (* x_m (* s_m s_m)))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x_m <= 1.2e+136) {
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	} else {
		tmp = t_0 / (c_m * ((x_m * c_m) * (x_m * (s_m * s_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (s_m * x_m)
    if (x_m <= 8d-7) then
        tmp = 1.0d0 / (t_1 * t_1)
    else if (x_m <= 1.2d+136) then
        tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))))
    else
        tmp = t_0 / (c_m * ((x_m * c_m) * (x_m * (s_m * s_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x_m <= 1.2e+136) {
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	} else {
		tmp = t_0 / (c_m * ((x_m * c_m) * (x_m * (s_m * s_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 8e-7:
		tmp = 1.0 / (t_1 * t_1)
	elif x_m <= 1.2e+136:
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))))
	else:
		tmp = t_0 / (c_m * ((x_m * c_m) * (x_m * (s_m * s_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 8e-7)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	elseif (x_m <= 1.2e+136)
		tmp = Float64(t_0 / Float64(x_m * Float64(x_m * Float64(c_m * Float64(s_m * Float64(s_m * c_m))))));
	else
		tmp = Float64(t_0 / Float64(c_m * Float64(Float64(x_m * c_m) * Float64(x_m * Float64(s_m * s_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 8e-7)
		tmp = 1.0 / (t_1 * t_1);
	elseif (x_m <= 1.2e+136)
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	else
		tmp = t_0 / (c_m * ((x_m * c_m) * (x_m * (s_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8e-7], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.2e+136], N[(t$95$0 / N[(x$95$m * N[(x$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(c$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.9999999999999996e-7

   1. Initial program 65.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 84.7%

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

   if 7.9999999999999996e-7 < x < 1.2e136

   1. Initial program 71.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 78.4

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

   4. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 78.4

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

   6. Applied rewrites 78.4%

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

   7. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 92.3

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

   9. Applied rewrites 92.3%

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

   if 1.2e136 < x

   1. Initial program 59.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 75.9

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

   4. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 75.9

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

   6. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 76.4

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

   8. Applied rewrites 76.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+136}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;x\_m \leq 3.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{c\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(s\_m \cdot \left(s\_m \cdot x\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* s_m x_m))))
   (if (<= x_m 8e-7)
     (/ 1.0 (* t_1 t_1))
     (if (<= x_m 3.2e+172)
       (/ t_0 (* x_m (* x_m (* c_m (* s_m (* s_m c_m))))))
       (/ t_0 (* c_m (* c_m (* x_m (* s_m (* s_m x_m))))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x_m <= 3.2e+172) {
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	} else {
		tmp = t_0 / (c_m * (c_m * (x_m * (s_m * (s_m * x_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (s_m * x_m)
    if (x_m <= 8d-7) then
        tmp = 1.0d0 / (t_1 * t_1)
    else if (x_m <= 3.2d+172) then
        tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))))
    else
        tmp = t_0 / (c_m * (c_m * (x_m * (s_m * (s_m * x_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x_m <= 3.2e+172) {
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	} else {
		tmp = t_0 / (c_m * (c_m * (x_m * (s_m * (s_m * x_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 8e-7:
		tmp = 1.0 / (t_1 * t_1)
	elif x_m <= 3.2e+172:
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))))
	else:
		tmp = t_0 / (c_m * (c_m * (x_m * (s_m * (s_m * x_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 8e-7)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	elseif (x_m <= 3.2e+172)
		tmp = Float64(t_0 / Float64(x_m * Float64(x_m * Float64(c_m * Float64(s_m * Float64(s_m * c_m))))));
	else
		tmp = Float64(t_0 / Float64(c_m * Float64(c_m * Float64(x_m * Float64(s_m * Float64(s_m * x_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 8e-7)
		tmp = 1.0 / (t_1 * t_1);
	elseif (x_m <= 3.2e+172)
		tmp = t_0 / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	else
		tmp = t_0 / (c_m * (c_m * (x_m * (s_m * (s_m * x_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8e-7], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 3.2e+172], N[(t$95$0 / N[(x$95$m * N[(x$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(c$95$m * N[(c$95$m * N[(x$95$m * N[(s$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.9999999999999996e-7

   1. Initial program 65.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 84.7%

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

   if 7.9999999999999996e-7 < x < 3.19999999999999985e172

   1. Initial program 71.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 79.7

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.3

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

   9. Applied rewrites 91.3%

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

   if 3.19999999999999985e172 < x

   1. Initial program 57.1%

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

      1. 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 91.4

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

   4. Applied rewrites 91.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   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 91.4

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

      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. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

   6. Applied rewrites 88.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{s\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<= x_m 2.2e-16)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* s_m (* x_m (* c_m (* x_m (* s_m c_m)))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 2.2e-16) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / (s_m * (x_m * (c_m * (x_m * (s_m * c_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if (x_m <= 2.2d-16) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x_m + x_m)) / (s_m * (x_m * (c_m * (x_m * (s_m * c_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 2.2e-16) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m + x_m)) / (s_m * (x_m * (c_m * (x_m * (s_m * c_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 2.2e-16:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x_m + x_m)) / (s_m * (x_m * (c_m * (x_m * (s_m * c_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 2.2e-16)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(s_m * Float64(x_m * Float64(c_m * Float64(x_m * Float64(s_m * c_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 2.2e-16)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x_m + x_m)) / (s_m * (x_m * (c_m * (x_m * (s_m * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.2e-16], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(x$95$m * N[(c$95$m * N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e-16

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0

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

      1. 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 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 84.6%

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

   if 2.2e-16 < 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. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 77.4

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

   4. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 77.4

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

   6. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 90.4

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 89.1

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

   8. Applied rewrites 89.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<= x_m 8e-7)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* (* x_m c_m) (* x_m (* s_m (* s_m c_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / ((x_m * c_m) * (x_m * (s_m * (s_m * c_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if (x_m <= 8d-7) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x_m + x_m)) / ((x_m * c_m) * (x_m * (s_m * (s_m * c_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m + x_m)) / ((x_m * c_m) * (x_m * (s_m * (s_m * c_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 8e-7:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x_m + x_m)) / ((x_m * c_m) * (x_m * (s_m * (s_m * c_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 8e-7)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(x_m * c_m) * Float64(x_m * Float64(s_m * Float64(s_m * c_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 8e-7)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x_m + x_m)) / ((x_m * c_m) * (x_m * (s_m * (s_m * c_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8e-7], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.9999999999999996e-7

   1. Initial program 65.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 84.7%

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

   if 7.9999999999999996e-7 < x

   1. Initial program 65.2%

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

      1. 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 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 91.4

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 90.2

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

   6. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 90.2

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

   8. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 85.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\_m\right)\\ \mathbf{if}\;x\_m \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{x\_m \cdot \left(x\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot c\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x_m))))
   (if (<= x_m 8e-7)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* x_m (* x_m (* c_m (* s_m (* s_m c_m)))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x_m)
    if (x_m <= 8d-7) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x_m + x_m)) / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x_m);
	double tmp;
	if (x_m <= 8e-7) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m + x_m)) / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (s_m * x_m)
	tmp = 0
	if x_m <= 8e-7:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x_m + x_m)) / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x_m))
	tmp = 0.0
	if (x_m <= 8e-7)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(x_m * Float64(x_m * Float64(c_m * Float64(s_m * Float64(s_m * c_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (s_m * x_m);
	tmp = 0.0;
	if (x_m <= 8e-7)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x_m + x_m)) / (x_m * (x_m * (c_m * (s_m * (s_m * c_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8e-7], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(x$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.9999999999999996e-7

   1. Initial program 65.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 84.7%

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

   if 7.9999999999999996e-7 < x

   1. Initial program 65.2%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 77.1

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

   6. Applied rewrites 77.1%

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

   7. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 80.0

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

   9. Applied rewrites 80.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;c\_m \leq 1.55 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(s\_m \cdot c\_m\right) \cdot \left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(s\_m \cdot x\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= c_m 1.55e-130)
   (/ 1.0 (* x_m (* (* s_m c_m) (* c_m (* s_m x_m)))))
   (/ 1.0 (* s_m (* c_m (* c_m (* x_m (* s_m x_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 1.55e-130) {
		tmp = 1.0 / (x_m * ((s_m * c_m) * (c_m * (s_m * x_m))));
	} else {
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (c_m <= 1.55d-130) then
        tmp = 1.0d0 / (x_m * ((s_m * c_m) * (c_m * (s_m * x_m))))
    else
        tmp = 1.0d0 / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (c_m <= 1.55e-130) {
		tmp = 1.0 / (x_m * ((s_m * c_m) * (c_m * (s_m * x_m))));
	} else {
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if c_m <= 1.55e-130:
		tmp = 1.0 / (x_m * ((s_m * c_m) * (c_m * (s_m * x_m))))
	else:
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (c_m <= 1.55e-130)
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(s_m * c_m) * Float64(c_m * Float64(s_m * x_m)))));
	else
		tmp = Float64(1.0 / Float64(s_m * Float64(c_m * Float64(c_m * Float64(x_m * Float64(s_m * x_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (c_m <= 1.55e-130)
		tmp = 1.0 / (x_m * ((s_m * c_m) * (c_m * (s_m * x_m))));
	else
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (s_m * x_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[c$95$m, 1.55e-130], N[(1.0 / N[(x$95$m * N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s$95$m * N[(c$95$m * N[(c$95$m * N[(x$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.55000000000000005e-130

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

      1. 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 65.1

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

   5. Applied rewrites 65.1%

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

   7. Applied rewrites 69.7%

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

   if 1.55000000000000005e-130 < c

   1. Initial program 60.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 71.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.55 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot c\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 65.2

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

5. Applied rewrites 65.2%

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

7. Applied rewrites 76.2%

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

Alternative 15: 76.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 65.2

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

5. Applied rewrites 65.2%

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

7. Applied rewrites 70.9%

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

8. Final simplification 70.9%

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

Alternative 16: 72.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 65.2

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

5. Applied rewrites 65.2%

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

7. Applied rewrites 67.6%

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

8. Final simplification 67.6%

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

Alternative 17: 71.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 65.2

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

5. Applied rewrites 65.2%

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

6. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))