mixedcos

Percentage Accurate: 66.9% → 98.6%

Time: 9.6s

Alternatives: 11

Speedup: 9.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 7.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\_m\right)}{{\left(s\_m \cdot x\_m\right)}^{2}}}{c\_m}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m + x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 x_m) s_m)))
   (if (<= x_m 7.4e-81)
     (/ (/ (/ (cos (* 2.0 x_m)) (pow (* s_m x_m) 2.0)) c_m) c_m)
     (/ (/ (cos (+ x_m x_m)) 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 * x_m) * s_m;
	double tmp;
	if (x_m <= 7.4e-81) {
		tmp = ((cos((2.0 * x_m)) / pow((s_m * x_m), 2.0)) / c_m) / c_m;
	} else {
		tmp = (cos((x_m + x_m)) / 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 * x_m) * s_m
    if (x_m <= 7.4d-81) then
        tmp = ((cos((2.0d0 * x_m)) / ((s_m * x_m) ** 2.0d0)) / c_m) / c_m
    else
        tmp = (cos((x_m + x_m)) / 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 * x_m) * s_m;
	double tmp;
	if (x_m <= 7.4e-81) {
		tmp = ((Math.cos((2.0 * x_m)) / Math.pow((s_m * x_m), 2.0)) / c_m) / c_m;
	} else {
		tmp = (Math.cos((x_m + x_m)) / 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 * x_m) * s_m
	tmp = 0
	if x_m <= 7.4e-81:
		tmp = ((math.cos((2.0 * x_m)) / math.pow((s_m * x_m), 2.0)) / c_m) / c_m
	else:
		tmp = (math.cos((x_m + x_m)) / 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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (x_m <= 7.4e-81)
		tmp = Float64(Float64(Float64(cos(Float64(2.0 * x_m)) / (Float64(s_m * x_m) ^ 2.0)) / c_m) / c_m);
	else
		tmp = Float64(Float64(cos(Float64(x_m + x_m)) / 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 * x_m) * s_m;
	tmp = 0.0;
	if (x_m <= 7.4e-81)
		tmp = ((cos((2.0 * x_m)) / ((s_m * x_m) ^ 2.0)) / c_m) / c_m;
	else
		tmp = (cos((x_m + x_m)) / 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 7.4e-81], N[(N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.39999999999999971e-81

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 89.8%

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

   if 7.39999999999999971e-81 < x

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 97.0

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

   8. Applied rewrites 97.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{\left(s \cdot x\right)}^{2}}}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c\_m \cdot {\left(s\_m \cdot x\_m\right)}^{2}}}{c\_m}\\ \end{array} \end{array} \]

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 x_m) s_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -2e-179)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (/ 1.0 (* c_m (pow (* s_m x_m) 2.0))) 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 * x_m) * s_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -2e-179) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / (c_m * pow((s_m * x_m), 2.0))) / 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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -2e-179)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(c_m * (Float64(s_m * x_m) ^ 2.0))) / c_m);
	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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-179], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(c$95$m * N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $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))) < -2e-179

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   if -2e-179 < (/.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 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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 78.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot {\left(s \cdot x\right)}^{2}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s\_m \cdot x\_m\right)}^{-2}}{c\_m}}{c\_m}\\ \end{array} \end{array} \]

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 x_m) s_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -2e-179)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (/ (pow (* s_m x_m) -2.0) c_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 * x_m) * s_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -2e-179) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (pow((s_m * x_m), -2.0) / c_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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -2e-179)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64((Float64(s_m * x_m) ^ -2.0) / c_m) / c_m);
	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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-179], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], -2.0], $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $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))) < -2e-179

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   if -2e-179 < (/.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 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. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 78.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s \cdot x\right)}^{-2}}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c\_m} \cdot \frac{-1}{\left(\left(s\_m \cdot x\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

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 x_m) s_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -2e-179)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (* (/ -1.0 c_m) (/ -1.0 (* (* (* s_m x_m) (* s_m x_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 * x_m) * s_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -2e-179) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (-1.0 / c_m) * (-1.0 / (((s_m * x_m) * (s_m * x_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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -2e-179)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(-1.0 / c_m) * Float64(-1.0 / Float64(Float64(Float64(s_m * x_m) * Float64(s_m * x_m)) * c_m)));
	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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-179], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / c$95$m), $MachinePrecision] * N[(-1.0 / N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2e-179

   1. Initial program 76.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   if -2e-179 < (/.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 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. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 78.5%

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

   11. Applied rewrites 78.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-179}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \frac{-1}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 7.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{c\_m \cdot {\left(s\_m \cdot x\_m\right)}^{2}}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m + x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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 x_m) s_m)))
   (if (<= x_m 7.4e-81)
     (/ (/ 1.0 (* c_m (pow (* s_m x_m) 2.0))) c_m)
     (/ (/ (cos (+ x_m x_m)) 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 * x_m) * s_m;
	double tmp;
	if (x_m <= 7.4e-81) {
		tmp = (1.0 / (c_m * pow((s_m * x_m), 2.0))) / c_m;
	} else {
		tmp = (cos((x_m + x_m)) / 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 * x_m) * s_m
    if (x_m <= 7.4d-81) then
        tmp = (1.0d0 / (c_m * ((s_m * x_m) ** 2.0d0))) / c_m
    else
        tmp = (cos((x_m + x_m)) / 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 * x_m) * s_m;
	double tmp;
	if (x_m <= 7.4e-81) {
		tmp = (1.0 / (c_m * Math.pow((s_m * x_m), 2.0))) / c_m;
	} else {
		tmp = (Math.cos((x_m + x_m)) / 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 * x_m) * s_m
	tmp = 0
	if x_m <= 7.4e-81:
		tmp = (1.0 / (c_m * math.pow((s_m * x_m), 2.0))) / c_m
	else:
		tmp = (math.cos((x_m + x_m)) / 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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (x_m <= 7.4e-81)
		tmp = Float64(Float64(1.0 / Float64(c_m * (Float64(s_m * x_m) ^ 2.0))) / c_m);
	else
		tmp = Float64(Float64(cos(Float64(x_m + x_m)) / 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 * x_m) * s_m;
	tmp = 0.0;
	if (x_m <= 7.4e-81)
		tmp = (1.0 / (c_m * ((s_m * x_m) ^ 2.0))) / c_m;
	else
		tmp = (cos((x_m + x_m)) / 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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 7.4e-81], N[(N[(1.0 / N[(c$95$m * N[Power[N[(s$95$m * x$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.39999999999999971e-81

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 78.6%

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

   if 7.39999999999999971e-81 < x

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 97.0

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

   8. Applied rewrites 97.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{c \cdot {\left(s \cdot x\right)}^{2}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{{\left(s\_m \cdot x\_m\right)}^{-2}}{c\_m}}{c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 x_m) s_m)))
   (if (<= x_m 1.6e-99)
     (/ (/ (pow (* s_m x_m) -2.0) c_m) c_m)
     (/ (cos (+ x_m x_m)) (* 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 * x_m) * s_m;
	double tmp;
	if (x_m <= 1.6e-99) {
		tmp = (pow((s_m * x_m), -2.0) / c_m) / c_m;
	} else {
		tmp = cos((x_m + x_m)) / (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 * x_m) * s_m
    if (x_m <= 1.6d-99) then
        tmp = (((s_m * x_m) ** (-2.0d0)) / c_m) / c_m
    else
        tmp = cos((x_m + x_m)) / (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 * x_m) * s_m;
	double tmp;
	if (x_m <= 1.6e-99) {
		tmp = (Math.pow((s_m * x_m), -2.0) / c_m) / c_m;
	} else {
		tmp = Math.cos((x_m + x_m)) / (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 * x_m) * s_m
	tmp = 0
	if x_m <= 1.6e-99:
		tmp = (math.pow((s_m * x_m), -2.0) / c_m) / c_m
	else:
		tmp = math.cos((x_m + x_m)) / (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(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (x_m <= 1.6e-99)
		tmp = Float64(Float64((Float64(s_m * x_m) ^ -2.0) / c_m) / c_m);
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / 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 * x_m) * s_m;
	tmp = 0.0;
	if (x_m <= 1.6e-99)
		tmp = (((s_m * x_m) ^ -2.0) / c_m) / c_m;
	else
		tmp = cos((x_m + x_m)) / (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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.6e-99], N[(N[(N[Power[N[(s$95$m * x$95$m), $MachinePrecision], -2.0], $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e-99

   1. Initial program 65.5%

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

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   7. Applied rewrites 78.2%

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

   if 1.6e-99 < x

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

Alternative 7: 79.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \mathbf{if}\;c\_m \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s\_m \cdot x\_m\right) \cdot c\_m}}{\left(c\_m \cdot x\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

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 (<= c_m 2e-74)
     (/ 1.0 (* t_0 t_0))
     (/ (/ 1.0 (* (* s_m x_m) c_m)) (* (* c_m x_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 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 2e-74) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = (1.0 / ((s_m * x_m) * c_m)) / ((c_m * x_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) :: tmp
    t_0 = (c_m * s_m) * x_m
    if (c_m <= 2d-74) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = (1.0d0 / ((s_m * x_m) * c_m)) / ((c_m * x_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 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 2e-74) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = (1.0 / ((s_m * x_m) * c_m)) / ((c_m * x_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 = (c_m * s_m) * x_m
	tmp = 0
	if c_m <= 2e-74:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = (1.0 / ((s_m * x_m) * c_m)) / ((c_m * x_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 = Float64(Float64(c_m * s_m) * x_m)
	tmp = 0.0
	if (c_m <= 2e-74)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(s_m * x_m) * c_m)) / Float64(Float64(c_m * x_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 = (c_m * s_m) * x_m;
	tmp = 0.0;
	if (c_m <= 2e-74)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = (1.0 / ((s_m * x_m) * c_m)) / ((c_m * x_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[(N[(c$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[c$95$m, 2e-74], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.99999999999999992e-74

   1. Initial program 68.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.6%

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

   10. Applied rewrites 70.8%

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

   if 1.99999999999999992e-74 < c

   1. Initial program 58.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.1

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

   9. Applied rewrites 81.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 7.0× speedup

Math

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

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 c_m) (/ -1.0 (* (* (* s_m x_m) (* s_m x_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 / c_m) * (-1.0 / (((s_m * x_m) * (s_m * x_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) / c_m) * ((-1.0d0) / (((s_m * x_m) * (s_m * x_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 / c_m) * (-1.0 / (((s_m * x_m) * (s_m * x_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 / c_m) * (-1.0 / (((s_m * x_m) * (s_m * x_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(Float64(-1.0 / c_m) * Float64(-1.0 / Float64(Float64(Float64(s_m * x_m) * Float64(s_m * x_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 / c_m) * (-1.0 / (((s_m * x_m) * (s_m * x_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[(N[(-1.0 / c$95$m), $MachinePrecision] * N[(-1.0 / N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. 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. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 60.0

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

5. Applied rewrites 60.0%

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

7. Applied rewrites 65.2%

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

9. Applied rewrites 71.6%

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

11. Applied rewrites 71.6%

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

12. Final simplification 71.6%

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

Alternative 9: 79.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(c\_m \cdot s\_m\right) \cdot x\_m\\ \mathbf{if}\;c\_m \leq 5.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 x_m) s_m)) (t_1 (* (* c_m s_m) x_m)))
   (if (<= c_m 5.1e-47) (/ 1.0 (* t_1 t_1)) (/ 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 * x_m) * s_m;
	double t_1 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 5.1e-47) {
		tmp = 1.0 / (t_1 * t_1);
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (c_m * x_m) * s_m
    t_1 = (c_m * s_m) * x_m
    if (c_m <= 5.1d-47) then
        tmp = 1.0d0 / (t_1 * t_1)
    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 * x_m) * s_m;
	double t_1 = (c_m * s_m) * x_m;
	double tmp;
	if (c_m <= 5.1e-47) {
		tmp = 1.0 / (t_1 * t_1);
	} 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 * x_m) * s_m
	t_1 = (c_m * s_m) * x_m
	tmp = 0
	if c_m <= 5.1e-47:
		tmp = 1.0 / (t_1 * t_1)
	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(Float64(c_m * x_m) * s_m)
	t_1 = Float64(Float64(c_m * s_m) * x_m)
	tmp = 0.0
	if (c_m <= 5.1e-47)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	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 * x_m) * s_m;
	t_1 = (c_m * s_m) * x_m;
	tmp = 0.0;
	if (c_m <= 5.1e-47)
		tmp = 1.0 / (t_1 * t_1);
	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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[c$95$m, 5.1e-47], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 5.1e-47

   1. Initial program 68.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.2%

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

   10. Applied rewrites 70.4%

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

   if 5.1e-47 < c

   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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 87.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.4% accurate, 9.0× speedup

Math

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

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

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unswap-sqr N/A

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

   7. unpow2 N/A

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

   8. unswap-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 95.5

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

5. Applied rewrites 95.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 76.5%

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

10. Applied rewrites 66.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 73.9%

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

14. Final simplification 73.9%

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

Alternative 11: 75.1% accurate, 9.0× speedup

Math

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

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 (* (* (* (* (* c_m x_m) s_m) c_m) x_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) {
	return 1.0 / (((((c_m * x_m) * s_m) * c_m) * x_m) * s_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 / (((((c_m * x_m) * s_m) * c_m) * x_m) * s_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 / (((((c_m * x_m) * s_m) * c_m) * x_m) * s_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 / (((((c_m * x_m) * s_m) * c_m) * x_m) * s_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(Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * c_m) * x_m) * s_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 / (((((c_m * x_m) * s_m) * c_m) * x_m) * s_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[(N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unswap-sqr N/A

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

   7. unpow2 N/A

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

   8. unswap-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 95.5

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

5. Applied rewrites 95.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 76.5%

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

10. Applied rewrites 66.0%

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

12. Applied rewrites 73.2%

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

13. Final simplification 73.2%

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))