mixedcos

Percentage Accurate: 67.2% → 97.6%

Time: 9.1s

Alternatives: 9

Speedup: 9.0×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \frac{\frac{{\left(\frac{s\_m}{\cos \left(x \cdot 2\right)}\right)}^{-1}}{c \cdot x}}{\left(c \cdot x\right) \cdot s\_m} \end{array} \]

FPCore

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

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	return (pow((s_m / cos((x * 2.0))), -1.0) / (c * x)) / ((c * x) * s_m);
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    code = (((s_m / cos((x * 2.0d0))) ** (-1.0d0)) / (c * x)) / ((c * x) * s_m)
end function

Java

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

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	return (math.pow((s_m / math.cos((x * 2.0))), -1.0) / (c * x)) / ((c * x) * s_m)

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	return Float64(Float64((Float64(s_m / cos(Float64(x * 2.0))) ^ -1.0) / Float64(c * x)) / Float64(Float64(c * x) * s_m))
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	tmp = (((s_m / cos((x * 2.0))) ^ -1.0) / (c * x)) / ((c * x) * s_m);
end

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 69.2

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

   18. lift-pow.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 69.2

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

4. Applied rewrites 69.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. pow2 N/A

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

   20. unpow-prod-down N/A

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

6. Applied rewrites 97.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 97.7

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 97.7

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

8. Applied rewrites 97.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 97.7

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 97.7

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

10. Applied rewrites 97.7%

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

11. Final simplification 97.7%

    \[\leadsto \frac{\frac{{\left(\frac{s}{\cos \left(x \cdot 2\right)}\right)}^{-1}}{c \cdot x}}{\left(c \cdot x\right) \cdot s} \]
12. Add Preprocessing

Alternative 2: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(x \cdot c\right) \cdot s\_m\\ t_1 := \left(x \cdot s\_m\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* x c) s_m)) (t_1 (* (* x s_m) c)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x)))
        -2e-113)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ 1.0 (* t_1 t_1)))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double t_1 = (x * s_m) * c;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -2e-113) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / (t_1 * t_1);
	}
	return tmp;
}

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(x * c) * s_m)
	t_1 = Float64(Float64(x * s_m) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -2e-113)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * s$95$m), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-113], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.99999999999999996e-113

   1. Initial program 58.8%

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

   3. Taylor expanded in x around 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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.7

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

   8. Applied rewrites 48.7%

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

   if -1.99999999999999996e-113 < (/.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 68.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.1%

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

   10. Applied rewrites 79.3%

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

   12. Applied rewrites 85.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(x \cdot c\right) \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s\right) \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(x \cdot s\_m\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot s\_m\right) \cdot x\right) \cdot \left(x \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* x s_m) c)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x)))
        -1e+82)
     (/ 1.0 (* (* (* (* c c) s_m) x) (* x s_m)))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (x * s_m) * c;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -1e+82) {
		tmp = 1.0 / ((((c * c) * s_m) * x) * (x * s_m));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * s_m) * c
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-1d+82)) then
        tmp = 1.0d0 / ((((c * c) * s_m) * x) * (x * s_m))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (x * s_m) * c;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -1e+82) {
		tmp = 1.0 / ((((c * c) * s_m) * x) * (x * s_m));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (x * s_m) * c
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -1e+82:
		tmp = 1.0 / ((((c * c) * s_m) * x) * (x * s_m))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(x * s_m) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -1e+82)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * c) * s_m) * x) * Float64(x * s_m)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (x * s_m) * c;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -1e+82)
		tmp = 1.0 / ((((c * c) * s_m) * x) * (x * s_m));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(x * s$95$m), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e+82], N[(1.0 / N[(N[(N[(N[(c * c), $MachinePrecision] * s$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999996e81

   1. Initial program 55.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 0.5%

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

   10. Applied rewrites 0.5%

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

   12. Applied rewrites 38.8%

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

   if -9.9999999999999996e81 < (/.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 68.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.4%

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

   10. Applied rewrites 78.6%

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

   12. Applied rewrites 85.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot s\right) \cdot x\right) \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s\right) \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 69.2

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

   18. lift-pow.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 69.2

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

4. Applied rewrites 69.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. pow2 N/A

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

   20. unpow-prod-down N/A

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

6. Applied rewrites 97.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 97.7

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 97.7

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

8. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 97.7

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

10. Applied rewrites 97.7%

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

Alternative 5: 97.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 69.2

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

   18. lift-pow.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 69.2

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

4. Applied rewrites 69.2%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. associate-/r* N/A

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

   18. lift-*.f64 N/A

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

   19. pow2 N/A

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

   20. unpow-prod-down N/A

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

6. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot 2\right)}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}} \]
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 97.7

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

8. Applied rewrites 97.7%

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

Alternative 6: 97.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 97.2

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

7. Applied rewrites 97.2%

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

Alternative 7: 78.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.4%

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

10. Applied rewrites 71.3%

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

12. Applied rewrites 77.2%

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

13. Final simplification 77.2%

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

Alternative 8: 78.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.4%

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

10. Applied rewrites 71.3%

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

12. Applied rewrites 77.2%

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

Alternative 9: 76.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.4%

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

10. Applied rewrites 71.3%

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

12. Applied rewrites 75.5%

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

Reproduce

herbie shell --seed 2024307 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))