mixedcos

Percentage Accurate: 66.8% → 97.4%

Time: 9.9s

Alternatives: 11

Speedup: 9.0×

• 31.7% 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: 66.8% 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: 97.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(c$95$m * 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 69.1%

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

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 74.7

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. lift-pow.f64 N/A

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

   19. pow2 N/A

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

   20. pow-prod-down N/A

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

   21. lower-pow.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 86.8

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

4. Applied rewrites 86.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 97.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2 N/A

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

   4. lift-+.f64 97.5

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

8. Applied rewrites 97.5%

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

9. Final simplification 97.5%

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

Alternative 2: 81.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s\_m}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(t\_0 \cdot s\_m\right) \cdot c\_m\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{-2}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* c_m x) s_m)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (* (* (pow s_m 2.0) x) x) (pow c_m 2.0)))
        -4e-33)
     (/ (fma -2.0 (* x x) 1.0) (* (* (* t_0 s_m) c_m) x))
     (pow t_0 -2.0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = (c_m * x) * s_m;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s_m, 2.0) * x) * x) * pow(c_m, 2.0))) <= -4e-33) {
		tmp = fma(-2.0, (x * x), 1.0) / (((t_0 * s_m) * c_m) * x);
	} else {
		tmp = pow(t_0, -2.0);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(Float64(c_m * x) * s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -4e-33)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(t_0 * s_m) * c_m) * x));
	else
		tmp = t_0 ^ -2.0;
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(c$95$m * x), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-33], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(t$95$0 * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[Power[t$95$0, -2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.0000000000000002e-33

   1. Initial program 73.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

   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 0.7%

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

   10. Applied rewrites 0.7%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 55.3

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

   13. Applied rewrites 55.3%

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

   if -4.0000000000000002e-33 < (/.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

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 68.1

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

   5. Applied rewrites 68.1%

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

   7. Applied rewrites 84.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot s\right) \cdot c\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(c \cdot x\right) \cdot s\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\right) \cdot s\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s\_m}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(t\_0 \cdot s\_m\right) \cdot c\_m\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* c_m x) s_m)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (* (* (pow s_m 2.0) x) x) (pow c_m 2.0)))
        -4e-33)
     (/ (fma -2.0 (* x x) 1.0) (* (* (* t_0 s_m) c_m) x))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = (c_m * x) * s_m;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s_m, 2.0) * x) * x) * pow(c_m, 2.0))) <= -4e-33) {
		tmp = fma(-2.0, (x * x), 1.0) / (((t_0 * s_m) * c_m) * x);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(Float64(c_m * x) * s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -4e-33)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(t_0 * s_m) * c_m) * x));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(c$95$m * x), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-33], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(t$95$0 * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.0000000000000002e-33

   1. Initial program 73.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

   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 0.7%

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

   10. Applied rewrites 0.7%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 55.3

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

   13. Applied rewrites 55.3%

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

   if -4.0000000000000002e-33 < (/.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. 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. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 74.5

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. pow-prod-down N/A

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

      21. lower-pow.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-/l/ N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 97.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot s\right) \cdot c\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s\_m}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq 10^{+298}:\\ \;\;\;\;\frac{1}{\left(\left(s\_m \cdot x\right) \cdot \left(s\_m \cdot x\right)\right) \cdot \left(c\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(s\_m \cdot x\right) \cdot c\_m\right) \cdot s\_m\right) \cdot c\_m\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x)) (* (* (* (pow s_m 2.0) x) x) (pow c_m 2.0)))
      1e+298)
   (/ 1.0 (* (* (* s_m x) (* s_m x)) (* c_m c_m)))
   (/ 1.0 (* (* (* (* (* s_m x) c_m) s_m) c_m) x))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s_m, 2.0) * x) * x) * pow(c_m, 2.0))) <= 1e+298) {
		tmp = 1.0 / (((s_m * x) * (s_m * x)) * (c_m * c_m));
	} else {
		tmp = 1.0 / (((((s_m * x) * c_m) * s_m) * c_m) * x);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((((s_m ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= 1d+298) then
        tmp = 1.0d0 / (((s_m * x) * (s_m * x)) * (c_m * c_m))
    else
        tmp = 1.0d0 / (((((s_m * x) * c_m) * s_m) * c_m) * x)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s_m, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= 1e+298) {
		tmp = 1.0 / (((s_m * x) * (s_m * x)) * (c_m * c_m));
	} else {
		tmp = 1.0 / (((((s_m * x) * c_m) * s_m) * c_m) * x);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s_m, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= 1e+298:
		tmp = 1.0 / (((s_m * x) * (s_m * x)) * (c_m * c_m))
	else:
		tmp = 1.0 / (((((s_m * x) * c_m) * s_m) * c_m) * x)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= 1e+298)
		tmp = Float64(1.0 / Float64(Float64(Float64(s_m * x) * Float64(s_m * x)) * Float64(c_m * c_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(s_m * x) * c_m) * s_m) * c_m) * x));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= 1e+298)
		tmp = 1.0 / (((s_m * x) * (s_m * x)) * (c_m * c_m));
	else
		tmp = 1.0 / (((((s_m * x) * c_m) * s_m) * c_m) * x);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+298], N[(1.0 / N[(N[(N[(s$95$m * x), $MachinePrecision] * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(s$95$m * x), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x), $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))) < 9.9999999999999996e297

   1. Initial program 79.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   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 68.1%

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

   10. Applied rewrites 65.4%

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

   if 9.9999999999999996e297 < (/.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 56.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

   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 80.3%

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

   10. Applied rewrites 76.7%

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

   12. Applied rewrites 78.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq 10^{+298}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot s\right) \cdot c\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;{s\_m}^{2} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\right)\right) \cdot \left(c\_m \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s\_m \cdot x\right)}^{-2}}{c\_m}}{c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<= (pow s_m 2.0) 5e+307)
   (/ (cos (+ x x)) (* (* (* s_m s_m) (* c_m x)) (* c_m x)))
   (/ (/ (pow (* s_m x) -2.0) c_m) c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if (pow(s_m, 2.0) <= 5e+307) {
		tmp = cos((x + x)) / (((s_m * s_m) * (c_m * x)) * (c_m * x));
	} else {
		tmp = (pow((s_m * x), -2.0) / c_m) / c_m;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((s_m ** 2.0d0) <= 5d+307) then
        tmp = cos((x + x)) / (((s_m * s_m) * (c_m * x)) * (c_m * x))
    else
        tmp = (((s_m * x) ** (-2.0d0)) / c_m) / c_m
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if (Math.pow(s_m, 2.0) <= 5e+307) {
		tmp = Math.cos((x + x)) / (((s_m * s_m) * (c_m * x)) * (c_m * x));
	} else {
		tmp = (Math.pow((s_m * x), -2.0) / c_m) / c_m;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if math.pow(s_m, 2.0) <= 5e+307:
		tmp = math.cos((x + x)) / (((s_m * s_m) * (c_m * x)) * (c_m * x))
	else:
		tmp = (math.pow((s_m * x), -2.0) / c_m) / c_m
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if ((s_m ^ 2.0) <= 5e+307)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x)) * Float64(c_m * x)));
	else
		tmp = Float64(Float64((Float64(s_m * x) ^ -2.0) / c_m) / c_m);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 5e+307)
		tmp = cos((x + x)) / (((s_m * s_m) * (c_m * x)) * (c_m * x));
	else
		tmp = (((s_m * x) ^ -2.0) / c_m) / c_m;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 5e+307], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(s$95$m * x), $MachinePrecision], -2.0], $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 5e307

   1. Initial program 75.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 98.7

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

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   6. 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 98.7

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 88.6%

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

   if 5e307 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 47.7%

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

   3. Taylor expanded in x around 0

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

      1. 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 50.0

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

   5. Applied rewrites 50.0%

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

   7. Applied rewrites 78.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot s\right) \cdot \left(c \cdot x\right)\right) \cdot \left(c \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(s \cdot x\right)}^{-2}}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 2.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{{\left(s\_m \cdot x\right)}^{-2}}{c\_m}}{c\_m}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{t\_0}{\left(x \cdot x\right) \cdot \left(\left(c\_m \cdot s\_m\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\right)\right) \cdot \left(c\_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= x 2.8e-63)
     (/ (/ (pow (* s_m x) -2.0) c_m) c_m)
     (if (<= x 1.1e+88)
       (/ t_0 (* (* x x) (* (* c_m s_m) (* c_m s_m))))
       (/ t_0 (* (* (* s_m s_m) (* c_m x)) (* c_m x)))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x + x));
	double tmp;
	if (x <= 2.8e-63) {
		tmp = (pow((s_m * x), -2.0) / c_m) / c_m;
	} else if (x <= 1.1e+88) {
		tmp = t_0 / ((x * x) * ((c_m * s_m) * (c_m * s_m)));
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x)) * (c_m * x));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (x <= 2.8d-63) then
        tmp = (((s_m * x) ** (-2.0d0)) / c_m) / c_m
    else if (x <= 1.1d+88) then
        tmp = t_0 / ((x * x) * ((c_m * s_m) * (c_m * s_m)))
    else
        tmp = t_0 / (((s_m * s_m) * (c_m * x)) * (c_m * x))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (x <= 2.8e-63) {
		tmp = (Math.pow((s_m * x), -2.0) / c_m) / c_m;
	} else if (x <= 1.1e+88) {
		tmp = t_0 / ((x * x) * ((c_m * s_m) * (c_m * s_m)));
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x)) * (c_m * x));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((x + x))
	tmp = 0
	if x <= 2.8e-63:
		tmp = (math.pow((s_m * x), -2.0) / c_m) / c_m
	elif x <= 1.1e+88:
		tmp = t_0 / ((x * x) * ((c_m * s_m) * (c_m * s_m)))
	else:
		tmp = t_0 / (((s_m * s_m) * (c_m * x)) * (c_m * x))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 2.8e-63)
		tmp = Float64(Float64((Float64(s_m * x) ^ -2.0) / c_m) / c_m);
	elseif (x <= 1.1e+88)
		tmp = Float64(t_0 / Float64(Float64(x * x) * Float64(Float64(c_m * s_m) * Float64(c_m * s_m))));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x)) * Float64(c_m * x)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (x <= 2.8e-63)
		tmp = (((s_m * x) ^ -2.0) / c_m) / c_m;
	elseif (x <= 1.1e+88)
		tmp = t_0 / ((x * x) * ((c_m * s_m) * (c_m * s_m)));
	else
		tmp = t_0 / (((s_m * s_m) * (c_m * x)) * (c_m * x));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.8e-63], N[(N[(N[Power[N[(s$95$m * x), $MachinePrecision], -2.0], $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision], If[LessEqual[x, 1.1e+88], N[(t$95$0 / N[(N[(x * x), $MachinePrecision] * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.8000000000000002e-63

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0

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

      1. 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 61.8

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

   5. Applied rewrites 61.8%

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

   7. Applied rewrites 74.3%

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

   if 2.8000000000000002e-63 < x < 1.10000000000000004e88

   1. Initial program 74.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. *-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 93.4

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

   5. Applied rewrites 93.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   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 93.4

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

   7. Applied rewrites 93.4%

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

   9. Applied rewrites 97.5%

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

   if 1.10000000000000004e88 < x

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   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 98.0

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

   7. Applied rewrites 98.0%

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

   9. Applied rewrites 77.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{{\left(s \cdot x\right)}^{-2}}{c}}{c}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot s\right) \cdot \left(c \cdot x\right)\right) \cdot \left(c \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(c$95$m * x), $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 69.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
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 97.3

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

7. Applied rewrites 97.3%

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

Alternative 8: 77.8% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 74.7

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. lift-pow.f64 N/A

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

   19. pow2 N/A

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

   20. pow-prod-down N/A

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

   21. lower-pow.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 86.8

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

4. Applied rewrites 86.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-/l/ N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 N/A

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 73.7%

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

10. Final simplification 73.7%

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

Alternative 9: 77.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 73.7%

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

Alternative 10: 75.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 73.7%

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

10. Applied rewrites 69.0%

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

12. Applied rewrites 70.3%

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

13. Final simplification 70.3%

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

Alternative 11: 73.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 73.7%

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

10. Applied rewrites 69.0%

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

11. Final simplification 69.0%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))