mixedcos

Percentage Accurate: 66.9% → 98.9%

Time: 9.5s

Alternatives: 8

Speedup: 9.0×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ \mathbf{if}\;x\_m \leq 10^{-46}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* c_m x_m) s_m)))
   (if (<= x_m 1e-46)
     (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))
     (/ (cos (+ x_m x_m)) (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (c_m * x_m) * s_m;
	double tmp;
	if (x_m <= 1e-46) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = cos((x_m + x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c_m * x_m) * s_m
    if (x_m <= 1d-46) then
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    else
        tmp = cos((x_m + x_m)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = (c_m * x_m) * s_m;
	double tmp;
	if (x_m <= 1e-46) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = Math.cos((x_m + x_m)) / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = (c_m * x_m) * s_m
	tmp = 0
	if x_m <= 1e-46:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	else:
		tmp = math.cos((x_m + x_m)) / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(c_m * x_m) * s_m)
	tmp = 0.0
	if (x_m <= 1e-46)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = (c_m * x_m) * s_m;
	tmp = 0.0;
	if (x_m <= 1e-46)
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	else
		tmp = cos((x_m + x_m)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1e-46], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000002e-46

   1. Initial program 70.5%

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

   3. 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 98.1

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

   5. Applied rewrites 98.1%

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

   6. 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 85.7%

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

   10. Applied rewrites 81.1%

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

   12. Applied rewrites 82.2%

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

   if 1.00000000000000002e-46 < x

   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 96.0

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

   5. Applied rewrites 96.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   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 96.0

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

   7. Applied rewrites 96.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-46}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s\_m}}{c\_m}}{s\_m \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
      -4e-112)
   (/ (/ (/ -2.0 s_m) c_m) (* s_m c_m))
   (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -4e-112) {
		tmp = ((-2.0 / s_m) / c_m) / (s_m * c_m);
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((((s_m ** 2.0d0) * x_m) * x_m) * (c_m ** 2.0d0))) <= (-4d-112)) then
        tmp = (((-2.0d0) / s_m) / c_m) / (s_m * c_m)
    else
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (((Math.pow(s_m, 2.0) * x_m) * x_m) * Math.pow(c_m, 2.0))) <= -4e-112) {
		tmp = ((-2.0 / s_m) / c_m) / (s_m * c_m);
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (((math.pow(s_m, 2.0) * x_m) * x_m) * math.pow(c_m, 2.0))) <= -4e-112:
		tmp = ((-2.0 / s_m) / c_m) / (s_m * c_m)
	else:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-112)
		tmp = Float64(Float64(Float64(-2.0 / s_m) / c_m) / Float64(s_m * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -4e-112)
		tmp = ((-2.0 / s_m) / c_m) / (s_m * c_m);
	else
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-112], N[(N[(N[(-2.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $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))) < -3.9999999999999998e-112

   1. Initial program 58.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   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 95.2

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

   10. Applied rewrites 35.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 41.6%

       \[\leadsto \frac{\frac{\frac{-2}{s}}{c}}{\color{blue}{s \cdot c}} \]

   if -3.9999999999999998e-112 < (/.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 71.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.4%

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

   10. Applied rewrites 81.4%

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

   12. Applied rewrites 82.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s}}{c}}{s \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 9.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 9.4e-39)
   (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))
   (/ (cos (+ x_m x_m)) (* (* (* (* (* c_m x_m) s_m) x_m) c_m) s_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 9.4e-39) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = cos((x_m + x_m)) / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 9.4d-39) then
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    else
        tmp = cos((x_m + x_m)) / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 9.4e-39) {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = Math.cos((x_m + x_m)) / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 9.4e-39:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	else:
		tmp = math.cos((x_m + x_m)) / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 9.4e-39)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * x_m) * c_m) * s_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 9.4e-39)
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	else
		tmp = cos((x_m + x_m)) / (((((c_m * x_m) * s_m) * x_m) * c_m) * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 9.4e-39], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.4000000000000005e-39

   1. Initial program 70.1%

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

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

   5. Applied rewrites 98.1%

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

   6. 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 85.8%

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

   10. Applied rewrites 81.0%

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

   12. Applied rewrites 82.0%

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

   if 9.4000000000000005e-39 < x

   1. Initial program 70.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 95.9

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

   5. Applied rewrites 95.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   6. 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 95.9

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

   7. Applied rewrites 95.9%

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

   9. Applied rewrites 85.6%

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

   11. Applied rewrites 93.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot x\right) \cdot c\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(s\_m \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= s_m 2e+153)
   (/ (cos (+ x_m x_m)) (* (* (* (* s_m s_m) x_m) c_m) (* c_m x_m)))
   (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 2e+153) {
		tmp = cos((x_m + x_m)) / ((((s_m * s_m) * x_m) * c_m) * (c_m * x_m));
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (s_m <= 2d+153) then
        tmp = cos((x_m + x_m)) / ((((s_m * s_m) * x_m) * c_m) * (c_m * x_m))
    else
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 2e+153) {
		tmp = Math.cos((x_m + x_m)) / ((((s_m * s_m) * x_m) * c_m) * (c_m * x_m));
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if s_m <= 2e+153:
		tmp = math.cos((x_m + x_m)) / ((((s_m * s_m) * x_m) * c_m) * (c_m * x_m))
	else:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (s_m <= 2e+153)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(s_m * s_m) * x_m) * c_m) * Float64(c_m * x_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (s_m <= 2e+153)
		tmp = cos((x_m + x_m)) / ((((s_m * s_m) * x_m) * c_m) * (c_m * x_m));
	else
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[s$95$m, 2e+153], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 2e153

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 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 84.2%

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

   if 2e153 < s

   1. Initial program 54.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-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 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.3%

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

   10. Applied rewrites 84.0%

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

   12. Applied rewrites 90.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot c\right) \cdot \left(c \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= s_m 5e+188)
   (/ (cos (+ x_m x_m)) (* (* (* (* c_m x_m) s_m) (* s_m x_m)) c_m))
   (/ 1.0 (* (* (* (* s_m x_m) c_m) (* s_m x_m)) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 5e+188) {
		tmp = cos((x_m + x_m)) / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (s_m <= 5d+188) then
        tmp = cos((x_m + x_m)) / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_m)
    else
        tmp = 1.0d0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 5e+188) {
		tmp = Math.cos((x_m + x_m)) / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_m);
	} else {
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if s_m <= 5e+188:
		tmp = math.cos((x_m + x_m)) / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_m)
	else:
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (s_m <= 5e+188)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * Float64(s_m * x_m)) * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s_m * x_m) * c_m) * Float64(s_m * x_m)) * c_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (s_m <= 5e+188)
		tmp = cos((x_m + x_m)) / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_m);
	else
		tmp = 1.0 / ((((s_m * x_m) * c_m) * (s_m * x_m)) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[s$95$m, 5e+188], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 5.0000000000000001e188

   1. Initial program 72.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-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 98.1

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

   5. Applied rewrites 98.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 98.1

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 92.2%

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

   if 5.0000000000000001e188 < s

   1. Initial program 46.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   6. 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 85.5%

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

   10. Applied rewrites 78.2%

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

   12. Applied rewrites 86.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 5 \cdot 10^{+188}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 97.5

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 78.4%

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

10. Applied rewrites 74.8%

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

12. Applied rewrites 75.6%

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

13. Final simplification 75.6%

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

Alternative 7: 76.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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 97.5

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 78.4%

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

10. Applied rewrites 74.8%

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

12. Applied rewrites 72.1%

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

13. Final simplification 72.1%

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

Alternative 8: 65.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.1%

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

3. Taylor expanded in x around 0

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

   1. 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 67.6

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

5. Applied rewrites 67.6%

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

7. Applied rewrites 71.8%

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

9. Applied rewrites 71.8%

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

11. Applied rewrites 68.0%

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

12. Final simplification 68.0%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))