mixedcos

Percentage Accurate: 67.3% → 89.6%

Time: 9.8s

Alternatives: 9

Speedup: 2.4×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.3% 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: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;c \leq 7.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{t\_0}{c \cdot \left(x \cdot \left(s\_m \cdot \left(s\_m \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{c}^{2} \cdot \left(x \cdot \left(s\_m \cdot \left(x \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))))
   (if (<= c 7.5e-53)
     (/ t_0 (* c (* x (* s_m (* s_m (* c x))))))
     (/ t_0 (* (pow c 2.0) (* x (* s_m (* x s_m))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = cos((2.0 * x));
	double tmp;
	if (c <= 7.5e-53) {
		tmp = t_0 / (c * (x * (s_m * (s_m * (c * x)))));
	} else {
		tmp = t_0 / (pow(c, 2.0) * (x * (s_m * (x * s_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((2.0d0 * x))
    if (c <= 7.5d-53) then
        tmp = t_0 / (c * (x * (s_m * (s_m * (c * x)))))
    else
        tmp = t_0 / ((c ** 2.0d0) * (x * (s_m * (x * s_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = Math.cos((2.0 * x));
	double tmp;
	if (c <= 7.5e-53) {
		tmp = t_0 / (c * (x * (s_m * (s_m * (c * x)))));
	} else {
		tmp = t_0 / (Math.pow(c, 2.0) * (x * (s_m * (x * s_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = math.cos((2.0 * x))
	tmp = 0
	if c <= 7.5e-53:
		tmp = t_0 / (c * (x * (s_m * (s_m * (c * x)))))
	else:
		tmp = t_0 / (math.pow(c, 2.0) * (x * (s_m * (x * s_m))))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (c <= 7.5e-53)
		tmp = Float64(t_0 / Float64(c * Float64(x * Float64(s_m * Float64(s_m * Float64(c * x))))));
	else
		tmp = Float64(t_0 / Float64((c ^ 2.0) * Float64(x * Float64(s_m * Float64(x * s_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = cos((2.0 * x));
	tmp = 0.0;
	if (c <= 7.5e-53)
		tmp = t_0 / (c * (x * (s_m * (s_m * (c * x)))));
	else
		tmp = t_0 / ((c ^ 2.0) * (x * (s_m * (x * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, 7.5e-53], N[(t$95$0 / N[(c * N[(x * N[(s$95$m * N[(s$95$m * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(s$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 7.5000000000000001e-53

   1. Initial program 73.9%

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

   3. Taylor expanded in c 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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.3

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

   5. Simplified 79.3%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.9

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

   8. Simplified 80.9%

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

   9. Taylor expanded in s around 0

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 85.0

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

   11. Simplified 85.0%

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

   12. Taylor expanded in c around 0

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 92.0

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

   14. Simplified 92.0%

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

   if 7.5000000000000001e-53 < c

   1. Initial program 73.1%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 79.3

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

   5. Simplified 79.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 7.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(s \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s\_m \cdot \left(c \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s_m 2.0))))) -1e-119)
   (/ -2.0 (* c (* c (* s_m s_m))))
   (/ 1.0 (* c (* s_m (* c (* s_m (* x x))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s_m, 2.0))))) <= -1e-119) {
		tmp = -2.0 / (c * (c * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= (-1d-119)) then
        tmp = (-2.0d0) / (c * (c * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c * (s_m * (c * (s_m * (x * x)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= -1e-119) {
		tmp = -2.0 / (c * (c * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= -1e-119:
		tmp = -2.0 / (c * (c * (s_m * s_m)))
	else:
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= -1e-119)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(s_m * Float64(c * Float64(s_m * Float64(x * x))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= -1e-119)
		tmp = -2.0 / (c * (c * (s_m * s_m)));
	else
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-119], N[(-2.0 / N[(c * N[(c * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(s$95$m * N[(c * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.00000000000000001e-119

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 33.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.4

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

   8. Simplified 46.4%

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

   if -1.00000000000000001e-119 < (/.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 71.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.7

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

   8. Simplified 75.7%

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

   9. Taylor expanded in s around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 76.5

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

   11. Simplified 76.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s_m 2.0))))) -1e-119)
   (/ -2.0 (* c (* c (* s_m s_m))))
   (/ 1.0 (* c (* c (* s_m (* s_m (* x x))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s_m, 2.0))))) <= -1e-119) {
		tmp = -2.0 / (c * (c * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c * (c * (s_m * (s_m * (x * x)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= (-1d-119)) then
        tmp = (-2.0d0) / (c * (c * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c * (c * (s_m * (s_m * (x * x)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= -1e-119) {
		tmp = -2.0 / (c * (c * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c * (c * (s_m * (s_m * (x * x)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= -1e-119:
		tmp = -2.0 / (c * (c * (s_m * s_m)))
	else:
		tmp = 1.0 / (c * (c * (s_m * (s_m * (x * x)))))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= -1e-119)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(s_m * Float64(s_m * Float64(x * x))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= -1e-119)
		tmp = -2.0 / (c * (c * (s_m * s_m)));
	else
		tmp = 1.0 / (c * (c * (s_m * (s_m * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-119], N[(-2.0 / N[(c * N[(c * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(c * N[(s$95$m * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.00000000000000001e-119

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 33.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.4

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

   8. Simplified 46.4%

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

   if -1.00000000000000001e-119 < (/.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 71.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-119}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := c \cdot \left(c \cdot \left(s\_m \cdot s\_m\right)\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s\_m}^{2}\right)\right)} \leq 0:\\ \;\;\;\;\frac{-2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.6666666666666666 \cdot \frac{x}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* c (* c (* s_m s_m)))))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s_m 2.0))))) 0.0)
     (/ -2.0 t_0)
     (* x (* 0.6666666666666666 (/ x t_0))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = c * (c * (s_m * s_m));
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s_m, 2.0))))) <= 0.0) {
		tmp = -2.0 / t_0;
	} else {
		tmp = x * (0.6666666666666666 * (x / t_0));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (c * (s_m * s_m))
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s_m ** 2.0d0))))) <= 0.0d0) then
        tmp = (-2.0d0) / t_0
    else
        tmp = x * (0.6666666666666666d0 * (x / t_0))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = c * (c * (s_m * s_m));
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s_m, 2.0))))) <= 0.0) {
		tmp = -2.0 / t_0;
	} else {
		tmp = x * (0.6666666666666666 * (x / t_0));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = c * (c * (s_m * s_m))
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s_m, 2.0))))) <= 0.0:
		tmp = -2.0 / t_0
	else:
		tmp = x * (0.6666666666666666 * (x / t_0))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(c * Float64(c * Float64(s_m * s_m)))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s_m ^ 2.0))))) <= 0.0)
		tmp = Float64(-2.0 / t_0);
	else
		tmp = Float64(x * Float64(0.6666666666666666 * Float64(x / t_0)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = c * (c * (s_m * s_m));
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s_m ^ 2.0))))) <= 0.0)
		tmp = -2.0 / t_0;
	else
		tmp = x * (0.6666666666666666 * (x / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(c * N[(c * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-2.0 / t$95$0), $MachinePrecision], N[(x * N[(0.6666666666666666 * N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -0.0

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

   5. Simplified 41.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 54.7

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

   8. Simplified 54.7%

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

   if -0.0 < (/.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 67.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \left(\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]

   5. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{s \cdot \left(\left(c \cdot c\right) \cdot s\right)} \cdot \mathsf{fma}\left(x, x \cdot -2, \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.6666666666666666, 1\right)\right)}{x \cdot x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{2}{3} \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{3} \cdot {x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{3}}{{c}^{2} \cdot {s}^{2}} \cdot {x}^{2}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{3} \cdot 1}}{{c}^{2} \cdot {s}^{2}} \cdot {x}^{2} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{2}{3} \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right)} \cdot {x}^{2} \]

      5. unpow2 N/A

         \[\leadsto \left(\frac{2}{3} \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{2}{3} \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) \cdot x\right) \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{2}{3} \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) \cdot x\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{2}{3} \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}\right) \cdot x\right)} \]

      9. associate-*r/ N/A

         \[\leadsto x \cdot \left(\color{blue}{\frac{\frac{2}{3} \cdot 1}{{c}^{2} \cdot {s}^{2}}} \cdot x\right) \]

      10. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{\color{blue}{\frac{2}{3}}}{{c}^{2} \cdot {s}^{2}} \cdot x\right) \]

      11. associate-*l/ N/A

          \[\leadsto x \cdot \color{blue}{\frac{\frac{2}{3} \cdot x}{{c}^{2} \cdot {s}^{2}}} \]

      12. associate-/l* N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{x}{{c}^{2} \cdot {s}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{x}{{c}^{2} \cdot {s}^{2}}\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto x \cdot \left(\frac{2}{3} \cdot \color{blue}{\frac{x}{{c}^{2} \cdot {s}^{2}}}\right) \]

      15. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{2}{3} \cdot \frac{x}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}}\right) \]

      16. associate-*l* N/A

          \[\leadsto x \cdot \left(\frac{2}{3} \cdot \frac{x}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}}\right) \]

      17. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{2}{3} \cdot \frac{x}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{2}{3} \cdot \frac{x}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}}\right) \]

      19. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{2}{3} \cdot \frac{x}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)}\right) \]

      20. lower-*.f64 48.5

          \[\leadsto x \cdot \left(0.6666666666666666 \cdot \frac{x}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)}\right) \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \frac{x}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 0:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.6666666666666666 \cdot \frac{x}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{c \cdot \left(s\_m \cdot \left(c \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+171}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s\_m \cdot \left(s\_m \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= x 2.2e-12)
   (/ 1.0 (* c (* s_m (* c (* s_m (* x x))))))
   (if (<= x 2.1e+171)
     (/ (cos (* 2.0 x)) (* c (* s_m (* s_m (* c (* x x))))))
     (/ 1.0 (* c (* x (* c (* x (* s_m s_m)))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.2e-12) {
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))));
	} else if (x <= 2.1e+171) {
		tmp = cos((2.0 * x)) / (c * (s_m * (s_m * (c * (x * x)))));
	} else {
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x <= 2.2d-12) then
        tmp = 1.0d0 / (c * (s_m * (c * (s_m * (x * x)))))
    else if (x <= 2.1d+171) then
        tmp = cos((2.0d0 * x)) / (c * (s_m * (s_m * (c * (x * x)))))
    else
        tmp = 1.0d0 / (c * (x * (c * (x * (s_m * s_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.2e-12) {
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))));
	} else if (x <= 2.1e+171) {
		tmp = Math.cos((2.0 * x)) / (c * (s_m * (s_m * (c * (x * x)))));
	} else {
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if x <= 2.2e-12:
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))))
	elif x <= 2.1e+171:
		tmp = math.cos((2.0 * x)) / (c * (s_m * (s_m * (c * (x * x)))))
	else:
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 2.2e-12)
		tmp = Float64(1.0 / Float64(c * Float64(s_m * Float64(c * Float64(s_m * Float64(x * x))))));
	elseif (x <= 2.1e+171)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(c * Float64(s_m * Float64(s_m * Float64(c * Float64(x * x))))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(x * Float64(c * Float64(x * Float64(s_m * s_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	tmp = 0.0;
	if (x <= 2.2e-12)
		tmp = 1.0 / (c * (s_m * (c * (s_m * (x * x)))));
	elseif (x <= 2.1e+171)
		tmp = cos((2.0 * x)) / (c * (s_m * (s_m * (c * (x * x)))));
	else
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[x, 2.2e-12], N[(1.0 / N[(c * N[(s$95$m * N[(c * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e+171], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(c * N[(s$95$m * N[(s$95$m * N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(x * N[(c * N[(x * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.19999999999999992e-12

   1. Initial program 73.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 74.1

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

   5. Simplified 74.1%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 74.2

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

   8. Simplified 74.2%

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

   9. Taylor expanded in s around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 75.2

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

   11. Simplified 75.2%

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

   if 2.19999999999999992e-12 < x < 2.1000000000000001e171

   1. Initial program 78.3%

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

   3. Taylor expanded in c 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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 83.8

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

   5. Simplified 83.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 88.4

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

   8. Simplified 88.4%

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

   if 2.1000000000000001e171 < x

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.1

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

   8. Simplified 69.1%

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

   9. Taylor expanded in s around 0

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 73.4

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

   11. Simplified 73.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+171}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+171}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(c \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= x 2.1e+171)
   (/ (cos (* 2.0 x)) (* c (* c (* s_m (* s_m (* x x))))))
   (/ 1.0 (* c (* x (* c (* x (* s_m s_m))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.1e+171) {
		tmp = cos((2.0 * x)) / (c * (c * (s_m * (s_m * (x * x)))));
	} else {
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x <= 2.1d+171) then
        tmp = cos((2.0d0 * x)) / (c * (c * (s_m * (s_m * (x * x)))))
    else
        tmp = 1.0d0 / (c * (x * (c * (x * (s_m * s_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.1e+171) {
		tmp = Math.cos((2.0 * x)) / (c * (c * (s_m * (s_m * (x * x)))));
	} else {
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if x <= 2.1e+171:
		tmp = math.cos((2.0 * x)) / (c * (c * (s_m * (s_m * (x * x)))))
	else:
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 2.1e+171)
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(c * Float64(c * Float64(s_m * Float64(s_m * Float64(x * x))))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(x * Float64(c * Float64(x * Float64(s_m * s_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	tmp = 0.0;
	if (x <= 2.1e+171)
		tmp = cos((2.0 * x)) / (c * (c * (s_m * (s_m * (x * x)))));
	else
		tmp = 1.0 / (c * (x * (c * (x * (s_m * s_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[x, 2.1e+171], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(c * N[(c * N[(s$95$m * N[(s$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(x * N[(c * N[(x * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1000000000000001e171

   1. Initial program 74.6%

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

   3. Taylor expanded in c 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. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 79.4

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

   5. Simplified 79.4%

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

   if 2.1000000000000001e171 < x

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.1

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

   8. Simplified 69.1%

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

   9. Taylor expanded in s around 0

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 73.4

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

   11. Simplified 73.4%

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

Alternative 7: 89.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in c 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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 78.6

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

5. Simplified 78.6%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 80.1

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

8. Simplified 80.1%

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

9. Taylor expanded in s around 0

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

    1. associate-*r* N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 84.3

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

11. Simplified 84.3%

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

12. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-*l* N/A

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

    10. lower-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 91.3

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

14. Simplified 91.3%

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

15. Final simplification 91.3%

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

Alternative 8: 85.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in c 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. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 78.6

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

5. Simplified 78.6%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 80.1

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

8. Simplified 80.1%

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

9. Taylor expanded in s around 0

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

    1. associate-*r* N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. associate-*l* N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 84.3

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

11. Simplified 84.3%

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

12. Taylor expanded in x around 0

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

    1. unpow2 N/A

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

    2. associate-*l* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 89.5

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

14. Simplified 89.5%

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

15. Final simplification 89.5%

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

Alternative 9: 29.5% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

5. Simplified 50.1%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 26.2

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

8. Simplified 26.2%

   \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))