mixedcos

Percentage Accurate: 65.6% → 97.3%

Time: 13.3s

Alternatives: 7

Speedup: 24.1×

• 30.4% 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: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c_m \leq 1.7 \cdot 10^{-267}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(\left(c_m \cdot s_m\right) \cdot \left(x \cdot \left(c_m \cdot s_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s_m}}{c_m} \cdot \frac{t_0}{c_m \cdot \left(x \cdot s_m\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))))
   (if (<= c_m 1.7e-267)
     (/ t_0 (* x (* (* c_m s_m) (* x (* c_m s_m)))))
     (* (/ (/ 1.0 (* x s_m)) c_m) (/ t_0 (* c_m (* x s_m)))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = cos((x * 2.0));
	double tmp;
	if (c_m <= 1.7e-267) {
		tmp = t_0 / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	} else {
		tmp = ((1.0 / (x * s_m)) / c_m) * (t_0 / (c_m * (x * s_m)));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    if (c_m <= 1.7d-267) then
        tmp = t_0 / (x * ((c_m * s_m) * (x * (c_m * s_m))))
    else
        tmp = ((1.0d0 / (x * s_m)) / c_m) * (t_0 / (c_m * (x * s_m)))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = Math.cos((x * 2.0));
	double tmp;
	if (c_m <= 1.7e-267) {
		tmp = t_0 / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	} else {
		tmp = ((1.0 / (x * s_m)) / c_m) * (t_0 / (c_m * (x * s_m)));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = math.cos((x * 2.0))
	tmp = 0
	if c_m <= 1.7e-267:
		tmp = t_0 / (x * ((c_m * s_m) * (x * (c_m * s_m))))
	else:
		tmp = ((1.0 / (x * s_m)) / c_m) * (t_0 / (c_m * (x * s_m)))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (c_m <= 1.7e-267)
		tmp = Float64(t_0 / Float64(x * Float64(Float64(c_m * s_m) * Float64(x * Float64(c_m * s_m)))));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s_m)) / c_m) * Float64(t_0 / Float64(c_m * Float64(x * s_m))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = cos((x * 2.0));
	tmp = 0.0;
	if (c_m <= 1.7e-267)
		tmp = t_0 / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	else
		tmp = ((1.0 / (x * s_m)) / c_m) * (t_0 / (c_m * (x * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 1.7e-267], N[(t$95$0 / N[(x * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(t$95$0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.7000000000000001e-267

   1. Initial program 68.6%

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

      1. *-un-lft-identity 68.6%

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

      2. add-sqr-sqrt 68.5%

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

      3. times-frac 68.5%

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

   3. Applied egg-rr 97.0%

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

      1. *-commutative 97.0%

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

      2. frac-2neg 97.0%

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

      3. frac-2neg 97.0%

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

      4. metadata-eval 97.0%

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

      5. frac-times 96.7%

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

      6. *-commutative 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*r* 96.1%

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

      9. distribute-rgt-neg-in 96.1%

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

      10. *-commutative 96.1%

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

      11. associate-*r* 98.8%

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

      12. distribute-rgt-neg-in 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in x around inf 63.4%

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

      1. *-commutative 63.4%

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

      2. unpow2 63.4%

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

      3. unpow2 63.4%

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

      4. unpow2 63.4%

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

      5. swap-sqr 80.2%

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

      6. swap-sqr 96.7%

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

      7. unpow2 96.7%

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

      8. *-commutative 96.7%

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

      9. associate-*l* 96.7%

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

      10. *-commutative 96.7%

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

   8. Simplified 96.7%

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

      1. unpow2 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*r* 94.1%

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

      4. associate-*r* 96.1%

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

      5. *-commutative 96.1%

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

      6. associate-*r* 93.5%

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

      7. associate-*r* 93.5%

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

      8. *-commutative 93.5%

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

      9. associate-*r* 96.2%

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

      10. *-commutative 96.2%

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

   10. Applied egg-rr 96.2%

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

   if 1.7000000000000001e-267 < c

   1. Initial program 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. add-sqr-sqrt 66.3%

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

      3. times-frac 66.3%

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

   3. Applied egg-rr 97.1%

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

   4. Taylor expanded in c around 0 97.1%

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

      1. associate-/r* 97.1%

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

      2. *-commutative 97.1%

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

      3. *-rgt-identity 97.1%

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

      4. associate-*r/ 97.2%

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

      5. associate-*l/ 97.2%

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

      6. *-lft-identity 97.2%

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

      7. *-commutative 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.7 \cdot 10^{-267}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{c} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 2: 97.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x \cdot s_m\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c_m \leq 1.5 \cdot 10^{-267}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(\left(c_m \cdot s_m\right) \cdot \left(x \cdot \left(c_m \cdot s_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))) (t_1 (cos (* x 2.0))))
   (if (<= c_m 1.5e-267)
     (/ t_1 (* x (* (* c_m s_m) (* x (* c_m s_m)))))
     (* (/ t_1 t_0) (/ 1.0 t_0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (c_m <= 1.5e-267) {
		tmp = t_1 / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	} else {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    t_1 = cos((x * 2.0d0))
    if (c_m <= 1.5d-267) then
        tmp = t_1 / (x * ((c_m * s_m) * (x * (c_m * s_m))))
    else
        tmp = (t_1 / t_0) * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (c_m <= 1.5e-267) {
		tmp = t_1 / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	} else {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if c_m <= 1.5e-267:
		tmp = t_1 / (x * ((c_m * s_m) * (x * (c_m * s_m))))
	else:
		tmp = (t_1 / t_0) * (1.0 / t_0)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (c_m <= 1.5e-267)
		tmp = Float64(t_1 / Float64(x * Float64(Float64(c_m * s_m) * Float64(x * Float64(c_m * s_m)))));
	else
		tmp = Float64(Float64(t_1 / t_0) * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (c_m <= 1.5e-267)
		tmp = t_1 / (x * ((c_m * s_m) * (x * (c_m * s_m))));
	else
		tmp = (t_1 / t_0) * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 1.5e-267], N[(t$95$1 / N[(x * N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.5e-267

   1. Initial program 68.6%

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

      1. *-un-lft-identity 68.6%

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

      2. add-sqr-sqrt 68.5%

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

      3. times-frac 68.5%

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

   3. Applied egg-rr 97.0%

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

      1. *-commutative 97.0%

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

      2. frac-2neg 97.0%

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

      3. frac-2neg 97.0%

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

      4. metadata-eval 97.0%

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

      5. frac-times 96.7%

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

      6. *-commutative 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*r* 96.1%

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

      9. distribute-rgt-neg-in 96.1%

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

      10. *-commutative 96.1%

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

      11. associate-*r* 98.8%

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

      12. distribute-rgt-neg-in 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in x around inf 63.4%

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

      1. *-commutative 63.4%

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

      2. unpow2 63.4%

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

      3. unpow2 63.4%

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

      4. unpow2 63.4%

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

      5. swap-sqr 80.2%

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

      6. swap-sqr 96.7%

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

      7. unpow2 96.7%

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

      8. *-commutative 96.7%

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

      9. associate-*l* 96.7%

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

      10. *-commutative 96.7%

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

   8. Simplified 96.7%

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

      1. unpow2 96.7%

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

      2. *-commutative 96.7%

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

      3. associate-*r* 94.1%

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

      4. associate-*r* 96.1%

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

      5. *-commutative 96.1%

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

      6. associate-*r* 93.5%

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

      7. associate-*r* 93.5%

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

      8. *-commutative 93.5%

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

      9. associate-*r* 96.2%

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

      10. *-commutative 96.2%

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

   10. Applied egg-rr 96.2%

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

   if 1.5e-267 < c

   1. Initial program 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. add-sqr-sqrt 66.3%

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

      3. times-frac 66.3%

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

   3. Applied egg-rr 97.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.5 \cdot 10^{-267}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 3: 87.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s_m}}{c_m} \cdot \frac{1}{c_m \cdot \left(x \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s_m \cdot \left(\left(x \cdot \left(c_m \cdot s_m\right)\right) \cdot \left(x \cdot c_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (if (<= x 1.25e-103)
   (* (/ (/ 1.0 (* x s_m)) c_m) (/ 1.0 (* c_m (* x s_m))))
   (/ (cos (* x 2.0)) (* s_m (* (* x (* c_m s_m)) (* x c_m))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double tmp;
	if (x <= 1.25e-103) {
		tmp = ((1.0 / (x * s_m)) / c_m) * (1.0 / (c_m * (x * s_m)));
	} else {
		tmp = cos((x * 2.0)) / (s_m * ((x * (c_m * s_m)) * (x * c_m)));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x <= 1.25d-103) then
        tmp = ((1.0d0 / (x * s_m)) / c_m) * (1.0d0 / (c_m * (x * s_m)))
    else
        tmp = cos((x * 2.0d0)) / (s_m * ((x * (c_m * s_m)) * (x * c_m)))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double tmp;
	if (x <= 1.25e-103) {
		tmp = ((1.0 / (x * s_m)) / c_m) * (1.0 / (c_m * (x * s_m)));
	} else {
		tmp = Math.cos((x * 2.0)) / (s_m * ((x * (c_m * s_m)) * (x * c_m)));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	tmp = 0
	if x <= 1.25e-103:
		tmp = ((1.0 / (x * s_m)) / c_m) * (1.0 / (c_m * (x * s_m)))
	else:
		tmp = math.cos((x * 2.0)) / (s_m * ((x * (c_m * s_m)) * (x * c_m)))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	tmp = 0.0
	if (x <= 1.25e-103)
		tmp = Float64(Float64(Float64(1.0 / Float64(x * s_m)) / c_m) * Float64(1.0 / Float64(c_m * Float64(x * s_m))));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(s_m * Float64(Float64(x * Float64(c_m * s_m)) * Float64(x * c_m))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	tmp = 0.0;
	if (x <= 1.25e-103)
		tmp = ((1.0 / (x * s_m)) / c_m) * (1.0 / (c_m * (x * s_m)));
	else
		tmp = cos((x * 2.0)) / (s_m * ((x * (c_m * s_m)) * (x * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := If[LessEqual[x, 1.25e-103], N[(N[(N[(1.0 / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(1.0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.24999999999999992e-103

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0 53.8%

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

      1. associate-/r* 53.9%

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

      2. *-commutative 53.9%

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

      3. unpow2 53.9%

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

      4. unpow2 53.9%

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

      5. swap-sqr 69.4%

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

      6. unpow2 69.4%

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

      7. associate-/r* 69.4%

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

      8. unpow2 69.4%

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

      9. unpow2 69.4%

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

      10. swap-sqr 81.5%

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

      11. unpow2 81.5%

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

      12. *-commutative 81.5%

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

   4. Simplified 81.5%

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

      1. pow-flip 81.6%

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

      2. *-commutative 81.6%

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

      3. pow-flip 81.5%

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

      4. unpow2 81.5%

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

      5. associate-/r* 81.6%

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

      6. un-div-inv 81.6%

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

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in c around 0 81.6%

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

      1. associate-/r* 96.6%

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

      2. *-commutative 96.6%

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

      3. *-rgt-identity 96.6%

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

      4. associate-*r/ 96.6%

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

      5. associate-*l/ 96.6%

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

      6. *-lft-identity 96.6%

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

      7. *-commutative 96.6%

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

   9. Simplified 81.7%

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

   if 1.24999999999999992e-103 < x

   1. Initial program 66.3%

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

      1. *-un-lft-identity 66.3%

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

      2. add-sqr-sqrt 66.3%

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

      3. times-frac 66.3%

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

   3. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. frac-2neg 97.8%

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

      3. frac-2neg 97.8%

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

      4. metadata-eval 97.8%

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

      5. frac-times 97.8%

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

      6. *-commutative 97.8%

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

      7. *-commutative 97.8%

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

      8. associate-*r* 96.4%

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

      9. distribute-rgt-neg-in 96.4%

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

      10. *-commutative 96.4%

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

      11. associate-*r* 98.1%

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

      12. distribute-rgt-neg-in 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in x around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. unpow2 62.2%

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

      3. unpow2 62.2%

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

      4. unpow2 62.2%

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

      5. swap-sqr 76.7%

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

      6. swap-sqr 97.8%

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

      7. unpow2 97.8%

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

      8. *-commutative 97.8%

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

      9. associate-*l* 96.0%

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

      10. *-commutative 96.0%

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

   8. Simplified 96.0%

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

      1. unpow2 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*r* 94.3%

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

      4. *-commutative 94.3%

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

      5. associate-*r* 89.7%

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

      6. associate-*r* 91.6%

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

      7. *-commutative 91.6%

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

      8. associate-*r* 90.1%

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

   10. Applied egg-rr 90.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{c} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 4: 96.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(c_m \cdot s_m\right)\\ \frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0} \end{array} \end{array} \]

FPCore

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

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = x * (c_m * s_m);
	return cos((x * 2.0)) / (t_0 * t_0);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = x * (c_m * s_m)
    code = cos((x * 2.0d0)) / (t_0 * t_0)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = x * (c_m * s_m);
	return Math.cos((x * 2.0)) / (t_0 * t_0);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = x * (c_m * s_m)
	return math.cos((x * 2.0)) / (t_0 * t_0)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(x * Float64(c_m * s_m))
	return Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = x * (c_m * s_m);
	tmp = cos((x * 2.0)) / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. *-un-lft-identity 67.6%

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

   2. add-sqr-sqrt 67.5%

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

   3. times-frac 67.5%

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

3. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. frac-2neg 97.0%

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

   3. frac-2neg 97.0%

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

   4. metadata-eval 97.0%

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

   5. frac-times 96.7%

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

   6. *-commutative 96.7%

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

   7. *-commutative 96.7%

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

   8. associate-*r* 96.2%

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

   9. distribute-rgt-neg-in 96.2%

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

   10. *-commutative 96.2%

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

   11. associate-*r* 98.7%

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

   12. distribute-rgt-neg-in 98.7%

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

5. Applied egg-rr 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 75.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

2. Taylor expanded in x around 0 53.4%

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

   1. associate-/r* 53.4%

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

   2. *-commutative 53.4%

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

   3. unpow2 53.4%

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

   4. unpow2 53.4%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 64.9%

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

   8. unpow2 64.9%

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

   9. unpow2 64.9%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

   12. *-commutative 76.5%

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

4. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*r* 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l* 75.3%

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

6. Applied egg-rr 75.3%

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

7. Final simplification 75.3%

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

Alternative 6: 78.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

2. Taylor expanded in x around 0 53.4%

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

   1. associate-/r* 53.4%

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

   2. *-commutative 53.4%

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

   3. unpow2 53.4%

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

   4. unpow2 53.4%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 64.9%

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

   8. unpow2 64.9%

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

   9. unpow2 64.9%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

   12. *-commutative 76.5%

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

4. Simplified 76.5%

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

   1. *-commutative 76.5%

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

   2. unpow2 76.5%

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

6. Applied egg-rr 76.5%

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

7. Final simplification 76.5%

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

Alternative 7: 77.6% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

2. Taylor expanded in x around 0 53.4%

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

   1. associate-/r* 53.4%

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

   2. *-commutative 53.4%

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

   3. unpow2 53.4%

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

   4. unpow2 53.4%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 64.9%

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

   8. unpow2 64.9%

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

   9. unpow2 64.9%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

   12. *-commutative 76.5%

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

4. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*r* 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l* 75.3%

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

6. Applied egg-rr 75.3%

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

8. Applied egg-rr 38.7%

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

10. Applied egg-rr 77.8%

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

11. Final simplification 77.8%

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

Reproduce

herbie shell --seed 2023337 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))