mixedcos

Percentage Accurate: 66.7% → 97.3%

Time: 22.0s

Alternatives: 12

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% 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

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 6.5 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x \cdot \left(c\_m \cdot s\_m\right)}}{c\_m \cdot s\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{t\_1}{t\_0}\\ \end{array} \end{array} \]

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 6.5e-260)
     (/ (/ (/ t_1 (* x (* c_m s_m))) (* c_m s_m)) x)
     (* (/ 1.0 t_0) (/ t_1 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 <= 6.5e-260) {
		tmp = ((t_1 / (x * (c_m * s_m))) / (c_m * s_m)) / x;
	} else {
		tmp = (1.0 / t_0) * (t_1 / 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 <= 6.5d-260) then
        tmp = ((t_1 / (x * (c_m * s_m))) / (c_m * s_m)) / x
    else
        tmp = (1.0d0 / t_0) * (t_1 / 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 <= 6.5e-260) {
		tmp = ((t_1 / (x * (c_m * s_m))) / (c_m * s_m)) / x;
	} else {
		tmp = (1.0 / t_0) * (t_1 / 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 <= 6.5e-260:
		tmp = ((t_1 / (x * (c_m * s_m))) / (c_m * s_m)) / x
	else:
		tmp = (1.0 / t_0) * (t_1 / 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 <= 6.5e-260)
		tmp = Float64(Float64(Float64(t_1 / Float64(x * Float64(c_m * s_m))) / Float64(c_m * s_m)) / x);
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / 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 <= 6.5e-260)
		tmp = ((t_1 / (x * (c_m * s_m))) / (c_m * s_m)) / x;
	else
		tmp = (1.0 / t_0) * (t_1 / 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, 6.5e-260], N[(N[(N[(t$95$1 / N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 6.50000000000000002e-260

   1. Initial program 66.0%

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

      1. *-un-lft-identity 66.0%

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

      2. add-sqr-sqrt 66.0%

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

      3. times-frac 66.0%

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

   4. Applied egg-rr 97.7%

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

      1. associate-*l/ 97.8%

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

      2. *-un-lft-identity 97.8%

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

      3. *-commutative 97.8%

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

      4. associate-*r* 94.4%

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

      5. associate-/r* 89.8%

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

      6. *-commutative 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in x around inf 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-*r* 91.7%

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

   9. Simplified 91.7%

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

   if 6.50000000000000002e-260 < c

   1. Initial program 60.9%

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

      1. *-un-lft-identity 60.9%

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

      2. add-sqr-sqrt 60.9%

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

      3. times-frac 61.0%

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

   4. Applied egg-rr 98.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 6.5 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;x \leq 1.04 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s\_m}}{\left(x \cdot c\_m\right) \cdot t\_0}\\ \end{array} \end{array} \]

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))))
   (if (<= x 1.04e-7)
     (/ (/ 1.0 t_0) t_0)
     (/ (/ (cos (* x 2.0)) s_m) (* (* x c_m) 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 tmp;
	if (x <= 1.04e-7) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (cos((x * 2.0)) / s_m) / ((x * c_m) * 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) :: tmp
    t_0 = c_m * (x * s_m)
    if (x <= 1.04d-7) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = (cos((x * 2.0d0)) / s_m) / ((x * c_m) * 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 tmp;
	if (x <= 1.04e-7) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = (Math.cos((x * 2.0)) / s_m) / ((x * c_m) * 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)
	tmp = 0
	if x <= 1.04e-7:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = (math.cos((x * 2.0)) / s_m) / ((x * c_m) * 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))
	tmp = 0.0
	if (x <= 1.04e-7)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / s_m) / Float64(Float64(x * c_m) * 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);
	tmp = 0.0;
	if (x <= 1.04e-7)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = (cos((x * 2.0)) / s_m) / ((x * c_m) * 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]}, If[LessEqual[x, 1.04e-7], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s$95$m), $MachinePrecision] / N[(N[(x * c$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.04e-7

   1. Initial program 60.7%

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

   3. Taylor expanded in x around 0 55.1%

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

      1. associate-/r* 55.4%

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

      2. *-commutative 55.4%

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

      3. unpow2 55.4%

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

      4. unpow2 55.4%

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

      5. swap-sqr 68.5%

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

      6. unpow2 68.5%

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

      7. associate-/r* 68.2%

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

      8. unpow2 68.2%

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

      9. unpow2 68.2%

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

      10. swap-sqr 88.9%

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

      11. unpow2 88.9%

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

      12. *-commutative 88.9%

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

   5. Simplified 88.9%

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

      1. pow-flip 89.3%

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

      2. *-commutative 89.3%

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

      3. pow-flip 88.9%

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

      4. unpow2 88.9%

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

      5. associate-/r* 89.3%

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

      6. un-div-inv 89.3%

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

   7. Applied egg-rr 89.3%

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

      1. un-div-inv 89.3%

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

   9. Applied egg-rr 89.3%

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

   if 1.04e-7 < x

   1. Initial program 72.2%

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

      1. associate-/r* 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*r* 61.9%

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

      4. unpow2 61.9%

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

      5. associate-/r* 63.5%

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

      6. add-sqr-sqrt 57.0%

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

      7. sqrt-unprod 45.1%

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

      8. swap-sqr 45.1%

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

      9. metadata-eval 45.1%

         \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      10. metadata-eval 45.1%

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

      11. swap-sqr 45.1%

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

      12. *-commutative 45.1%

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

      13. *-commutative 45.1%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 63.5%

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

      16. add-cube-cbrt 63.4%

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

   4. Applied egg-rr 96.4%

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

      1. /-rgt-identity 96.4%

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

      2. associate-*r* 97.6%

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

      3. associate-/l* 97.6%

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

   6. Applied egg-rr 97.6%

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

   8. Applied egg-rr 91.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.04 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 9.5 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{t\_0}{c\_m \cdot \left(s\_m \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{c\_m}}{c\_m \cdot \left(x \cdot s\_m\right)}}{x \cdot s\_m}\\ \end{array} \end{array} \]

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 9.5e-194)
     (/ (/ t_0 (* c_m (* s_m (* x (* c_m s_m))))) x)
     (/ (/ (/ t_0 c_m) (* c_m (* x s_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 <= 9.5e-194) {
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x;
	} else {
		tmp = ((t_0 / c_m) / (c_m * (x * s_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 <= 9.5d-194) then
        tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x
    else
        tmp = ((t_0 / c_m) / (c_m * (x * s_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 <= 9.5e-194) {
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x;
	} else {
		tmp = ((t_0 / c_m) / (c_m * (x * s_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 <= 9.5e-194:
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x
	else:
		tmp = ((t_0 / c_m) / (c_m * (x * s_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 <= 9.5e-194)
		tmp = Float64(Float64(t_0 / Float64(c_m * Float64(s_m * Float64(x * Float64(c_m * s_m))))) / x);
	else
		tmp = Float64(Float64(Float64(t_0 / c_m) / Float64(c_m * Float64(x * s_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 <= 9.5e-194)
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x;
	else
		tmp = ((t_0 / c_m) / (c_m * (x * s_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, 9.5e-194], N[(N[(t$95$0 / N[(c$95$m * N[(s$95$m * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(t$95$0 / c$95$m), $MachinePrecision] / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 9.50000000000000009e-194

   1. Initial program 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. add-sqr-sqrt 65.2%

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

      3. times-frac 65.2%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-*r* 94.0%

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

      5. associate-/r* 89.7%

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

      6. *-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. *-un-lft-identity 89.7%

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

      2. associate-/l/ 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*r* 89.0%

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

      5. *-commutative 89.0%

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

      6. associate-*r* 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. *-lft-identity 89.0%

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

      2. *-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*r* 90.8%

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

   10. Simplified 90.8%

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

   if 9.50000000000000009e-194 < c

   1. Initial program 61.7%

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

      1. associate-/r* 62.1%

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

      2. *-commutative 62.1%

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

      3. associate-*r* 58.7%

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

      4. unpow2 58.7%

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

      5. associate-/r* 58.3%

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

      6. add-sqr-sqrt 23.6%

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

      7. sqrt-unprod 49.8%

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

      8. swap-sqr 49.8%

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

      9. metadata-eval 49.8%

         \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      10. metadata-eval 49.8%

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

      11. swap-sqr 49.8%

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

      12. *-commutative 49.8%

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

      13. *-commutative 49.8%

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

      14. sqrt-unprod 29.8%

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

      15. add-sqr-sqrt 58.3%

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

      16. add-cube-cbrt 58.2%

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

   4. Applied egg-rr 96.6%

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

      1. /-rgt-identity 96.6%

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

      2. associate-*r* 94.6%

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

      3. associate-/l* 94.6%

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

   6. Applied egg-rr 94.6%

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

   8. Applied egg-rr 95.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 9.5 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{c \cdot \left(x \cdot s\right)}}{x \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 9 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{t\_0}{c\_m \cdot \left(s\_m \cdot \left(x \cdot \left(c\_m \cdot s\_m\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{c\_m \cdot \left(x \cdot s\_m\right)}}{c\_m}}{x \cdot s\_m}\\ \end{array} \end{array} \]

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 9e-194)
     (/ (/ t_0 (* c_m (* s_m (* x (* c_m s_m))))) x)
     (/ (/ (/ t_0 (* c_m (* x s_m))) 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 <= 9e-194) {
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x;
	} else {
		tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 9d-194) then
        tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x
    else
        tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 9e-194) {
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x;
	} else {
		tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 9e-194:
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x
	else:
		tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 9e-194)
		tmp = Float64(Float64(t_0 / Float64(c_m * Float64(s_m * Float64(x * Float64(c_m * s_m))))) / x);
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(c_m * Float64(x * s_m))) / 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 <= 9e-194)
		tmp = (t_0 / (c_m * (s_m * (x * (c_m * s_m))))) / x;
	else
		tmp = ((t_0 / (c_m * (x * s_m))) / 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, 9e-194], N[(N[(t$95$0 / N[(c$95$m * N[(s$95$m * N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(t$95$0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 8.9999999999999997e-194

   1. Initial program 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. add-sqr-sqrt 65.2%

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

      3. times-frac 65.2%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-*r* 94.0%

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

      5. associate-/r* 89.7%

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

      6. *-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

      1. *-un-lft-identity 89.7%

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

      2. associate-/l/ 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*r* 89.0%

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

      5. *-commutative 89.0%

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

      6. associate-*r* 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. *-lft-identity 89.0%

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

      2. *-commutative 89.0%

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

      3. *-commutative 89.0%

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

      4. associate-*r* 90.8%

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

   10. Simplified 90.8%

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

   if 8.9999999999999997e-194 < c

   1. Initial program 61.7%

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

      1. *-un-lft-identity 61.7%

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

      2. add-sqr-sqrt 61.7%

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

      3. times-frac 61.7%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l/ 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. associate-/r* 95.6%

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

      4. *-commutative 95.6%

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

   6. Applied egg-rr 95.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 9 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c}}{x \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 10^{-193}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{x \cdot \left(c\_m \cdot s\_m\right)}}{c\_m \cdot s\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{c\_m \cdot \left(x \cdot s\_m\right)}}{c\_m}}{x \cdot s\_m}\\ \end{array} \end{array} \]

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 1e-193)
     (/ (/ (/ t_0 (* x (* c_m s_m))) (* c_m s_m)) x)
     (/ (/ (/ t_0 (* c_m (* x s_m))) 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 <= 1e-193) {
		tmp = ((t_0 / (x * (c_m * s_m))) / (c_m * s_m)) / x;
	} else {
		tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 1d-193) then
        tmp = ((t_0 / (x * (c_m * s_m))) / (c_m * s_m)) / x
    else
        tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 1e-193) {
		tmp = ((t_0 / (x * (c_m * s_m))) / (c_m * s_m)) / x;
	} else {
		tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 1e-193:
		tmp = ((t_0 / (x * (c_m * s_m))) / (c_m * s_m)) / x
	else:
		tmp = ((t_0 / (c_m * (x * s_m))) / 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 <= 1e-193)
		tmp = Float64(Float64(Float64(t_0 / Float64(x * Float64(c_m * s_m))) / Float64(c_m * s_m)) / x);
	else
		tmp = Float64(Float64(Float64(t_0 / Float64(c_m * Float64(x * s_m))) / 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 <= 1e-193)
		tmp = ((t_0 / (x * (c_m * s_m))) / (c_m * s_m)) / x;
	else
		tmp = ((t_0 / (c_m * (x * s_m))) / 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, 1e-193], N[(N[(N[(t$95$0 / N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(t$95$0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 1e-193

   1. Initial program 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. add-sqr-sqrt 65.2%

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

      3. times-frac 65.2%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-*r* 94.0%

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

      5. associate-/r* 89.7%

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

      6. *-commutative 89.7%

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

   6. Applied egg-rr 89.7%

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

   7. Taylor expanded in x around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r* 91.5%

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

   9. Simplified 91.5%

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

   if 1e-193 < c

   1. Initial program 61.7%

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

      1. *-un-lft-identity 61.7%

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

      2. add-sqr-sqrt 61.7%

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

      3. times-frac 61.7%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l/ 98.0%

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

      2. *-un-lft-identity 98.0%

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

      3. associate-/r* 95.6%

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

      4. *-commutative 95.6%

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

   6. Applied egg-rr 95.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 10^{-193}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c}}{x \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.5% accurate, 2.7× speedup

Math

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

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 (* s_m (* x c_m))) (/ (/ (cos (* x 2.0)) 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) {
	return (1.0 / (s_m * (x * c_m))) * ((cos((x * 2.0)) / s_m) / (x * c_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 / (s_m * (x * c_m))) * ((cos((x * 2.0d0)) / s_m) / (x * c_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 / (s_m * (x * c_m))) * ((Math.cos((x * 2.0)) / s_m) / (x * c_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 / (s_m * (x * c_m))) * ((math.cos((x * 2.0)) / s_m) / (x * c_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(Float64(1.0 / Float64(s_m * Float64(x * c_m))) * Float64(Float64(cos(Float64(x * 2.0)) / s_m) / Float64(x * c_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 / (s_m * (x * c_m))) * ((cos((x * 2.0)) / s_m) / (x * c_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[(N[(1.0 / N[(s$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / s$95$m), $MachinePrecision] / N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. *-un-lft-identity 63.7%

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

   2. add-sqr-sqrt 63.7%

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

   3. times-frac 63.7%

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

4. Applied egg-rr 97.9%

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

   1. *-un-lft-identity 97.9%

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

   2. associate-*r* 95.8%

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

   3. times-frac 95.3%

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

   4. *-commutative 95.3%

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

6. Applied egg-rr 95.3%

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

   1. associate-*l/ 95.8%

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

   2. *-lft-identity 95.8%

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

8. Simplified 95.8%

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

   1. /-rgt-identity 97.0%

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

   2. associate-*r* 96.5%

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

   3. associate-/l* 96.5%

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

10. Applied egg-rr 97.4%

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

    1. div-inv 97.4%

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

    2. *-commutative 97.4%

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

    3. inv-pow 97.4%

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

    4. pow-flip 97.4%

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

    5. metadata-eval 97.4%

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

    6. pow1 97.4%

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

12. Applied egg-rr 97.4%

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

13. Final simplification 97.4%

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

Alternative 7: 77.4% accurate, 17.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 5.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(c\_m \cdot s\_m\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c\_m \cdot s\_m\right) \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* c_m (* x s_m))))
   (if (<= c_m 5.5e-34)
     (/ 1.0 (* x (* (* c_m s_m) t_0)))
     (/ 1.0 (* (* c_m s_m) (* x 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 tmp;
	if (c_m <= 5.5e-34) {
		tmp = 1.0 / (x * ((c_m * s_m) * t_0));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x * 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) :: tmp
    t_0 = c_m * (x * s_m)
    if (c_m <= 5.5d-34) then
        tmp = 1.0d0 / (x * ((c_m * s_m) * t_0))
    else
        tmp = 1.0d0 / ((c_m * s_m) * (x * 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 tmp;
	if (c_m <= 5.5e-34) {
		tmp = 1.0 / (x * ((c_m * s_m) * t_0));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x * 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)
	tmp = 0
	if c_m <= 5.5e-34:
		tmp = 1.0 / (x * ((c_m * s_m) * t_0))
	else:
		tmp = 1.0 / ((c_m * s_m) * (x * 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))
	tmp = 0.0
	if (c_m <= 5.5e-34)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(c_m * s_m) * t_0)));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(x * 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);
	tmp = 0.0;
	if (c_m <= 5.5e-34)
		tmp = 1.0 / (x * ((c_m * s_m) * t_0));
	else
		tmp = 1.0 / ((c_m * s_m) * (x * 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]}, If[LessEqual[c$95$m, 5.5e-34], N[(1.0 / N[(x * N[(N[(c$95$m * s$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 5.50000000000000014e-34

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0 53.8%

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

      1. associate-/r* 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

      4. unpow2 53.2%

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

      5. swap-sqr 64.3%

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

      6. unpow2 64.3%

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

      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 78.9%

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

      11. unpow2 78.9%

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

      12. *-commutative 78.9%

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

   5. Simplified 78.9%

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

      1. unpow2 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-*r* 76.7%

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

      4. associate-*r* 73.4%

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

   7. Applied egg-rr 73.4%

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

   if 5.50000000000000014e-34 < c

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0 48.8%

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

      1. associate-/r* 49.4%

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

      2. *-commutative 49.4%

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

      3. unpow2 49.4%

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

      4. unpow2 49.4%

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

      5. swap-sqr 61.8%

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

      6. unpow2 61.8%

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

      7. associate-/r* 61.1%

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

      8. unpow2 61.1%

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

      9. unpow2 61.1%

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

      10. swap-sqr 80.7%

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

      11. unpow2 80.7%

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

      12. *-commutative 80.7%

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

   5. Simplified 80.7%

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

      1. unpow2 80.7%

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

      2. associate-*r* 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-*l* 74.6%

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

   7. Applied egg-rr 74.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 17.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \mathbf{if}\;c\_m \leq 8.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(c\_m \cdot s\_m\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s\_m \cdot \left(\left(x \cdot c\_m\right) \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* c_m (* x s_m))))
   (if (<= c_m 8.2e-18)
     (/ 1.0 (* x (* (* c_m s_m) t_0)))
     (/ 1.0 (* s_m (* (* x c_m) 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 tmp;
	if (c_m <= 8.2e-18) {
		tmp = 1.0 / (x * ((c_m * s_m) * t_0));
	} else {
		tmp = 1.0 / (s_m * ((x * c_m) * 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) :: tmp
    t_0 = c_m * (x * s_m)
    if (c_m <= 8.2d-18) then
        tmp = 1.0d0 / (x * ((c_m * s_m) * t_0))
    else
        tmp = 1.0d0 / (s_m * ((x * c_m) * 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 tmp;
	if (c_m <= 8.2e-18) {
		tmp = 1.0 / (x * ((c_m * s_m) * t_0));
	} else {
		tmp = 1.0 / (s_m * ((x * c_m) * 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)
	tmp = 0
	if c_m <= 8.2e-18:
		tmp = 1.0 / (x * ((c_m * s_m) * t_0))
	else:
		tmp = 1.0 / (s_m * ((x * c_m) * 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))
	tmp = 0.0
	if (c_m <= 8.2e-18)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(c_m * s_m) * t_0)));
	else
		tmp = Float64(1.0 / Float64(s_m * Float64(Float64(x * c_m) * 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);
	tmp = 0.0;
	if (c_m <= 8.2e-18)
		tmp = 1.0 / (x * ((c_m * s_m) * t_0));
	else
		tmp = 1.0 / (s_m * ((x * c_m) * 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]}, If[LessEqual[c$95$m, 8.2e-18], N[(1.0 / N[(x * N[(N[(c$95$m * s$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s$95$m * N[(N[(x * c$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 8.1999999999999995e-18

   1. Initial program 66.7%

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

   3. Taylor expanded in x around 0 54.2%

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

      1. associate-/r* 53.7%

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

      2. *-commutative 53.7%

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

      3. unpow2 53.7%

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

      4. unpow2 53.7%

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

      5. swap-sqr 64.4%

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

      6. unpow2 64.4%

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

      7. associate-/r* 65.0%

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

      8. unpow2 65.0%

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

      9. unpow2 65.0%

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

      10. swap-sqr 78.6%

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

      11. unpow2 78.6%

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

      12. *-commutative 78.6%

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

   5. Simplified 78.6%

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

      1. unpow2 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*r* 76.5%

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

      4. associate-*r* 73.3%

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

   7. Applied egg-rr 73.3%

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

   if 8.1999999999999995e-18 < c

   1. Initial program 56.2%

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

   3. Taylor expanded in x around 0 47.5%

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

      1. associate-/r* 48.1%

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

      2. *-commutative 48.1%

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

      3. unpow2 48.1%

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

      4. unpow2 48.1%

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

      5. swap-sqr 61.3%

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

      6. unpow2 61.3%

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

      7. associate-/r* 60.6%

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

      8. unpow2 60.6%

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

      9. unpow2 60.6%

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

      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%

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

   5. Simplified 81.5%

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

      1. *-commutative 81.5%

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

      2. *-commutative 81.5%

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

      3. *-commutative 81.5%

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

      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* 81.4%

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

      6. associate-*r* 81.3%

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

   7. Applied egg-rr 81.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.8% 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 63.7%

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

3. Taylor expanded in x around 0 52.3%

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

   1. associate-/r* 52.1%

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

   2. *-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. swap-sqr 63.5%

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

   6. unpow2 63.5%

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

   7. associate-/r* 63.7%

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

   8. unpow2 63.7%

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

   9. unpow2 63.7%

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

   10. swap-sqr 79.4%

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

   11. unpow2 79.4%

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

   12. *-commutative 79.4%

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

5. Simplified 79.4%

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

   1. unpow2 79.4%

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

   2. associate-*r* 77.2%

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

   3. *-commutative 77.2%

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

   4. associate-*l* 75.1%

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

7. Applied egg-rr 75.1%

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

8. Final simplification 75.1%

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

Alternative 10: 79.5% accurate, 24.1× speedup

Math

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

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 (* (* x s_m) (* 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 * ((x * s_m) * (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 * ((x * s_m) * (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 * ((x * s_m) * (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 * ((x * s_m) * (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(c_m * Float64(Float64(x * s_m) * 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 * ((x * s_m) * (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[(c$95$m * N[(N[(x * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in x around 0 52.3%

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

   1. associate-/r* 52.1%

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

   2. *-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. swap-sqr 63.5%

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

   6. unpow2 63.5%

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

   7. associate-/r* 63.7%

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

   8. unpow2 63.7%

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

   9. unpow2 63.7%

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

   10. swap-sqr 79.4%

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

   11. unpow2 79.4%

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

   12. *-commutative 79.4%

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

5. Simplified 79.4%

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

   1. *-commutative 79.4%

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

   2. *-commutative 79.4%

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

   3. *-commutative 79.4%

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

   4. unpow2 79.4%

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

   5. *-commutative 79.4%

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

   6. associate-*r* 77.3%

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

7. Applied egg-rr 77.3%

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

8. Final simplification 77.3%

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

Alternative 11: 79.5% accurate, 24.1× speedup

Math

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

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) (* (* x s_m) (* 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) / ((x * s_m) * (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) / ((x * s_m) * (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) / ((x * s_m) * (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) / ((x * s_m) * (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(Float64(1.0 / c_m) / Float64(Float64(x * s_m) * 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) / ((x * s_m) * (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[(N[(1.0 / c$95$m), $MachinePrecision] / N[(N[(x * s$95$m), $MachinePrecision] * N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. *-un-lft-identity 63.7%

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

   2. add-sqr-sqrt 63.7%

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

   3. times-frac 63.7%

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

4. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-/r* 98.0%

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

   3. frac-times 94.0%

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

   4. div-inv 94.0%

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

   5. *-commutative 94.0%

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

6. Applied egg-rr 94.0%

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

7. Taylor expanded in x around 0 77.6%

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

8. Final simplification 77.6%

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

Alternative 12: 80.3% accurate, 24.1× speedup

Math

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

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 (* 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(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 = 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[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 63.7%

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

3. Taylor expanded in x around 0 52.3%

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

   1. associate-/r* 52.1%

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

   2. *-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. swap-sqr 63.5%

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

   6. unpow2 63.5%

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

   7. associate-/r* 63.7%

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

   8. unpow2 63.7%

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

   9. unpow2 63.7%

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

   10. swap-sqr 79.4%

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

   11. unpow2 79.4%

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

   12. *-commutative 79.4%

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

5. Simplified 79.4%

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

   1. pow-flip 79.8%

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

   2. *-commutative 79.8%

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

   3. pow-flip 79.4%

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

   4. unpow2 79.4%

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

   5. associate-/r* 79.8%

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

   6. un-div-inv 79.7%

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

7. Applied egg-rr 79.7%

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

   1. un-div-inv 79.8%

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

9. Applied egg-rr 79.8%

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

10. Final simplification 79.8%

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

Reproduce

herbie shell --seed 2024035 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))