mixedcos

Percentage Accurate: 66.9% → 97.5%

Time: 12.8s

Alternatives: 9

Speedup: 24.1×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 1.5× speedup

Math

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

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
 (* (pow (* s_m (* x c_m)) -2.0) (cos (* x 2.0))))

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) {
	return pow((s_m * (x * c_m)), -2.0) * cos((x * 2.0));
}

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
    code = ((s_m * (x * c_m)) ** (-2.0d0)) * cos((x * 2.0d0))
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 Math.pow((s_m * (x * c_m)), -2.0) * Math.cos((x * 2.0));
}

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):
	return math.pow((s_m * (x * c_m)), -2.0) * math.cos((x * 2.0))

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)
	return Float64((Float64(s_m * Float64(x * c_m)) ^ -2.0) * cos(Float64(x * 2.0)))
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)
	tmp = ((s_m * (x * c_m)) ^ -2.0) * cos((x * 2.0));
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_] := N[(N[Power[N[(s$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in c around 0 60.1%

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

   1. unpow2 60.1%

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

   2. rem-square-sqrt 60.1%

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

   3. swap-sqr 67.7%

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

   4. unpow2 67.7%

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

   5. unpow2 67.7%

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

   6. unpow2 67.7%

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

   7. unswap-sqr 86.9%

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

   8. rem-sqrt-square 96.6%

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

5. Simplified 96.6%

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

   1. pow-prod-down 78.1%

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

   2. associate-/r* 78.4%

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

   3. add-sqr-sqrt 44.3%

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

   4. fabs-sqr 44.3%

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

   5. add-sqr-sqrt 78.4%

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

   6. pow-prod-down 60.3%

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

   7. associate-/r* 60.1%

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

   8. pow1 60.1%

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

   9. metadata-eval 60.1%

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

   10. sqrt-pow2 60.1%

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

   11. pow-prod-down 67.7%

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

   12. div-inv 67.7%

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

   13. *-commutative 67.7%

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

7. Applied egg-rr 97.7%

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

8. Final simplification 97.7%

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

Alternative 2: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x \cdot c\_m\right)\\ \frac{\cos \left(x \cdot 2\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in c around 0 60.1%

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

   1. unpow2 60.1%

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

   2. rem-square-sqrt 60.1%

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

   3. swap-sqr 67.7%

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

   4. unpow2 67.7%

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

   5. unpow2 67.7%

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

   6. unpow2 67.7%

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

   7. unswap-sqr 86.9%

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

   8. rem-sqrt-square 96.6%

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

5. Simplified 96.6%

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

   1. pow-prod-down 68.0%

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

   2. *-commutative 68.0%

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

   3. unpow2 68.0%

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

   4. sqr-abs 68.0%

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

   5. unpow2 68.0%

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

   6. swap-sqr 79.8%

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

   7. associate-*l* 78.8%

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

   8. associate-*l* 79.0%

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

7. Applied egg-rr 97.1%

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

8. Final simplification 97.1%

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

Alternative 3: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x \cdot c\_m\right)\\ \frac{\frac{\cos \left(x \cdot 2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

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 (* s_m (* x c_m)))) (/ (/ (cos (* x 2.0)) t_0) t_0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in c around 0 60.1%

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

   1. unpow2 60.1%

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

   2. rem-square-sqrt 60.1%

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

   3. swap-sqr 67.7%

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

   4. unpow2 67.7%

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

   5. unpow2 67.7%

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

   6. unpow2 67.7%

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

   7. unswap-sqr 86.9%

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

   8. rem-sqrt-square 96.6%

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

5. Simplified 96.6%

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

   1. pow-prod-down 78.1%

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

   2. associate-/r* 78.4%

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

   3. add-sqr-sqrt 44.3%

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

   4. fabs-sqr 44.3%

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

   5. add-sqr-sqrt 78.4%

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

   6. pow-prod-down 60.3%

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

   7. associate-/r* 60.1%

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

   8. pow1 60.1%

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

   9. metadata-eval 60.1%

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

   10. sqrt-pow2 60.1%

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

   11. pow-prod-down 67.7%

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

   12. div-inv 67.7%

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

   13. *-commutative 67.7%

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

7. Applied egg-rr 97.7%

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

   1. *-commutative 97.7%

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

   2. metadata-eval 97.7%

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

   3. pow-flip 97.1%

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

   4. pow2 97.1%

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

   5. associate-/r/ 97.1%

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

   6. clear-num 97.1%

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

   7. associate-/r* 97.6%

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

   8. *-commutative 97.6%

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

9. Applied egg-rr 97.6%

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

10. Final simplification 97.6%

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

Alternative 4: 79.7% accurate, 3.0× speedup

Math

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

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 (pow (* c_m (* s_m x)) -2.0))

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) {
	return pow((c_m * (s_m * x)), -2.0);
}

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
    code = (c_m * (s_m * x)) ** (-2.0d0)
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 Math.pow((c_m * (s_m * x)), -2.0);
}

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):
	return math.pow((c_m * (s_m * x)), -2.0)

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)
	return Float64(c_m * Float64(s_m * x)) ^ -2.0
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)
	tmp = (c_m * (s_m * x)) ^ -2.0;
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_] := N[Power[N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in c around 0 60.1%

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

   1. unpow2 60.1%

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

   2. rem-square-sqrt 60.1%

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

   3. swap-sqr 67.7%

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

   4. unpow2 67.7%

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

   5. unpow2 67.7%

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

   6. unpow2 67.7%

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

   7. unswap-sqr 86.9%

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

   8. rem-sqrt-square 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in x around 0 68.0%

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

   1. associate-/r* 68.0%

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

   2. unpow2 68.0%

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

   3. sqr-abs 68.0%

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

   4. unpow2 68.0%

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

   5. associate-/r* 68.0%

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

   6. *-commutative 68.0%

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

   7. unpow2 68.0%

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

   8. unpow2 68.0%

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

   9. swap-sqr 79.8%

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

   10. associate-*r* 78.8%

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

   11. associate-*r* 79.0%

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

   12. unpow2 79.0%

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

   13. /-rgt-identity 79.0%

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

   14. unpow2 79.0%

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

   15. associate-/l* 78.9%

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

   16. associate-/l* 79.3%

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

   17. associate-*l/ 79.3%

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

   18. unpow-1 79.3%

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

   19. unpow-1 79.3%

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

8. Simplified 80.2%

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

9. Final simplification 80.2%

   \[\leadsto {\left(c \cdot \left(s \cdot x\right)\right)}^{-2} \]
10. Add Preprocessing

Alternative 5: 76.4% accurate, 24.1× speedup

Math

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

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 (* (* s_m c_m) (* x (* c_m (* s_m x))))))

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 * c_m) * (x * (c_m * (s_m * x))));
}

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 * c_m) * (x * (c_m * (s_m * x))))
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 * c_m) * (x * (c_m * (s_m * x))));
}

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 * c_m) * (x * (c_m * (s_m * x))))

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0 56.4%

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

   1. unpow2 56.4%

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

   2. rem-square-sqrt 56.4%

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

   3. swap-sqr 62.4%

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

   4. unpow2 62.4%

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

   5. unpow2 62.4%

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

   6. unpow2 62.4%

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

   7. unswap-sqr 74.5%

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

   8. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. pow-prod-down 68.0%

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

   2. *-commutative 68.0%

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

   3. unpow2 68.0%

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

   4. sqr-abs 68.0%

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

   5. unpow2 68.0%

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

   6. swap-sqr 79.8%

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

   7. associate-*l* 78.8%

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

   8. associate-*l* 79.0%

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

7. Applied egg-rr 79.0%

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

   1. /-rgt-identity 79.0%

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

   2. associate-*l* 78.3%

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

   3. associate-/l* 78.3%

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

   4. *-commutative 78.3%

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

   5. associate-*r* 74.6%

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

   6. pow2 74.6%

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

9. Applied egg-rr 74.6%

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

    1. associate-/r/ 74.6%

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

    2. /-rgt-identity 74.6%

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

    3. *-commutative 74.6%

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

    4. associate-*r* 69.0%

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

    5. unpow2 69.0%

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

    6. swap-sqr 79.0%

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

    7. associate-*r* 78.8%

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

    8. *-commutative 78.8%

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

    9. associate-*r* 77.5%

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

    10. associate-*l* 77.0%

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

    11. associate-*r* 77.6%

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

    12. *-commutative 77.6%

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

    13. *-commutative 77.6%

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

11. Applied egg-rr 77.6%

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

12. Final simplification 77.6%

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

Alternative 6: 77.5% accurate, 24.1× speedup

Math

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

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 (* (* s_m (* x c_m)) (* c_m (* s_m x)))))

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)) * (c_m * (s_m * x)));
}

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)) * (c_m * (s_m * x)))
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)) * (c_m * (s_m * x)));
}

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)) * (c_m * (s_m * x)))

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0 56.4%

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

   1. unpow2 56.4%

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

   2. rem-square-sqrt 56.4%

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

   3. swap-sqr 62.4%

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

   4. unpow2 62.4%

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

   5. unpow2 62.4%

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

   6. unpow2 62.4%

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

   7. unswap-sqr 74.5%

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

   8. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. pow-prod-down 68.0%

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

   2. *-commutative 68.0%

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

   3. unpow2 68.0%

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

   4. sqr-abs 68.0%

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

   5. unpow2 68.0%

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

   6. swap-sqr 79.8%

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

   7. associate-*l* 78.8%

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

   8. associate-*l* 79.0%

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

7. Applied egg-rr 79.0%

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

8. Taylor expanded in s around 0 78.8%

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

9. Final simplification 78.8%

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

Alternative 7: 77.7% accurate, 24.1× speedup

Math

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

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x c_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 = s_m * (x * c_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 = s_m * (x * c_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 = s_m * (x * c_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 = s_m * (x * c_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(s_m * Float64(x * c_m))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = s_m * (x * c_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[(s$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0 56.4%

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

   1. unpow2 56.4%

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

   2. rem-square-sqrt 56.4%

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

   3. swap-sqr 62.4%

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

   4. unpow2 62.4%

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

   5. unpow2 62.4%

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

   6. unpow2 62.4%

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

   7. unswap-sqr 74.5%

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

   8. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. pow-prod-down 68.0%

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

   2. *-commutative 68.0%

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

   3. unpow2 68.0%

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

   4. sqr-abs 68.0%

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

   5. unpow2 68.0%

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

   6. swap-sqr 79.8%

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

   7. associate-*l* 78.8%

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

   8. associate-*l* 79.0%

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

7. Applied egg-rr 79.0%

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

8. Final simplification 79.0%

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

Alternative 8: 78.1% accurate, 24.1× speedup

Math

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

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) (* s_m x)))

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) / (s_m * x);
}

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) / (s_m * x)
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) / (s_m * x);
}

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) / (s_m * x)

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0 56.4%

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

   1. unpow2 56.4%

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

   2. rem-square-sqrt 56.4%

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

   3. swap-sqr 62.4%

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

   4. unpow2 62.4%

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

   5. unpow2 62.4%

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

   6. unpow2 62.4%

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

   7. unswap-sqr 74.5%

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

   8. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. add-sqr-sqrt 79.8%

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

   2. pow2 79.8%

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

   3. sqrt-div 79.8%

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

   4. metadata-eval 79.8%

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

   5. sqrt-pow1 80.1%

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

   6. metadata-eval 80.1%

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

   7. pow1 80.1%

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

   8. add-sqr-sqrt 44.0%

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

   9. fabs-sqr 44.0%

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

   10. add-sqr-sqrt 80.1%

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

   11. *-commutative 80.1%

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

   12. associate-*l* 79.3%

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

7. Applied egg-rr 79.3%

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

8. Taylor expanded in s around 0 80.1%

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

   1. *-commutative 80.1%

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

   2. associate-*r* 79.3%

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

   3. /-rgt-identity 79.3%

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

   4. associate-/l* 79.1%

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

   5. associate-/l* 79.1%

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

   6. *-lft-identity 79.1%

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

   7. *-commutative 79.1%

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

10. Simplified 79.1%

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

    1. associate-/l/ 79.3%

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

    2. *-commutative 79.3%

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

    3. pow2 79.3%

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

    4. div-inv 79.3%

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

    5. associate-*r* 79.1%

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

    6. *-commutative 79.1%

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

    7. associate-/r* 76.9%

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

    8. associate-*r* 77.5%

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

    9. *-commutative 77.5%

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

    10. *-commutative 77.5%

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

    11. *-commutative 77.5%

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

12. Applied egg-rr 77.5%

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

13. Final simplification 77.5%

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

Alternative 9: 79.1% accurate, 24.1× speedup

Math

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

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0 56.4%

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

   1. unpow2 56.4%

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

   2. rem-square-sqrt 56.4%

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

   3. swap-sqr 62.4%

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

   4. unpow2 62.4%

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

   5. unpow2 62.4%

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

   6. unpow2 62.4%

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

   7. unswap-sqr 74.5%

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

   8. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. add-sqr-sqrt 79.8%

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

   2. pow2 79.8%

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

   3. sqrt-div 79.8%

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

   4. metadata-eval 79.8%

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

   5. sqrt-pow1 80.1%

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

   6. metadata-eval 80.1%

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

   7. pow1 80.1%

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

   8. add-sqr-sqrt 44.0%

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

   9. fabs-sqr 44.0%

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

   10. add-sqr-sqrt 80.1%

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

   11. *-commutative 80.1%

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

   12. associate-*l* 79.3%

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

7. Applied egg-rr 79.3%

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

8. Taylor expanded in s around 0 80.1%

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

   1. *-commutative 80.1%

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

   2. associate-*r* 79.3%

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

   3. /-rgt-identity 79.3%

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

   4. associate-/l* 79.1%

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

   5. associate-/l* 79.1%

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

   6. *-lft-identity 79.1%

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

   7. *-commutative 79.1%

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

10. Simplified 79.1%

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

    1. associate-/l/ 79.3%

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

    2. *-commutative 79.3%

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

    3. pow2 79.3%

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

    4. div-inv 79.3%

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

    5. associate-*r* 79.1%

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

    6. associate-/r* 77.9%

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

    7. associate-*r* 78.9%

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

    8. *-commutative 78.9%

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

    9. *-commutative 78.9%

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

    10. *-commutative 78.9%

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

12. Applied egg-rr 78.9%

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

13. Final simplification 78.9%

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

Reproduce

herbie shell --seed 2024027 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))