mixedcos

Percentage Accurate: 66.1% → 99.5%

Time: 14.2s

Alternatives: 9

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.1% 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: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 1.22 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)} \cdot \frac{\frac{1}{c\_m}}{x\_m \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{t\_0} \cdot \frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* x_m c_m))))
   (if (<= x_m 1.22e-53)
     (* (/ 1.0 (* c_m (* x_m s_m))) (/ (/ 1.0 c_m) (* x_m s_m)))
     (* (/ (cos (* x_m 2.0)) t_0) (/ 1.0 t_0)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 1.22e-53) {
		tmp = (1.0 / (c_m * (x_m * s_m))) * ((1.0 / c_m) / (x_m * s_m));
	} else {
		tmp = (cos((x_m * 2.0)) / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    if (x_m <= 1.22d-53) then
        tmp = (1.0d0 / (c_m * (x_m * s_m))) * ((1.0d0 / c_m) / (x_m * s_m))
    else
        tmp = (cos((x_m * 2.0d0)) / t_0) * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 1.22e-53) {
		tmp = (1.0 / (c_m * (x_m * s_m))) * ((1.0 / c_m) / (x_m * s_m));
	} else {
		tmp = (Math.cos((x_m * 2.0)) / t_0) * (1.0 / t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 1.22e-53:
		tmp = (1.0 / (c_m * (x_m * s_m))) * ((1.0 / c_m) / (x_m * s_m))
	else:
		tmp = (math.cos((x_m * 2.0)) / t_0) * (1.0 / t_0)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 1.22e-53)
		tmp = Float64(Float64(1.0 / Float64(c_m * Float64(x_m * s_m))) * Float64(Float64(1.0 / c_m) / Float64(x_m * s_m)));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 1.22e-53)
		tmp = (1.0 / (c_m * (x_m * s_m))) * ((1.0 / c_m) / (x_m * s_m));
	else
		tmp = (cos((x_m * 2.0)) / t_0) * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.22e-53], N[(N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.22000000000000003e-53

   1. Initial program 68.6%

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

      1. *-un-lft-identity 68.6%

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

      2. add-sqr-sqrt 68.5%

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

      3. times-frac 68.5%

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

      4. sqrt-prod 68.5%

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

      5. sqrt-pow1 43.6%

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

      6. metadata-eval 43.6%

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

      7. pow1 43.6%

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

      8. *-commutative 43.6%

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

      9. associate-*r* 39.7%

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

      10. unpow2 39.7%

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

      11. pow-prod-down 43.6%

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

      12. sqrt-pow1 50.4%

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

      13. metadata-eval 50.4%

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

      14. pow1 50.4%

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

      15. *-commutative 50.4%

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

   4. Applied egg-rr 94.5%

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

   5. Taylor expanded in x around 0 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/l/ 81.5%

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

      3. associate-/l/ 81.4%

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

      4. associate-/l/ 80.7%

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

   7. Simplified 80.7%

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

   8. Taylor expanded in s around 0 81.8%

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

      1. associate-/r* 81.9%

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

   10. Simplified 81.9%

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

   if 1.22000000000000003e-53 < x

   1. Initial program 55.7%

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

      1. *-un-lft-identity 55.7%

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

      2. add-sqr-sqrt 55.7%

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

      3. times-frac 55.7%

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

      4. sqrt-prod 55.8%

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

      5. sqrt-pow1 39.8%

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

      6. metadata-eval 39.8%

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

      7. pow1 39.8%

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

      8. *-commutative 39.8%

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

      9. associate-*r* 37.9%

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

      10. unpow2 37.9%

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

      11. pow-prod-down 39.9%

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

      12. sqrt-pow1 38.2%

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

      13. metadata-eval 38.2%

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

      14. pow1 38.2%

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

      15. *-commutative 38.2%

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

   4. Applied egg-rr 97.0%

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

      1. *-commutative 97.0%

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

      2. div-inv 97.0%

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

      3. associate-/r* 96.9%

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

      4. associate-/l/ 94.4%

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

      5. *-commutative 94.4%

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

   6. Applied egg-rr 94.4%

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

      1. div-inv 94.4%

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

      2. times-frac 96.9%

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

      3. associate-*r* 93.0%

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

      4. associate-/r* 93.1%

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

      5. associate-*r* 95.7%

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

   8. Applied egg-rr 95.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;c\_m \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{t\_1}{c\_m}}{\left(x\_m \cdot s\_m\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{t\_0}}{c\_m}}{x\_m \cdot s\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= c_m 4.1e-193)
     (/ (/ t_1 c_m) (* (* x_m s_m) t_0))
     (/ (/ (/ t_1 t_0) c_m) (* x_m s_m)))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = cos((x_m * 2.0));
	double tmp;
	if (c_m <= 4.1e-193) {
		tmp = (t_1 / c_m) / ((x_m * s_m) * t_0);
	} else {
		tmp = ((t_1 / t_0) / c_m) / (x_m * s_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    t_1 = cos((x_m * 2.0d0))
    if (c_m <= 4.1d-193) then
        tmp = (t_1 / c_m) / ((x_m * s_m) * t_0)
    else
        tmp = ((t_1 / t_0) / c_m) / (x_m * s_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double t_1 = Math.cos((x_m * 2.0));
	double tmp;
	if (c_m <= 4.1e-193) {
		tmp = (t_1 / c_m) / ((x_m * s_m) * t_0);
	} else {
		tmp = ((t_1 / t_0) / c_m) / (x_m * s_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	t_1 = math.cos((x_m * 2.0))
	tmp = 0
	if c_m <= 4.1e-193:
		tmp = (t_1 / c_m) / ((x_m * s_m) * t_0)
	else:
		tmp = ((t_1 / t_0) / c_m) / (x_m * s_m)
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	t_1 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (c_m <= 4.1e-193)
		tmp = Float64(Float64(t_1 / c_m) / Float64(Float64(x_m * s_m) * t_0));
	else
		tmp = Float64(Float64(Float64(t_1 / t_0) / c_m) / Float64(x_m * s_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	t_1 = cos((x_m * 2.0));
	tmp = 0.0;
	if (c_m <= 4.1e-193)
		tmp = (t_1 / c_m) / ((x_m * s_m) * t_0);
	else
		tmp = ((t_1 / t_0) / c_m) / (x_m * s_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 4.1e-193], N[(N[(t$95$1 / c$95$m), $MachinePrecision] / N[(N[(x$95$m * s$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 / t$95$0), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 4.10000000000000003e-193

   1. Initial program 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. add-sqr-sqrt 65.3%

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

      3. times-frac 65.3%

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

      4. sqrt-prod 65.4%

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

      5. sqrt-pow1 30.2%

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

      6. metadata-eval 30.2%

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

      7. pow1 30.2%

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

      8. *-commutative 30.2%

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

      9. associate-*r* 28.3%

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

      10. unpow2 28.3%

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

      11. pow-prod-down 30.2%

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

      12. sqrt-pow1 45.9%

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

      13. metadata-eval 45.9%

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

      14. pow1 45.9%

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

      15. *-commutative 45.9%

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

   4. Applied egg-rr 95.0%

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

      1. *-commutative 95.0%

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

      2. div-inv 95.0%

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

      3. associate-/r* 95.0%

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

      4. associate-/l/ 92.4%

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

      5. *-commutative 92.4%

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

   6. Applied egg-rr 92.4%

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

   if 4.10000000000000003e-193 < c

   1. Initial program 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. add-sqr-sqrt 64.4%

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

      3. times-frac 64.4%

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

      4. sqrt-prod 64.5%

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

      5. sqrt-pow1 64.5%

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

      6. metadata-eval 64.5%

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

      7. pow1 64.5%

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

      8. *-commutative 64.5%

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

      9. associate-*r* 58.6%

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

      10. unpow2 58.6%

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

      11. pow-prod-down 64.5%

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

      12. sqrt-pow1 49.0%

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

      13. metadata-eval 49.0%

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

      14. pow1 49.0%

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

      15. *-commutative 49.0%

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

   4. Applied egg-rr 95.5%

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

      1. associate-*l/ 95.5%

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

      2. *-un-lft-identity 95.5%

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

      3. associate-/r* 94.6%

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

      4. *-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 4.1 \cdot 10^{-193}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c}}{x \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \frac{1}{t\_0} \cdot \frac{\cos \left(x\_m \cdot 2\right)}{t\_0} \end{array} \end{array} \]

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	return (1.0 / t_0) * (cos((x_m * 2.0)) / t_0);
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x_m * s_m)
    code = (1.0d0 / t_0) * (cos((x_m * 2.0d0)) / t_0)
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	return (1.0 / t_0) * (Math.cos((x_m * 2.0)) / t_0);
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	return (1.0 / t_0) * (math.cos((x_m * 2.0)) / t_0)

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	return Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x_m * 2.0)) / t_0))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = (1.0 / t_0) * (cos((x_m * 2.0)) / t_0);
end

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. *-un-lft-identity 65.0%

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

   2. add-sqr-sqrt 65.0%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. sqrt-pow1 42.5%

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

   6. metadata-eval 42.5%

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

   7. pow1 42.5%

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

   8. *-commutative 42.5%

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

   9. associate-*r* 39.2%

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

   10. unpow2 39.2%

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

   11. pow-prod-down 42.6%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.2%

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

5. Final simplification 95.2%

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

Alternative 4: 93.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. *-un-lft-identity 65.0%

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

   2. add-sqr-sqrt 65.0%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. sqrt-pow1 42.5%

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

   6. metadata-eval 42.5%

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

   7. pow1 42.5%

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

   8. *-commutative 42.5%

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

   9. associate-*r* 39.2%

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

   10. unpow2 39.2%

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

   11. pow-prod-down 42.6%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.2%

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

   1. *-commutative 95.2%

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

   2. div-inv 95.2%

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

   3. associate-/r* 95.2%

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

   4. associate-/l/ 92.7%

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

   5. *-commutative 92.7%

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

6. Applied egg-rr 92.7%

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

7. Final simplification 92.7%

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

Alternative 5: 76.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;s\_m \leq 7.8 \cdot 10^{+201}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= s_m 7.8e+201)
     (/ 1.0 (* (* x_m c_m) (* s_m t_0)))
     (/ 1.0 (* (* c_m s_m) (* x_m t_0))))))

C

x_m = fabs(x);
c_m = fabs(c);
s_m = fabs(s);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (s_m <= 7.8e+201) {
		tmp = 1.0 / ((x_m * c_m) * (s_m * t_0));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * t_0));
	}
	return tmp;
}

Fortran

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (s_m <= 7.8d+201) then
        tmp = 1.0d0 / ((x_m * c_m) * (s_m * t_0))
    else
        tmp = 1.0d0 / ((c_m * s_m) * (x_m * t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
s_m = Math.abs(s);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (s_m <= 7.8e+201) {
		tmp = 1.0 / ((x_m * c_m) * (s_m * t_0));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if s_m <= 7.8e+201:
		tmp = 1.0 / ((x_m * c_m) * (s_m * t_0))
	else:
		tmp = 1.0 / ((c_m * s_m) * (x_m * t_0))
	return tmp

Julia

x_m = abs(x)
c_m = abs(c)
s_m = abs(s)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (s_m <= 7.8e+201)
		tmp = Float64(1.0 / Float64(Float64(x_m * c_m) * Float64(s_m * t_0)));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(x_m * t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
c_m = abs(c);
s_m = abs(s);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (s_m <= 7.8e+201)
		tmp = 1.0 / ((x_m * c_m) * (s_m * t_0));
	else
		tmp = 1.0 / ((c_m * s_m) * (x_m * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s$95$m, 7.8e+201], N[(1.0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 7.8000000000000001e201

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0 55.2%

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

      1. associate-/r* 54.8%

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

      2. *-commutative 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. swap-sqr 66.0%

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

      6. unpow2 66.0%

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

      7. associate-/r* 66.4%

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

      8. unpow2 66.4%

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

      9. unpow2 66.4%

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

      10. swap-sqr 77.1%

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

      11. unpow2 77.1%

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

      12. *-commutative 77.1%

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

   5. Simplified 77.1%

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

      1. *-commutative 77.1%

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

      2. *-commutative 77.1%

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

      3. *-commutative 77.1%

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

      4. unpow2 77.1%

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

      5. associate-*r* 76.5%

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

      6. associate-*l* 75.3%

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

   7. Applied egg-rr 75.3%

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

   if 7.8000000000000001e201 < s

   1. Initial program 34.8%

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

   3. Taylor expanded in x around 0 34.6%

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

      1. associate-/r* 34.6%

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

      2. *-commutative 34.6%

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

      3. unpow2 34.6%

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

      4. unpow2 34.6%

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

      5. swap-sqr 48.4%

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

      6. unpow2 48.4%

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

      7. associate-/r* 48.4%

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

      8. unpow2 48.4%

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

      9. unpow2 48.4%

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

      10. swap-sqr 87.3%

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

      11. unpow2 87.3%

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

      12. *-commutative 87.3%

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

   5. Simplified 87.3%

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

      1. unpow2 87.3%

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

      2. associate-*r* 87.4%

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

      3. *-commutative 87.4%

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

      4. associate-*l* 87.2%

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

   7. Applied egg-rr 87.2%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 7.8 \cdot 10^{+201}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. *-un-lft-identity 65.0%

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

   2. add-sqr-sqrt 65.0%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. sqrt-pow1 42.5%

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

   6. metadata-eval 42.5%

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

   7. pow1 42.5%

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

   8. *-commutative 42.5%

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

   9. associate-*r* 39.2%

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

   10. unpow2 39.2%

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

   11. pow-prod-down 42.6%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.2%

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

5. Taylor expanded in x around 0 77.7%

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

   1. *-commutative 77.7%

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

   2. associate-/l/ 77.5%

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

   3. associate-/l/ 77.5%

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

   4. associate-/l/ 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in s around 0 77.7%

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

   1. associate-/r* 77.8%

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

10. Simplified 77.8%

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

11. Final simplification 77.8%

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

Alternative 7: 79.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. *-un-lft-identity 65.0%

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

   2. add-sqr-sqrt 65.0%

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

   3. times-frac 65.0%

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

   4. sqrt-prod 65.0%

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

   5. sqrt-pow1 42.5%

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

   6. metadata-eval 42.5%

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

   7. pow1 42.5%

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

   8. *-commutative 42.5%

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

   9. associate-*r* 39.2%

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

   10. unpow2 39.2%

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

   11. pow-prod-down 42.6%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.2%

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

   1. clear-num 95.2%

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

   2. un-div-inv 95.2%

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

   3. *-commutative 95.2%

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

   4. associate-/r* 94.9%

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

   5. associate-/l* 94.9%

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

   6. *-commutative 94.9%

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

6. Applied egg-rr 94.9%

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

7. Taylor expanded in x around 0 77.5%

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

8. Final simplification 77.5%

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

Alternative 8: 79.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

3. Taylor expanded in x around 0 54.0%

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

   1. associate-/r* 53.6%

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

   2. *-commutative 53.6%

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

   3. unpow2 53.6%

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

   4. unpow2 53.6%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 65.3%

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

   8. unpow2 65.3%

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

   9. unpow2 65.3%

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

   10. swap-sqr 77.7%

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

   11. unpow2 77.7%

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

   12. *-commutative 77.7%

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

5. Simplified 77.7%

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

   1. *-commutative 77.7%

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

   2. *-commutative 77.7%

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

   3. *-commutative 77.7%

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

   4. unpow2 77.7%

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

7. Applied egg-rr 77.7%

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

Alternative 9: 75.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

3. Taylor expanded in x around 0 54.0%

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

   1. associate-/r* 53.6%

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

   2. *-commutative 53.6%

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

   3. unpow2 53.6%

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

   4. unpow2 53.6%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 65.3%

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

   8. unpow2 65.3%

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

   9. unpow2 65.3%

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

   10. swap-sqr 77.7%

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

   11. unpow2 77.7%

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

   12. *-commutative 77.7%

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

5. Simplified 77.7%

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

   1. unpow2 77.7%

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

   2. associate-*r* 77.3%

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

   3. *-commutative 77.3%

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

   4. associate-*l* 74.7%

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

7. Applied egg-rr 74.7%

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

Reproduce

herbie shell --seed 2024106 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))