mixedcos

Percentage Accurate: 66.3% → 99.1%

Time: 15.7s

Alternatives: 5

Speedup: 24.1×

• 31.8% 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: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m \cdot 2\right)\\ t_1 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{\frac{{\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)}^{2}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

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 (cos (* x_m 2.0))) (t_1 (* s_m (* x_m c_m))))
   (if (<= x_m 4.4e+84)
     (/ 1.0 (/ (pow (* c_m (* x_m s_m)) 2.0) t_0))
     (/ (/ t_0 t_1) t_1))))

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 = cos((x_m * 2.0));
	double t_1 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 4.4e+84) {
		tmp = 1.0 / (pow((c_m * (x_m * s_m)), 2.0) / t_0);
	} else {
		tmp = (t_0 / t_1) / t_1;
	}
	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 = cos((x_m * 2.0d0))
    t_1 = s_m * (x_m * c_m)
    if (x_m <= 4.4d+84) then
        tmp = 1.0d0 / (((c_m * (x_m * s_m)) ** 2.0d0) / t_0)
    else
        tmp = (t_0 / t_1) / t_1
    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 = Math.cos((x_m * 2.0));
	double t_1 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 4.4e+84) {
		tmp = 1.0 / (Math.pow((c_m * (x_m * s_m)), 2.0) / t_0);
	} else {
		tmp = (t_0 / t_1) / t_1;
	}
	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 = math.cos((x_m * 2.0))
	t_1 = s_m * (x_m * c_m)
	tmp = 0
	if x_m <= 4.4e+84:
		tmp = 1.0 / (math.pow((c_m * (x_m * s_m)), 2.0) / t_0)
	else:
		tmp = (t_0 / t_1) / t_1
	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 = cos(Float64(x_m * 2.0))
	t_1 = Float64(s_m * Float64(x_m * c_m))
	tmp = 0.0
	if (x_m <= 4.4e+84)
		tmp = Float64(1.0 / Float64((Float64(c_m * Float64(x_m * s_m)) ^ 2.0) / t_0));
	else
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	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 = cos((x_m * 2.0));
	t_1 = s_m * (x_m * c_m);
	tmp = 0.0;
	if (x_m <= 4.4e+84)
		tmp = 1.0 / (((c_m * (x_m * s_m)) ^ 2.0) / t_0);
	else
		tmp = (t_0 / t_1) / t_1;
	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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4.4e+84], N[(1.0 / N[(N[Power[N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.3999999999999997e84

   1. Initial program 68.2%

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

      1. clear-num 68.2%

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

      2. inv-pow 68.2%

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

   4. Applied egg-rr 96.3%

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

      1. unpow-1 96.3%

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

      2. *-commutative 96.3%

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

   6. Simplified 96.3%

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

   if 4.3999999999999997e84 < x

   1. Initial program 67.1%

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

      1. clear-num 67.1%

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

      2. inv-pow 67.1%

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

   4. Applied egg-rr 95.6%

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

      1. unpow-1 95.6%

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

      2. *-commutative 95.6%

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

   6. Simplified 95.6%

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

      1. associate-/r/ 95.6%

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

      2. pow-flip 95.6%

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

      3. associate-*r* 97.6%

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

      4. metadata-eval 97.6%

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

      5. unpow-prod-down 70.8%

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

      6. metadata-eval 70.8%

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

      7. pow-flip 70.8%

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

      8. associate-/r/ 72.2%

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

      9. clear-num 72.1%

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

      10. add-sqr-sqrt 72.1%

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

      11. associate-*l* 72.2%

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

      12. sqrt-div 72.2%

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

      13. sqrt-pow1 72.2%

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

      14. metadata-eval 72.2%

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

      15. inv-pow 72.2%

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

      16. sqrt-pow1 55.7%

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

      17. metadata-eval 55.7%

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

      18. pow1 55.7%

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

   8. Applied egg-rr 97.5%

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

      1. associate-*l/ 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-/l/ 88.1%

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

      4. associate-*r* 87.3%

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

      5. div-inv 87.3%

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

      6. times-frac 91.7%

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

      7. associate-/l/ 91.6%

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

      8. times-frac 93.5%

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

      9. *-commutative 93.5%

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

      10. *-un-lft-identity 93.5%

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

      11. associate-*r* 95.6%

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

      12. *-commutative 95.6%

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

      13. associate-*r* 97.5%

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

      14. *-commutative 97.5%

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

      15. *-commutative 97.5%

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

   10. Applied egg-rr 97.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{\frac{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}{\cos \left(x \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x\_m \cdot 2\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

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))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 2.6e-11)
     (/ 1.0 (* t_1 t_1))
     (/ (/ (cos (* x_m 2.0)) 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 = s_m * (x_m * c_m);
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.6e-11) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = (cos((x_m * 2.0)) / t_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) :: t_1
    real(8) :: tmp
    t_0 = s_m * (x_m * c_m)
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 2.6d-11) then
        tmp = 1.0d0 / (t_1 * t_1)
    else
        tmp = (cos((x_m * 2.0d0)) / t_0) / 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 t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.6e-11) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = (Math.cos((x_m * 2.0)) / t_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)
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 2.6e-11:
		tmp = 1.0 / (t_1 * t_1)
	else:
		tmp = (math.cos((x_m * 2.0)) / t_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))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 2.6e-11)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(cos(Float64(x_m * 2.0)) / t_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);
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 2.6e-11)
		tmp = 1.0 / (t_1 * t_1);
	else
		tmp = (cos((x_m * 2.0)) / t_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]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.6e-11], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e-11

   1. Initial program 67.9%

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

   3. Taylor expanded in x around 0 57.0%

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

      1. associate-/r* 56.9%

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

      2. *-commutative 56.9%

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

      3. unpow2 56.9%

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

      4. unpow2 56.9%

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

      5. swap-sqr 73.9%

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

      6. unpow2 73.9%

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

      7. associate-/r* 73.9%

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

      8. unpow2 73.9%

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

      9. unpow2 73.9%

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

      10. swap-sqr 85.3%

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

      11. unpow2 85.3%

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

      12. *-commutative 85.3%

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

   5. Simplified 85.3%

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

      1. unpow2 85.3%

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

   7. Applied egg-rr 85.3%

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

   if 2.6000000000000001e-11 < x

   1. Initial program 68.3%

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

      1. clear-num 68.3%

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

      2. inv-pow 68.3%

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

   4. Applied egg-rr 96.6%

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

      1. unpow-1 96.6%

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

      2. *-commutative 96.6%

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

   6. Simplified 96.6%

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

      1. associate-/r/ 96.6%

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

      2. pow-flip 96.6%

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

      3. associate-*r* 98.1%

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

      4. metadata-eval 98.1%

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

      5. unpow-prod-down 75.5%

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

      6. metadata-eval 75.5%

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

      7. pow-flip 75.5%

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

      8. associate-/r/ 76.9%

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

      9. clear-num 76.8%

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

      10. add-sqr-sqrt 76.8%

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

      11. associate-*l* 76.9%

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

      12. sqrt-div 76.9%

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

      13. sqrt-pow1 76.9%

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

      14. metadata-eval 76.9%

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

      15. inv-pow 76.9%

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

      16. sqrt-pow1 58.5%

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

      17. metadata-eval 58.5%

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

      18. pow1 58.5%

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

   8. Applied egg-rr 98.0%

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

      1. associate-*l/ 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-/l/ 91.1%

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

      4. associate-*r* 90.6%

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

      5. div-inv 90.5%

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

      6. times-frac 90.9%

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

      7. associate-/l/ 90.8%

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

      8. times-frac 95.1%

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

      9. *-commutative 95.1%

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

      10. *-un-lft-identity 95.1%

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

      11. associate-*r* 96.7%

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

      12. *-commutative 96.7%

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

      13. associate-*r* 98.0%

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

      14. *-commutative 98.0%

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

      15. *-commutative 98.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 2.7× speedup

Math

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

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)))) (/ (/ (cos (* x_m 2.0)) 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 (cos((x_m * 2.0)) / 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 = (cos((x_m * 2.0d0)) / 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 (Math.cos((x_m * 2.0)) / 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 (math.cos((x_m * 2.0)) / 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(Float64(cos(Float64(x_m * 2.0)) / 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 = (cos((x_m * 2.0)) / 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[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 68.0%

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

   1. *-un-lft-identity 68.0%

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

   2. associate-*r* 69.9%

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

   3. times-frac 69.9%

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

   4. *-commutative 69.9%

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

   5. associate-*r* 68.8%

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

   6. pow-prod-down 87.4%

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

4. Applied egg-rr 87.4%

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

   1. frac-times 87.2%

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

   2. *-un-lft-identity 87.2%

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

   3. associate-*l* 75.6%

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

   4. unpow2 75.6%

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

   5. swap-sqr 96.7%

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

   6. associate-*r* 93.5%

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

   7. associate-*r* 96.1%

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

   8. associate-/r* 96.1%

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

   9. *-commutative 96.1%

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

6. Applied egg-rr 96.1%

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

7. Final simplification 96.1%

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

Alternative 4: 76.8% 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 68.0%

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

3. Taylor expanded in x around 0 54.9%

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

   1. associate-/r* 54.9%

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

   2. *-commutative 54.9%

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

   3. unpow2 54.9%

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

   4. unpow2 54.9%

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

   5. swap-sqr 68.6%

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

   6. unpow2 68.6%

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

   7. associate-/r* 68.6%

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

   8. unpow2 68.6%

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

   9. unpow2 68.6%

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

   10. swap-sqr 77.5%

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

   11. unpow2 77.5%

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

   12. *-commutative 77.5%

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

5. Simplified 77.5%

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

   1. unpow2 77.5%

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

   2. associate-*r* 75.8%

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

   3. associate-*l* 74.2%

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

7. Applied egg-rr 74.2%

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

8. Final simplification 74.2%

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

Alternative 5: 79.8% accurate, 24.1× speedup

Math

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

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 68.0%

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

3. Taylor expanded in x around 0 54.9%

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

   1. associate-/r* 54.9%

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

   2. *-commutative 54.9%

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

   3. unpow2 54.9%

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

   4. unpow2 54.9%

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

   5. swap-sqr 68.6%

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

   6. unpow2 68.6%

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

   7. associate-/r* 68.6%

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

   8. unpow2 68.6%

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

   9. unpow2 68.6%

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

   10. swap-sqr 77.5%

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

   11. unpow2 77.5%

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

   12. *-commutative 77.5%

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

5. Simplified 77.5%

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

   1. unpow2 77.5%

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

7. Applied egg-rr 77.5%

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

8. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2024077 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))