mixedcos

Percentage Accurate: 66.2% → 98.0%

Time: 17.9s

Alternatives: 7

Speedup: 24.1×

• 31.2% 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: 66.2% 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: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(s_m \cdot c_m\right)\\ \mathbf{if}\;{c_m}^{2} \leq 10^{-42}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s_m\right) \cdot c_m\right) \cdot \left(s_m \cdot \left(x \cdot c_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* x (* s_m c_m))))
   (if (<= (pow c_m 2.0) 1e-42)
     (/ (/ (cos (* 2.0 x)) t_0) t_0)
     (/ 1.0 (* (* (* x s_m) c_m) (* s_m (* x c_m)))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = x * (s_m * c_m);
	double tmp;
	if (pow(c_m, 2.0) <= 1e-42) {
		tmp = (cos((2.0 * x)) / t_0) / t_0;
	} else {
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s_m * c_m)
    if ((c_m ** 2.0d0) <= 1d-42) then
        tmp = (cos((2.0d0 * x)) / t_0) / t_0
    else
        tmp = 1.0d0 / (((x * s_m) * c_m) * (s_m * (x * c_m)))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = x * (s_m * c_m);
	double tmp;
	if (Math.pow(c_m, 2.0) <= 1e-42) {
		tmp = (Math.cos((2.0 * x)) / t_0) / t_0;
	} else {
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = x * (s_m * c_m)
	tmp = 0
	if math.pow(c_m, 2.0) <= 1e-42:
		tmp = (math.cos((2.0 * x)) / t_0) / t_0
	else:
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(x * Float64(s_m * c_m))
	tmp = 0.0
	if ((c_m ^ 2.0) <= 1e-42)
		tmp = Float64(Float64(cos(Float64(2.0 * x)) / t_0) / t_0);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * s_m) * c_m) * Float64(s_m * Float64(x * c_m))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = x * (s_m * c_m);
	tmp = 0.0;
	if ((c_m ^ 2.0) <= 1e-42)
		tmp = (cos((2.0 * x)) / t_0) / t_0;
	else
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[c$95$m, 2.0], $MachinePrecision], 1e-42], N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(N[(N[(x * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c 2) < 1.00000000000000004e-42

   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. add-sqr-sqrt 54.0%

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

      2. add-sqr-sqrt 54.0%

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

      3. times-frac 54.0%

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

      4. sqrt-prod 54.0%

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

      5. unpow2 54.0%

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

      6. sqrt-prod 31.0%

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

      7. add-sqr-sqrt 33.4%

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

      8. *-commutative 33.4%

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

      9. associate-*l* 31.7%

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

      10. pow2 31.7%

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

      11. sqrt-prod 31.7%

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

   4. Applied egg-rr 53.1%

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

      1. unpow2 53.1%

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

      2. unpow2 53.1%

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

      3. unpow2 53.1%

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

      4. unswap-sqr 71.9%

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

      5. rem-sqrt-square 77.7%

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

   6. Simplified 77.7%

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

   8. Applied egg-rr 98.4%

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

   if 1.00000000000000004e-42 < (pow.f64 c 2)

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0 56.7%

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

      1. unpow2 56.7%

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

      2. rem-square-sqrt 56.7%

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

      3. swap-sqr 63.1%

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

      4. unpow2 63.1%

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

      5. unpow2 63.1%

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

      6. unpow2 63.1%

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

      7. unswap-sqr 80.6%

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

      8. rem-sqrt-square 88.4%

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

   5. Simplified 88.4%

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

      1. pow1 88.4%

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

      2. metadata-eval 88.4%

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

      3. sqrt-pow1 88.4%

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

      4. pow2 88.4%

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

      5. add-sqr-sqrt 88.4%

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

      6. unpow2 88.4%

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

      7. add-sqr-sqrt 42.1%

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

      8. fabs-sqr 42.1%

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

      9. add-sqr-sqrt 68.6%

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

      10. associate-*r* 68.0%

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

      11. add-sqr-sqrt 35.7%

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

      12. fabs-sqr 35.7%

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

      13. add-sqr-sqrt 84.1%

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

      14. associate-*r* 88.3%

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

   7. Applied egg-rr 88.3%

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

      1. pow2 88.3%

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

      2. associate-*r* 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-*l* 92.8%

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

   9. Applied egg-rr 92.8%

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

       1. unpow2 92.8%

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

       2. associate-*r* 88.6%

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

       3. *-commutative 88.6%

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

       4. associate-*l* 87.9%

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

       5. associate-*r* 87.7%

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

       6. *-commutative 87.7%

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

       7. *-commutative 87.7%

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

   11. Applied egg-rr 87.7%

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

       1. *-commutative 87.7%

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

       2. *-commutative 87.7%

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

       3. associate-*l* 88.4%

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

       4. associate-*r* 88.6%

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

       5. *-commutative 88.6%

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

       6. *-commutative 88.6%

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

       7. *-commutative 88.6%

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

   13. Simplified 88.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 10^{-42}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s\right) \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(s_m \cdot c_m\right)\\ \mathbf{if}\;x \leq 9.5 \cdot 10^{-115}:\\ \;\;\;\;{\left(\left(x \cdot s_m\right) \cdot c_m\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* x (* s_m c_m))))
   (if (<= x 9.5e-115)
     (pow (* (* x s_m) c_m) -2.0)
     (/ (cos (* 2.0 x)) (* t_0 t_0)))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = x * (s_m * c_m);
	double tmp;
	if (x <= 9.5e-115) {
		tmp = pow(((x * s_m) * c_m), -2.0);
	} else {
		tmp = cos((2.0 * x)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s_m * c_m)
    if (x <= 9.5d-115) then
        tmp = ((x * s_m) * c_m) ** (-2.0d0)
    else
        tmp = cos((2.0d0 * x)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = x * (s_m * c_m);
	double tmp;
	if (x <= 9.5e-115) {
		tmp = Math.pow(((x * s_m) * c_m), -2.0);
	} else {
		tmp = Math.cos((2.0 * x)) / (t_0 * t_0);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = x * (s_m * c_m)
	tmp = 0
	if x <= 9.5e-115:
		tmp = math.pow(((x * s_m) * c_m), -2.0)
	else:
		tmp = math.cos((2.0 * x)) / (t_0 * t_0)
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(x * Float64(s_m * c_m))
	tmp = 0.0
	if (x <= 9.5e-115)
		tmp = Float64(Float64(x * s_m) * c_m) ^ -2.0;
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = x * (s_m * c_m);
	tmp = 0.0;
	if (x <= 9.5e-115)
		tmp = ((x * s_m) * c_m) ^ -2.0;
	else
		tmp = cos((2.0 * x)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9.5e-115], N[Power[N[(N[(x * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], -2.0], $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.4999999999999996e-115

   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 52.3%

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

      1. unpow2 52.3%

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

      2. rem-square-sqrt 52.3%

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

      3. swap-sqr 58.9%

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

      4. unpow2 58.9%

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

      5. unpow2 58.9%

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

      6. unpow2 58.9%

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

      7. unswap-sqr 79.6%

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

      8. rem-sqrt-square 83.9%

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

   5. Simplified 83.9%

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

      1. pow1 83.9%

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

      2. metadata-eval 83.9%

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

      3. sqrt-pow1 83.9%

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

      4. pow2 83.9%

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

      5. add-sqr-sqrt 83.9%

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

      6. expm1-log1p-u 83.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)\right)} \]

      7. expm1-udef 75.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)} - 1} \]

      8. pow-flip 75.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left|s \cdot x\right|\right)}^{\left(-2\right)}}\right)} - 1 \]

      9. add-sqr-sqrt 49.4%

         \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \left|\color{blue}{\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}}\right|\right)}^{\left(-2\right)}\right)} - 1 \]

      10. fabs-sqr 49.4%

          \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

      11. add-sqr-sqrt 75.6%

          \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

      12. associate-*r* 78.2%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{\left(-2\right)}\right)} - 1 \]

      13. metadata-eval 78.2%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{-2}}\right)} - 1 \]

   7. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\left(c \cdot s\right) \cdot x\right)}^{-2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 82.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(c \cdot s\right) \cdot x\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 83.9%

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

      3. associate-*l* 83.9%

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

   9. Simplified 83.9%

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

   if 9.4999999999999996e-115 < x

   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 c around 0 62.4%

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

      1. unpow2 62.4%

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

      2. rem-square-sqrt 62.4%

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

      3. swap-sqr 66.6%

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

      4. unpow2 66.6%

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

      5. unpow2 66.6%

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

      6. unpow2 66.6%

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

      7. unswap-sqr 82.8%

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

      8. rem-sqrt-square 99.6%

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

   5. Simplified 99.6%

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

      1. pow1 77.2%

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

      2. metadata-eval 77.2%

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

      3. sqrt-pow1 77.2%

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

      4. pow2 77.2%

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

      5. add-sqr-sqrt 77.2%

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

      6. unpow2 77.2%

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

      7. add-sqr-sqrt 37.5%

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

      8. fabs-sqr 37.5%

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

      9. add-sqr-sqrt 65.5%

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

      10. associate-*r* 65.1%

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

      11. add-sqr-sqrt 37.2%

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

      12. fabs-sqr 37.2%

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

      13. add-sqr-sqrt 76.9%

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

      14. associate-*r* 76.9%

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

   7. Applied egg-rr 97.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-115}:\\ \;\;\;\;{\left(\left(x \cdot s\right) \cdot c\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in c around 0 58.2%

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

   1. unpow2 58.2%

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

   2. rem-square-sqrt 58.2%

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

   3. swap-sqr 65.1%

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

   4. unpow2 65.1%

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

   5. unpow2 65.1%

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

   6. unpow2 65.1%

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

   7. unswap-sqr 87.3%

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

   8. rem-sqrt-square 96.9%

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

5. Simplified 96.9%

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

6. Taylor expanded in c around 0 76.7%

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

   1. unpow2 76.7%

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

   2. unpow2 76.7%

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

   3. sqr-abs 76.7%

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

   4. swap-sqr 96.9%

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

   5. associate-*l* 94.3%

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

   6. associate-*l* 96.6%

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

   7. unpow2 96.6%

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

   8. associate-*l* 96.9%

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

8. Simplified 96.9%

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

   1. div-inv 96.9%

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

   2. associate-*r* 96.6%

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

   3. pow2 96.6%

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

   4. associate-/r* 96.9%

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

   5. associate-*r/ 96.8%

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

   6. associate-*r* 94.5%

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

   7. *-commutative 94.5%

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

   8. times-frac 90.9%

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

   9. *-commutative 90.9%

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

   10. associate-*r* 92.0%

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

   11. *-commutative 92.0%

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

   12. associate-*l* 91.7%

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

10. Applied egg-rr 91.7%

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

    1. *-commutative 91.7%

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

    2. associate-/r/ 95.3%

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

    3. associate-/l/ 95.2%

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

    4. associate-/r* 95.3%

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

    5. associate-/r/ 95.4%

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

    6. associate-*l/ 95.4%

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

    7. *-lft-identity 95.4%

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

    8. *-commutative 95.4%

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

    9. associate-*r* 96.7%

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

    10. *-commutative 96.7%

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

12. Simplified 96.7%

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

13. Final simplification 96.7%

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

Alternative 4: 79.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in x around 0 54.2%

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

   1. unpow2 54.2%

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

   2. rem-square-sqrt 54.2%

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

   3. swap-sqr 60.1%

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

   4. unpow2 60.1%

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

   5. unpow2 60.1%

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

   6. unpow2 60.1%

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

   7. unswap-sqr 77.1%

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

   8. rem-sqrt-square 82.0%

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

5. Simplified 82.0%

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

   1. pow1 82.0%

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

   2. metadata-eval 82.0%

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

   3. sqrt-pow1 82.0%

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

   4. pow2 82.0%

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

   5. add-sqr-sqrt 82.0%

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

   6. expm1-log1p-u 81.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)\right)} \]

   7. expm1-udef 73.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(c \cdot \left|s \cdot x\right|\right)}^{2}}\right)} - 1} \]

   8. pow-flip 73.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot \left|s \cdot x\right|\right)}^{\left(-2\right)}}\right)} - 1 \]

   9. add-sqr-sqrt 44.8%

      \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \left|\color{blue}{\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}}\right|\right)}^{\left(-2\right)}\right)} - 1 \]

   10. fabs-sqr 44.8%

       \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(\sqrt{s \cdot x} \cdot \sqrt{s \cdot x}\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

   11. add-sqr-sqrt 73.5%

       \[\leadsto e^{\mathsf{log1p}\left({\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)}^{\left(-2\right)}\right)} - 1 \]

   12. associate-*r* 75.3%

       \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{\left(-2\right)}\right)} - 1 \]

   13. metadata-eval 75.3%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\left(c \cdot s\right) \cdot x\right)}^{\color{blue}{-2}}\right)} - 1 \]

7. Applied egg-rr 75.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\left(c \cdot s\right) \cdot x\right)}^{-2}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 81.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(c \cdot s\right) \cdot x\right)}^{-2}\right)\right)} \]

   2. expm1-log1p 81.9%

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

   3. associate-*l* 82.0%

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

9. Simplified 82.0%

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

10. Final simplification 82.0%

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

Alternative 5: 79.4% accurate, 17.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := x \cdot \left(s_m \cdot c_m\right)\\ \mathbf{if}\;c_m \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s_m\right) \cdot c_m\right) \cdot \left(s_m \cdot \left(x \cdot c_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* x (* s_m c_m))))
   (if (<= c_m 2e-46)
     (/ 1.0 (* t_0 t_0))
     (/ 1.0 (* (* (* x s_m) c_m) (* s_m (* x c_m)))))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = x * (s_m * c_m);
	double tmp;
	if (c_m <= 2e-46) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)));
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (s_m * c_m)
    if (c_m <= 2d-46) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = 1.0d0 / (((x * s_m) * c_m) * (s_m * (x * c_m)))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = x * (s_m * c_m);
	double tmp;
	if (c_m <= 2e-46) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = x * (s_m * c_m)
	tmp = 0
	if c_m <= 2e-46:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)))
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(x * Float64(s_m * c_m))
	tmp = 0.0
	if (c_m <= 2e-46)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x * s_m) * c_m) * Float64(s_m * Float64(x * c_m))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = x * (s_m * c_m);
	tmp = 0.0;
	if (c_m <= 2e-46)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = 1.0 / (((x * s_m) * c_m) * (s_m * (x * c_m)));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(x * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 2e-46], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.00000000000000005e-46

   1. Initial program 66.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. 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. unpow2 54.0%

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

      2. rem-square-sqrt 54.0%

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

      3. swap-sqr 60.4%

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

      4. unpow2 60.4%

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

      5. unpow2 60.4%

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

      6. unpow2 60.4%

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

      7. unswap-sqr 78.3%

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

      8. rem-sqrt-square 82.2%

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

   5. Simplified 82.2%

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

      1. pow1 82.2%

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

      2. metadata-eval 82.2%

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

      3. sqrt-pow1 82.2%

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

      4. pow2 82.2%

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

      5. add-sqr-sqrt 82.2%

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

      6. unpow2 82.2%

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

      7. add-sqr-sqrt 51.8%

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

      8. fabs-sqr 51.8%

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

      9. add-sqr-sqrt 62.4%

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

      10. associate-*r* 61.9%

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

      11. add-sqr-sqrt 46.5%

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

      12. fabs-sqr 46.5%

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

      13. add-sqr-sqrt 80.1%

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

      14. associate-*r* 80.7%

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

   7. Applied egg-rr 80.7%

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

   if 2.00000000000000005e-46 < c

   1. Initial program 63.0%

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

   3. Taylor expanded in x around 0 54.8%

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

      1. unpow2 54.8%

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

      2. rem-square-sqrt 54.8%

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

      3. swap-sqr 59.3%

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

      4. unpow2 59.3%

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

      5. unpow2 59.3%

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

      6. unpow2 59.3%

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

      7. unswap-sqr 73.6%

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

      8. rem-sqrt-square 81.4%

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

   5. Simplified 81.4%

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

      1. pow1 81.4%

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

      2. metadata-eval 81.4%

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

      3. sqrt-pow1 81.4%

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

      4. pow2 81.4%

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

      5. add-sqr-sqrt 81.4%

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

      6. unpow2 81.4%

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

      7. add-sqr-sqrt 36.9%

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

      8. fabs-sqr 36.9%

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

      9. add-sqr-sqrt 58.6%

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

      10. associate-*r* 58.7%

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

      11. add-sqr-sqrt 27.2%

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

      12. fabs-sqr 27.2%

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

      13. add-sqr-sqrt 78.7%

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

      14. associate-*r* 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. pow2 85.6%

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

      2. associate-*r* 81.4%

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

      3. *-commutative 81.4%

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

      4. associate-*l* 88.5%

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

   9. Applied egg-rr 88.5%

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

       1. unpow2 88.5%

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

       2. associate-*r* 81.6%

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

       3. *-commutative 81.6%

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

       4. associate-*l* 81.6%

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

       5. associate-*r* 85.7%

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

       6. *-commutative 85.7%

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

       7. *-commutative 85.7%

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

   11. Applied egg-rr 85.7%

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

       1. *-commutative 85.7%

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

       2. *-commutative 85.7%

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

       3. associate-*l* 85.6%

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

       4. associate-*r* 81.6%

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

       5. *-commutative 81.6%

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

       6. *-commutative 81.6%

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

       7. *-commutative 81.6%

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

   13. Simplified 81.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s\right) \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in x around 0 54.2%

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

   1. unpow2 54.2%

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

   2. rem-square-sqrt 54.2%

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

   3. swap-sqr 60.1%

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

   4. unpow2 60.1%

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

   5. unpow2 60.1%

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

   6. unpow2 60.1%

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

   7. unswap-sqr 77.1%

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

   8. rem-sqrt-square 82.0%

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

5. Simplified 82.0%

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

   1. pow1 82.0%

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

   2. metadata-eval 82.0%

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

   3. sqrt-pow1 82.0%

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

   4. pow2 82.0%

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

   5. add-sqr-sqrt 82.0%

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

   6. unpow2 82.0%

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

   7. add-sqr-sqrt 48.1%

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

   8. fabs-sqr 48.1%

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

   9. add-sqr-sqrt 61.5%

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

   10. associate-*r* 61.1%

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

   11. add-sqr-sqrt 41.7%

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

   12. fabs-sqr 41.7%

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

   13. add-sqr-sqrt 79.8%

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

   14. associate-*r* 81.9%

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

7. Applied egg-rr 81.9%

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

   1. associate-/r* 81.8%

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

   2. *-un-lft-identity 81.8%

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

   3. associate-*r* 79.8%

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

   4. times-frac 78.2%

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

   5. associate-*r* 80.4%

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

   6. *-commutative 80.4%

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

   7. associate-*l* 79.4%

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

9. Applied egg-rr 79.4%

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

    1. associate-/l/ 79.3%

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

    2. associate-*r* 80.4%

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

    3. *-commutative 80.4%

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

11. Simplified 80.4%

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

12. Final simplification 80.4%

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

Alternative 7: 77.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in x around 0 54.2%

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

   1. unpow2 54.2%

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

   2. rem-square-sqrt 54.2%

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

   3. swap-sqr 60.1%

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

   4. unpow2 60.1%

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

   5. unpow2 60.1%

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

   6. unpow2 60.1%

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

   7. unswap-sqr 77.1%

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

   8. rem-sqrt-square 82.0%

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

5. Simplified 82.0%

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

   1. pow1 82.0%

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

   2. metadata-eval 82.0%

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

   3. sqrt-pow1 82.0%

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

   4. pow2 82.0%

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

   5. add-sqr-sqrt 82.0%

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

   6. unpow2 82.0%

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

   7. add-sqr-sqrt 48.1%

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

   8. fabs-sqr 48.1%

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

   9. add-sqr-sqrt 61.5%

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

   10. associate-*r* 61.1%

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

   11. add-sqr-sqrt 41.7%

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

   12. fabs-sqr 41.7%

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

   13. add-sqr-sqrt 79.8%

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

   14. associate-*r* 81.9%

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

7. Applied egg-rr 81.9%

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

   1. pow2 81.9%

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

   2. associate-*r* 82.0%

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

   3. *-commutative 82.0%

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

   4. associate-*l* 83.1%

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

9. Applied egg-rr 83.1%

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

    1. unpow2 83.1%

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

    2. associate-*r* 81.0%

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

    3. *-commutative 81.0%

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

    4. associate-*l* 79.2%

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

    5. associate-*r* 79.1%

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

    6. *-commutative 79.1%

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

    7. *-commutative 79.1%

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

11. Applied egg-rr 79.1%

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

    1. *-commutative 79.1%

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

    2. *-commutative 79.1%

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

    3. associate-*l* 80.8%

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

    4. associate-*r* 81.0%

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

    5. *-commutative 81.0%

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

    6. *-commutative 81.0%

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

    7. *-commutative 81.0%

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

13. Simplified 81.0%

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

14. Final simplification 81.0%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))