mixedcos

Percentage Accurate: 66.4% → 99.4%

Time: 12.8s

Alternatives: 11

Speedup: 24.1×

• 31.5% 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: 66.4% 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: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ t_1 := \cos \left(x\_m \cdot 2\right)\\ \mathbf{if}\;x\_m \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{s\_m}}{c\_m} \cdot \frac{t\_1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* s_m (* x_m c_m))) (t_1 (cos (* x_m 2.0))))
   (if (<= x_m 4.8e+27)
     (* (/ (/ (/ 1.0 x_m) s_m) c_m) (/ t_1 (* c_m (* x_m s_m))))
     (/ t_1 (* 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 = cos((x_m * 2.0));
	double tmp;
	if (x_m <= 4.8e+27) {
		tmp = (((1.0 / x_m) / s_m) / c_m) * (t_1 / (c_m * (x_m * s_m)));
	} else {
		tmp = t_1 / (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 = cos((x_m * 2.0d0))
    if (x_m <= 4.8d+27) then
        tmp = (((1.0d0 / x_m) / s_m) / c_m) * (t_1 / (c_m * (x_m * s_m)))
    else
        tmp = t_1 / (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 = Math.cos((x_m * 2.0));
	double tmp;
	if (x_m <= 4.8e+27) {
		tmp = (((1.0 / x_m) / s_m) / c_m) * (t_1 / (c_m * (x_m * s_m)));
	} else {
		tmp = t_1 / (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 = math.cos((x_m * 2.0))
	tmp = 0
	if x_m <= 4.8e+27:
		tmp = (((1.0 / x_m) / s_m) / c_m) * (t_1 / (c_m * (x_m * s_m)))
	else:
		tmp = t_1 / (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 = cos(Float64(x_m * 2.0))
	tmp = 0.0
	if (x_m <= 4.8e+27)
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / s_m) / c_m) * Float64(t_1 / Float64(c_m * Float64(x_m * s_m))));
	else
		tmp = Float64(t_1 / Float64(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 = cos((x_m * 2.0));
	tmp = 0.0;
	if (x_m <= 4.8e+27)
		tmp = (((1.0 / x_m) / s_m) / c_m) * (t_1 / (c_m * (x_m * s_m)));
	else
		tmp = t_1 / (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[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 4.8e+27], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(t$95$1 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999995e27

   1. Initial program 60.7%

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

      1. associate-/r* 60.7%

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

      2. *-commutative 60.7%

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

      3. unpow2 60.7%

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

      4. sqr-neg 60.7%

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

      5. unpow2 60.7%

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

      6. cos-neg 60.7%

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

      7. *-commutative 60.7%

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

      8. distribute-rgt-neg-in 60.7%

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

      9. metadata-eval 60.7%

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

      10. unpow2 60.7%

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

      11. sqr-neg 60.7%

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

      12. unpow2 60.7%

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

      13. associate-*r* 53.9%

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

      14. unpow2 53.9%

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

      15. *-commutative 53.9%

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

   3. Simplified 53.9%

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

      1. associate-/l/ 54.0%

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

      2. div-inv 53.9%

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

      3. add-sqr-sqrt 29.5%

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

      4. sqrt-unprod 46.3%

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

      5. *-commutative 46.3%

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

      6. *-commutative 46.3%

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

      7. swap-sqr 46.3%

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

      8. metadata-eval 46.3%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

      9. metadata-eval 46.3%

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

      10. swap-sqr 46.3%

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

      11. sqrt-unprod 23.0%

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

      12. add-sqr-sqrt 53.9%

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

      13. *-commutative 53.9%

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

      14. unpow2 53.9%

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

      15. associate-*r* 60.7%

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

      16. *-commutative 60.7%

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

      17. *-commutative 60.7%

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

      18. add-sqr-sqrt 60.7%

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

      19. pow2 60.7%

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

   6. Applied egg-rr 96.9%

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

      1. associate-*r/ 96.9%

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

      2. *-rgt-identity 96.9%

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

      3. *-commutative 96.9%

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

   8. Simplified 96.9%

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

      1. *-commutative 96.9%

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

      2. unpow-prod-down 71.1%

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

      3. associate-/l/ 71.2%

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

      4. *-rgt-identity 71.2%

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

      5. pow2 71.2%

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

      6. frac-times 71.5%

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

      7. unpow2 71.5%

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

      8. *-commutative 71.5%

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

      9. times-frac 96.7%

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

      10. associate-/r* 96.8%

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

      11. *-commutative 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in x around inf 96.8%

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

   if 4.79999999999999995e27 < x

   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. Step-by-step derivation

      1. associate-/r* 65.0%

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

      2. *-commutative 65.0%

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

      3. unpow2 65.0%

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

      4. sqr-neg 65.0%

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

      5. unpow2 65.0%

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

      6. cos-neg 65.0%

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

      7. *-commutative 65.0%

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

      8. distribute-rgt-neg-in 65.0%

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

      9. metadata-eval 65.0%

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

      10. unpow2 65.0%

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

      11. sqr-neg 65.0%

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

      12. unpow2 65.0%

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

      13. associate-*r* 63.0%

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

      14. unpow2 63.0%

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

      15. *-commutative 63.0%

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

   3. Simplified 63.0%

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

      1. associate-/l/ 63.0%

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

      2. div-inv 63.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 39.2%

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

      5. *-commutative 39.2%

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

      6. *-commutative 39.2%

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

      7. swap-sqr 39.2%

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

      8. metadata-eval 39.2%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

      9. metadata-eval 39.2%

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

      10. swap-sqr 39.2%

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

      11. sqrt-unprod 61.6%

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

      12. add-sqr-sqrt 63.0%

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

      13. *-commutative 63.0%

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

      14. unpow2 63.0%

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

      15. associate-*r* 64.9%

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

      16. *-commutative 64.9%

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

      17. *-commutative 64.9%

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

      18. add-sqr-sqrt 64.9%

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

      19. pow2 64.9%

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

   6. Applied egg-rr 96.4%

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

      1. associate-*r/ 96.6%

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

      2. *-rgt-identity 96.6%

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

      3. *-commutative 96.6%

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

   8. Simplified 96.6%

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

      1. unpow2 96.6%

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

      2. *-commutative 96.6%

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

      3. *-commutative 96.6%

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

      4. *-commutative 96.6%

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

      5. associate-*l* 91.9%

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

      6. *-commutative 91.9%

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

      7. associate-*l* 94.9%

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

   10. Applied egg-rr 94.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s}}{c} \cdot \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(x\_m \cdot c\_m\right)\\ \mathbf{if}\;x\_m \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{s\_m}}{c\_m} \cdot \frac{1}{c\_m \cdot \left(x\_m \cdot s\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot 2\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* s_m (* x_m c_m))))
   (if (<= x_m 3.6e-16)
     (* (/ (/ (/ 1.0 x_m) s_m) c_m) (/ 1.0 (* 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 = s_m * (x_m * c_m);
	double tmp;
	if (x_m <= 3.6e-16) {
		tmp = (((1.0 / x_m) / s_m) / c_m) * (1.0 / (c_m * (x_m * s_m)));
	} 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) :: tmp
    t_0 = s_m * (x_m * c_m)
    if (x_m <= 3.6d-16) then
        tmp = (((1.0d0 / x_m) / s_m) / c_m) * (1.0d0 / (c_m * (x_m * s_m)))
    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 tmp;
	if (x_m <= 3.6e-16) {
		tmp = (((1.0 / x_m) / s_m) / c_m) * (1.0 / (c_m * (x_m * s_m)));
	} 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)
	tmp = 0
	if x_m <= 3.6e-16:
		tmp = (((1.0 / x_m) / s_m) / c_m) * (1.0 / (c_m * (x_m * s_m)))
	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))
	tmp = 0.0
	if (x_m <= 3.6e-16)
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / s_m) / c_m) * Float64(1.0 / Float64(c_m * Float64(x_m * s_m))));
	else
		tmp = Float64(cos(Float64(x_m * 2.0)) / Float64(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);
	tmp = 0.0;
	if (x_m <= 3.6e-16)
		tmp = (((1.0 / x_m) / s_m) / c_m) * (1.0 / (c_m * (x_m * s_m)));
	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]}, If[LessEqual[x$95$m, 3.6e-16], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] * N[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.59999999999999983e-16

   1. Initial program 59.5%

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

      1. associate-/r* 59.6%

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

      2. *-commutative 59.6%

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

      3. unpow2 59.6%

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

      4. sqr-neg 59.6%

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

      5. unpow2 59.6%

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

      6. cos-neg 59.6%

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

      7. *-commutative 59.6%

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

      8. distribute-rgt-neg-in 59.6%

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

      9. metadata-eval 59.6%

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

      10. unpow2 59.6%

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

      11. sqr-neg 59.6%

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

      12. unpow2 59.6%

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

      13. associate-*r* 52.3%

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

      14. unpow2 52.3%

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

      15. *-commutative 52.3%

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

   3. Simplified 52.3%

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

      1. associate-/l/ 52.3%

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

      2. div-inv 52.3%

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

      3. add-sqr-sqrt 31.6%

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

      4. sqrt-unprod 44.1%

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

      5. *-commutative 44.1%

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

      6. *-commutative 44.1%

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

      7. swap-sqr 44.1%

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

      8. metadata-eval 44.1%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

      9. metadata-eval 44.1%

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

      10. swap-sqr 44.1%

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

      11. sqrt-unprod 19.2%

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

      12. add-sqr-sqrt 52.3%

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

      13. *-commutative 52.3%

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

      14. unpow2 52.3%

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

      15. associate-*r* 59.6%

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

      16. *-commutative 59.6%

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

      17. *-commutative 59.6%

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

      18. add-sqr-sqrt 59.5%

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

      19. pow2 59.5%

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

   6. Applied egg-rr 96.7%

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

      1. associate-*r/ 96.7%

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

      2. *-rgt-identity 96.7%

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

      3. *-commutative 96.7%

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

   8. Simplified 96.7%

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

      1. *-commutative 96.7%

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

      2. unpow-prod-down 70.7%

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

      3. associate-/l/ 70.7%

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

      4. *-rgt-identity 70.7%

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

      5. pow2 70.7%

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

      6. frac-times 71.0%

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

      7. unpow2 71.0%

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

      8. *-commutative 71.0%

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

      9. times-frac 96.5%

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

      10. associate-/r* 96.6%

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

      11. *-commutative 96.6%

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

   10. Applied egg-rr 96.6%

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

   11. Taylor expanded in x around inf 96.6%

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

   12. Taylor expanded in x around 0 84.1%

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

   if 3.59999999999999983e-16 < x

   1. Initial program 67.3%

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

      1. associate-/r* 67.2%

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

      2. *-commutative 67.2%

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

      3. unpow2 67.2%

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

      4. sqr-neg 67.2%

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

      5. unpow2 67.2%

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

      6. cos-neg 67.2%

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

      7. *-commutative 67.2%

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

      8. distribute-rgt-neg-in 67.2%

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

      9. metadata-eval 67.2%

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

      10. unpow2 67.2%

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

      11. sqr-neg 67.2%

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

      12. unpow2 67.2%

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

      13. associate-*r* 65.6%

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

      14. unpow2 65.6%

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

      15. *-commutative 65.6%

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

   3. Simplified 65.6%

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

      1. associate-/l/ 65.6%

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

      2. div-inv 65.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 46.1%

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

      5. *-commutative 46.1%

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

      6. *-commutative 46.1%

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

      7. swap-sqr 46.1%

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

      8. metadata-eval 46.1%

         \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

      9. metadata-eval 46.1%

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

      10. swap-sqr 46.1%

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

      11. sqrt-unprod 64.5%

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

      12. add-sqr-sqrt 65.6%

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

      13. *-commutative 65.6%

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

      14. unpow2 65.6%

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

      15. associate-*r* 67.2%

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

      16. *-commutative 67.2%

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

      17. *-commutative 67.2%

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

      18. add-sqr-sqrt 67.2%

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

      19. pow2 67.2%

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

   6. Applied egg-rr 97.0%

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

      1. associate-*r/ 97.1%

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

      2. *-rgt-identity 97.1%

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

      3. *-commutative 97.1%

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

   8. Simplified 97.1%

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

      1. unpow2 97.1%

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

      2. *-commutative 97.1%

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

      3. *-commutative 97.1%

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

      4. *-commutative 97.1%

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

      5. associate-*l* 93.3%

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

      6. *-commutative 93.3%

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

      7. associate-*l* 95.7%

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

   10. Applied egg-rr 95.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{s}}{c} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

   1. associate-/l/ 56.0%

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

   2. div-inv 56.0%

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

   3. add-sqr-sqrt 22.7%

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

   4. sqrt-unprod 44.7%

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

   5. *-commutative 44.7%

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

   6. *-commutative 44.7%

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

   7. swap-sqr 44.7%

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

   8. metadata-eval 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   9. metadata-eval 44.7%

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

   10. swap-sqr 44.7%

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

   11. sqrt-unprod 31.9%

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

   12. add-sqr-sqrt 56.0%

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

   13. *-commutative 56.0%

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

   14. unpow2 56.0%

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

   15. associate-*r* 61.7%

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

   16. *-commutative 61.7%

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

   17. *-commutative 61.7%

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

   18. add-sqr-sqrt 61.7%

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

   19. pow2 61.7%

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

6. Applied egg-rr 96.8%

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

   1. associate-*r/ 96.8%

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

   2. *-rgt-identity 96.8%

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

   3. *-commutative 96.8%

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

8. Simplified 96.8%

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

   1. *-commutative 96.8%

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

   2. unpow-prod-down 71.6%

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

   3. associate-/l/ 71.6%

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

   4. *-rgt-identity 71.6%

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

   5. pow2 71.6%

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

   6. frac-times 71.9%

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

   7. unpow2 71.9%

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

   8. *-commutative 71.9%

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

   9. times-frac 96.7%

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

   10. associate-/r* 96.7%

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

   11. *-commutative 96.7%

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

10. Applied egg-rr 96.7%

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

11. Taylor expanded in x around inf 96.8%

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

12. Taylor expanded in x around 0 78.5%

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

13. Final simplification 78.5%

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

Alternative 4: 79.9% accurate, 20.9× speedup

Math

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

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 (/ 1.0 (* c_m (* x_m s_m))))) (* 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 = 1.0 / (c_m * (x_m * s_m));
	return 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 = 1.0d0 / (c_m * (x_m * s_m))
    code = 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 = 1.0 / (c_m * (x_m * s_m));
	return 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 = 1.0 / (c_m * (x_m * s_m))
	return 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(1.0 / Float64(c_m * Float64(x_m * s_m)))
	return 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 = 1.0 / (c_m * (x_m * s_m));
	tmp = 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[(1.0 / N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

   1. associate-/l/ 56.0%

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

   2. div-inv 56.0%

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

   3. add-sqr-sqrt 22.7%

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

   4. sqrt-unprod 44.7%

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

   5. *-commutative 44.7%

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

   6. *-commutative 44.7%

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

   7. swap-sqr 44.7%

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

   8. metadata-eval 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   9. metadata-eval 44.7%

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

   10. swap-sqr 44.7%

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

   11. sqrt-unprod 31.9%

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

   12. add-sqr-sqrt 56.0%

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

   13. *-commutative 56.0%

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

   14. unpow2 56.0%

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

   15. associate-*r* 61.7%

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

   16. *-commutative 61.7%

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

   17. *-commutative 61.7%

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

   18. add-sqr-sqrt 61.7%

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

   19. pow2 61.7%

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

6. Applied egg-rr 96.8%

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

   1. associate-*r/ 96.8%

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

   2. *-rgt-identity 96.8%

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

   3. *-commutative 96.8%

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

8. Simplified 96.8%

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

   1. unpow2 96.8%

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

   2. associate-*r* 94.2%

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

   3. associate-*l* 91.8%

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

   4. *-commutative 91.8%

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

   5. *-commutative 91.8%

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

   6. *-commutative 91.8%

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

   7. associate-*l* 93.5%

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

10. Applied egg-rr 93.5%

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

11. Taylor expanded in x around 0 76.4%

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

    1. inv-pow 76.4%

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

    2. associate-*r* 78.3%

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

    3. *-commutative 78.3%

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

    4. *-commutative 78.3%

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

    5. associate-*l* 76.5%

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

    6. *-commutative 76.5%

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

    7. associate-*l* 78.1%

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

    8. unpow-prod-down 78.1%

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

    9. inv-pow 78.1%

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

    10. *-commutative 78.1%

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

    11. associate-*r* 76.7%

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

    12. *-commutative 76.7%

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

    13. inv-pow 76.7%

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

    14. *-commutative 76.7%

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

    15. associate-*r* 78.5%

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

    16. *-commutative 78.5%

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

13. Applied egg-rr 78.5%

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

Alternative 5: 78.2% accurate, 22.4× speedup

Math

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

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 x_m) s_m) (* c_m (* c_m (* x_m (- s_m))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in x around 0 50.6%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 61.8%

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

   8. unpow2 61.8%

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

   9. unpow2 61.8%

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

   10. swap-sqr 78.5%

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

   11. unpow2 78.5%

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

   12. *-commutative 78.5%

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

   13. associate-*l* 78.1%

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

7. Simplified 78.1%

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

   1. pow-flip 78.1%

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

   2. associate-*r* 78.5%

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

   3. *-commutative 78.5%

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

   4. pow-flip 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sqrt-div 78.5%

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

   7. metadata-eval 78.5%

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

   8. sqrt-pow1 53.0%

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

   9. metadata-eval 53.0%

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

   10. pow1 53.0%

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

   11. *-commutative 53.0%

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

   12. *-commutative 53.0%

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

   13. associate-*l* 52.0%

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

   14. sqrt-div 52.0%

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

   15. metadata-eval 52.0%

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

   16. sqrt-pow1 76.9%

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

   17. metadata-eval 76.9%

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

   18. pow1 76.9%

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

   19. *-commutative 76.9%

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

   20. *-commutative 76.9%

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

   21. associate-*l* 78.3%

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

9. Applied egg-rr 78.3%

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

    1. metadata-eval 78.3%

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

    2. associate-*r* 76.9%

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

    3. *-commutative 76.9%

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

    4. frac-times 76.8%

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

    5. *-commutative 76.8%

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

    6. associate-/l/ 76.8%

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

    7. div-inv 76.8%

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

    8. clear-num 76.8%

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

    9. metadata-eval 76.8%

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

    10. associate-*r* 78.5%

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

    11. *-commutative 78.5%

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

    12. frac-times 78.5%

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

    13. *-commutative 78.5%

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

    14. associate-/l/ 78.5%

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

    15. div-inv 78.4%

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

    16. frac-2neg 78.4%

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

    17. frac-times 75.1%

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

11. Applied egg-rr 73.6%

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

12. Taylor expanded in s around 0 75.2%

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

13. Final simplification 75.2%

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

Alternative 6: 78.7% accurate, 24.1× speedup

Math

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

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 (* x_m (* s_m c_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 = x_m * (s_m * c_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 = x_m * (s_m * c_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 = x_m * (s_m * c_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 = x_m * (s_m * c_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(x_m * Float64(s_m * c_m))
	return Float64(Float64(1.0 / 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 = x_m * (s_m * c_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[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in x around 0 50.6%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 61.8%

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

   8. unpow2 61.8%

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

   9. unpow2 61.8%

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

   10. swap-sqr 78.5%

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

   11. unpow2 78.5%

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

   12. *-commutative 78.5%

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

   13. associate-*l* 78.1%

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

7. Simplified 78.1%

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

   1. pow-flip 78.1%

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

   2. associate-*r* 78.5%

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

   3. *-commutative 78.5%

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

   4. pow-flip 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sqrt-div 78.5%

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

   7. metadata-eval 78.5%

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

   8. sqrt-pow1 53.0%

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

   9. metadata-eval 53.0%

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

   10. pow1 53.0%

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

   11. *-commutative 53.0%

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

   12. *-commutative 53.0%

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

   13. associate-*l* 52.0%

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

   14. sqrt-div 52.0%

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

   15. metadata-eval 52.0%

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

   16. sqrt-pow1 76.9%

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

   17. metadata-eval 76.9%

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

   18. pow1 76.9%

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

   19. *-commutative 76.9%

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

   20. *-commutative 76.9%

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

   21. associate-*l* 78.3%

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

9. Applied egg-rr 78.3%

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

    1. un-div-inv 78.3%

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

    2. *-commutative 78.3%

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

    3. associate-*r* 76.5%

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

    4. *-commutative 76.5%

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

    5. associate-*r* 78.1%

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

11. Applied egg-rr 78.1%

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

12. Final simplification 78.1%

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

Alternative 7: 77.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in x around 0 50.6%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 61.8%

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

   8. unpow2 61.8%

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

   9. unpow2 61.8%

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

   10. swap-sqr 78.5%

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

   11. unpow2 78.5%

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

   12. *-commutative 78.5%

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

   13. associate-*l* 78.1%

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

7. Simplified 78.1%

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

   1. pow-flip 78.1%

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

   2. associate-*r* 78.5%

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

   3. *-commutative 78.5%

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

   4. pow-flip 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sqrt-div 78.5%

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

   7. metadata-eval 78.5%

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

   8. sqrt-pow1 53.0%

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

   9. metadata-eval 53.0%

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

   10. pow1 53.0%

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

   11. *-commutative 53.0%

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

   12. *-commutative 53.0%

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

   13. associate-*l* 52.0%

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

   14. sqrt-div 52.0%

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

   15. metadata-eval 52.0%

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

   16. sqrt-pow1 76.9%

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

   17. metadata-eval 76.9%

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

   18. pow1 76.9%

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

   19. *-commutative 76.9%

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

   20. *-commutative 76.9%

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

   21. associate-*l* 78.3%

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

9. Applied egg-rr 78.3%

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

    1. metadata-eval 78.3%

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

    2. associate-*r* 76.9%

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

    3. *-commutative 76.9%

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

    4. frac-times 76.8%

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

    5. *-commutative 76.8%

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

    6. associate-/l/ 76.8%

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

    7. div-inv 76.8%

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

    8. clear-num 76.8%

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

    9. metadata-eval 76.8%

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

    10. associate-*r* 78.5%

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

    11. *-commutative 78.5%

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

    12. frac-times 78.5%

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

    13. *-commutative 78.5%

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

    14. associate-/l/ 78.5%

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

    15. div-inv 78.4%

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

    16. frac-times 75.1%

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

    17. *-un-lft-identity 75.1%

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

    18. associate-/l/ 75.2%

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

    19. *-commutative 75.2%

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

11. Applied egg-rr 73.6%

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

12. Final simplification 73.6%

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

Alternative 8: 76.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in x around 0 50.6%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 61.8%

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

   8. unpow2 61.8%

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

   9. unpow2 61.8%

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

   10. swap-sqr 78.5%

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

   11. unpow2 78.5%

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

   12. *-commutative 78.5%

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

   13. associate-*l* 78.1%

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

7. Simplified 78.1%

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

   1. pow-flip 78.1%

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

   2. associate-*r* 78.5%

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

   3. *-commutative 78.5%

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

   4. pow-flip 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sqrt-div 78.5%

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

   7. metadata-eval 78.5%

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

   8. sqrt-pow1 53.0%

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

   9. metadata-eval 53.0%

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

   10. pow1 53.0%

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

   11. *-commutative 53.0%

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

   12. *-commutative 53.0%

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

   13. associate-*l* 52.0%

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

   14. sqrt-div 52.0%

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

   15. metadata-eval 52.0%

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

   16. sqrt-pow1 76.9%

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

   17. metadata-eval 76.9%

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

   18. pow1 76.9%

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

   19. *-commutative 76.9%

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

   20. *-commutative 76.9%

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

   21. associate-*l* 78.3%

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

9. Applied egg-rr 78.3%

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

    1. metadata-eval 78.3%

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

    2. associate-*r* 76.9%

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

    3. *-commutative 76.9%

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

    4. frac-times 76.8%

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

    5. *-commutative 76.8%

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

    6. associate-/l/ 76.8%

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

    7. div-inv 76.8%

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

    8. clear-num 76.8%

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

    9. metadata-eval 76.8%

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

    10. associate-*r* 78.5%

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

    11. *-commutative 78.5%

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

    12. frac-times 78.5%

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

    13. *-commutative 78.5%

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

    14. associate-/l/ 78.5%

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

    15. div-inv 78.4%

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

    16. associate-/l/ 76.6%

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

    17. frac-times 75.0%

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

    18. *-un-lft-identity 75.0%

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

11. Applied egg-rr 76.6%

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

12. Final simplification 76.6%

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

Alternative 9: 76.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

5. Taylor expanded in x around 0 50.6%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 61.9%

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

   6. unpow2 61.9%

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

   7. associate-/r* 61.8%

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

   8. unpow2 61.8%

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

   9. unpow2 61.8%

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

   10. swap-sqr 78.5%

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

   11. unpow2 78.5%

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

   12. *-commutative 78.5%

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

   13. associate-*l* 78.1%

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

7. Simplified 78.1%

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

   1. pow-flip 78.1%

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

   2. associate-*r* 78.5%

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

   3. *-commutative 78.5%

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

   4. pow-flip 78.5%

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

   5. add-sqr-sqrt 78.5%

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

   6. sqrt-div 78.5%

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

   7. metadata-eval 78.5%

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

   8. sqrt-pow1 53.0%

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

   9. metadata-eval 53.0%

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

   10. pow1 53.0%

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

   11. *-commutative 53.0%

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

   12. *-commutative 53.0%

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

   13. associate-*l* 52.0%

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

   14. sqrt-div 52.0%

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

   15. metadata-eval 52.0%

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

   16. sqrt-pow1 76.9%

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

   17. metadata-eval 76.9%

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

   18. pow1 76.9%

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

   19. *-commutative 76.9%

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

   20. *-commutative 76.9%

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

   21. associate-*l* 78.3%

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

9. Applied egg-rr 78.3%

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

    1. metadata-eval 78.3%

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

    2. associate-*r* 76.9%

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

    3. *-commutative 76.9%

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

    4. frac-times 76.8%

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

    5. *-commutative 76.8%

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

    6. associate-/l/ 76.8%

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

    7. div-inv 76.8%

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

    8. clear-num 76.8%

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

    9. metadata-eval 76.8%

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

    10. associate-*r* 78.5%

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

    11. *-commutative 78.5%

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

    12. frac-times 78.5%

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

    13. *-commutative 78.5%

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

    14. associate-/l/ 78.5%

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

    15. div-inv 78.4%

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

    16. associate-/l/ 76.6%

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

    17. frac-times 75.0%

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

    18. *-un-lft-identity 75.0%

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

11. Applied egg-rr 76.6%

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

    1. *-commutative 76.6%

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

    2. *-commutative 76.6%

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

    3. *-commutative 76.6%

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

    4. associate-*r* 75.2%

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

13. Simplified 75.2%

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

14. Final simplification 75.2%

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

Alternative 10: 74.6% accurate, 24.1× speedup

Math

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

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 (* (* x_m c_m) (* s_m (* s_m (* x_m c_m))))))

C

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

Fortran

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

Java

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

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
s_m = math.fabs(s)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 1.0 / ((x_m * c_m) * (s_m * (s_m * (x_m * c_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(x_m * c_m) * Float64(s_m * Float64(s_m * Float64(x_m * c_m)))))
end

MATLAB

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

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

   7. *-commutative 61.7%

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

   8. distribute-rgt-neg-in 61.7%

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

   9. metadata-eval 61.7%

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

   10. unpow2 61.7%

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

   11. sqr-neg 61.7%

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

   12. unpow2 61.7%

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

   13. associate-*r* 56.0%

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

   14. unpow2 56.0%

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

   15. *-commutative 56.0%

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

3. Simplified 56.0%

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

   1. associate-/l/ 56.0%

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

   2. div-inv 56.0%

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

   3. add-sqr-sqrt 22.7%

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

   4. sqrt-unprod 44.7%

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

   5. *-commutative 44.7%

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

   6. *-commutative 44.7%

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

   7. swap-sqr 44.7%

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

   8. metadata-eval 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   9. metadata-eval 44.7%

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

   10. swap-sqr 44.7%

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

   11. sqrt-unprod 31.9%

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

   12. add-sqr-sqrt 56.0%

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

   13. *-commutative 56.0%

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

   14. unpow2 56.0%

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

   15. associate-*r* 61.7%

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

   16. *-commutative 61.7%

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

   17. *-commutative 61.7%

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

   18. add-sqr-sqrt 61.7%

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

   19. pow2 61.7%

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

6. Applied egg-rr 96.8%

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

   1. associate-*r/ 96.8%

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

   2. *-rgt-identity 96.8%

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

   3. *-commutative 96.8%

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

8. Simplified 96.8%

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

   1. unpow2 96.8%

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

   2. associate-*r* 94.2%

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

   3. associate-*l* 91.8%

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

   4. *-commutative 91.8%

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

   5. *-commutative 91.8%

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

   6. *-commutative 91.8%

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

   7. associate-*l* 93.5%

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

10. Applied egg-rr 93.5%

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

11. Taylor expanded in x around 0 76.4%

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

Alternative 11: 74.5% accurate, 24.1× speedup

Math

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

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 (* (* x_m c_m) (* s_m (* c_m (* x_m s_m))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 61.7%

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

   2. *-commutative 61.7%

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

   3. unpow2 61.7%

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

   4. sqr-neg 61.7%

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

   5. unpow2 61.7%

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

   6. cos-neg 61.7%

      \[\leadsto \frac{\frac{\color{blue}{\cos \left(-2 \cdot x\right)}}{{\left(-c\right)}^{2}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   7. *-commutative 61.7%

      \[\leadsto \frac{\frac{\cos \left(-\color{blue}{x \cdot 2}\right)}{{\left(-c\right)}^{2}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   8. distribute-rgt-neg-in 61.7%

      \[\leadsto \frac{\frac{\cos \color{blue}{\left(x \cdot \left(-2\right)\right)}}{{\left(-c\right)}^{2}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   9. metadata-eval 61.7%

      \[\leadsto \frac{\frac{\cos \left(x \cdot \color{blue}{-2}\right)}{{\left(-c\right)}^{2}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   10. unpow2 61.7%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{\color{blue}{\left(-c\right) \cdot \left(-c\right)}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   11. sqr-neg 61.7%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{\color{blue}{c \cdot c}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   12. unpow2 61.7%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{\color{blue}{{c}^{2}}}}{x \cdot \left(x \cdot {s}^{2}\right)} \]

   13. associate-*r* 56.0%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{\left(x \cdot x\right) \cdot {s}^{2}}} \]

   14. unpow2 56.0%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{{x}^{2}} \cdot {s}^{2}} \]

   15. *-commutative 56.0%

       \[\leadsto \frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{\color{blue}{{s}^{2} \cdot {x}^{2}}} \]

3. Simplified 56.0%

   \[\leadsto \color{blue}{\frac{\frac{\cos \left(x \cdot -2\right)}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/l/ 56.0%

      \[\leadsto \color{blue}{\frac{\cos \left(x \cdot -2\right)}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}}} \]

   2. div-inv 56.0%

      \[\leadsto \color{blue}{\cos \left(x \cdot -2\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}}} \]

   3. add-sqr-sqrt 22.7%

      \[\leadsto \cos \color{blue}{\left(\sqrt{x \cdot -2} \cdot \sqrt{x \cdot -2}\right)} \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   4. sqrt-unprod 44.7%

      \[\leadsto \cos \color{blue}{\left(\sqrt{\left(x \cdot -2\right) \cdot \left(x \cdot -2\right)}\right)} \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   5. *-commutative 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{\left(-2 \cdot x\right)} \cdot \left(x \cdot -2\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   6. *-commutative 44.7%

      \[\leadsto \cos \left(\sqrt{\left(-2 \cdot x\right) \cdot \color{blue}{\left(-2 \cdot x\right)}}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   7. swap-sqr 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{\left(-2 \cdot -2\right) \cdot \left(x \cdot x\right)}}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   8. metadata-eval 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   9. metadata-eval 44.7%

      \[\leadsto \cos \left(\sqrt{\color{blue}{\left(2 \cdot 2\right)} \cdot \left(x \cdot x\right)}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   10. swap-sqr 44.7%

       \[\leadsto \cos \left(\sqrt{\color{blue}{\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}}\right) \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   11. sqrt-unprod 31.9%

       \[\leadsto \cos \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)} \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   12. add-sqr-sqrt 56.0%

       \[\leadsto \cos \color{blue}{\left(2 \cdot x\right)} \cdot \frac{1}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

   13. *-commutative 56.0%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\color{blue}{\left({x}^{2} \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   14. unpow2 56.0%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\left(\color{blue}{\left(x \cdot x\right)} \cdot {s}^{2}\right) \cdot {c}^{2}} \]

   15. associate-*r* 61.7%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

   16. *-commutative 61.7%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   17. *-commutative 61.7%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   18. add-sqr-sqrt 61.7%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\color{blue}{\sqrt{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot \sqrt{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}}} \]

   19. pow2 61.7%

       \[\leadsto \cos \left(2 \cdot x\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\right)}^{2}}} \]

6. Applied egg-rr 96.8%

   \[\leadsto \color{blue}{\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 96.8%

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right) \cdot 1}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]

   2. *-rgt-identity 96.8%

      \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}} \]

   3. *-commutative 96.8%

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}} \]

8. Simplified 96.8%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \]
9. Step-by-step derivation

   1. unpow2 96.8%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}} \]

   2. associate-*r* 94.2%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right)} \cdot \left(c \cdot \left(x \cdot s\right)\right)} \]

   3. associate-*l* 91.8%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(c \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}} \]

   4. *-commutative 91.8%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot c\right)} \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]

   5. *-commutative 91.8%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot c\right)}\right)} \]

   6. *-commutative 91.8%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot c\right)\right)} \]

   7. associate-*l* 93.5%

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot c\right) \cdot \left(s \cdot \color{blue}{\left(s \cdot \left(x \cdot c\right)\right)}\right)} \]

10. Applied egg-rr 93.5%

    \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}} \]

11. Taylor expanded in x around 0 76.4%

    \[\leadsto \frac{\color{blue}{1}}{\left(x \cdot c\right) \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)} \]

12. Taylor expanded in s around 0 75.3%

    \[\leadsto \frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

13. Final simplification 75.3%

    \[\leadsto \frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024153 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))