mixedcos

Percentage Accurate: 66.9% → 99.2%

Time: 13.8s

Alternatives: 9

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 81.9%

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

      1. *-un-lft-identity 81.9%

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

      2. add-sqr-sqrt 81.9%

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

      3. times-frac 81.9%

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

      4. sqrt-prod 81.9%

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

      5. sqrt-pow1 61.7%

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

      6. metadata-eval 61.7%

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

      7. pow1 61.7%

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

      8. *-commutative 61.7%

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

      9. associate-*r* 57.4%

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

      10. unpow2 57.4%

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

      11. pow-prod-down 61.7%

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

      12. sqrt-prod 61.7%

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

   4. Applied egg-rr 95.6%

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

      1. associate-*l/ 95.6%

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

      2. *-lft-identity 95.6%

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

      3. unpow2 95.6%

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

      4. rem-sqrt-square 95.6%

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

      5. unpow2 95.6%

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

      6. rem-sqrt-square 99.7%

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

   6. Simplified 99.7%

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

      1. associate-/r* 99.8%

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

      2. add-sqr-sqrt 55.4%

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

      3. fabs-sqr 55.4%

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

      4. add-sqr-sqrt 66.7%

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

      5. div-inv 66.6%

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

      6. *-commutative 66.6%

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

   8. Applied egg-rr 66.6%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. sqrt-prod 0.0%

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

      5. sqrt-pow1 0.0%

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

      6. metadata-eval 0.0%

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

      7. pow1 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r* 0.0%

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

      10. unpow2 0.0%

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

      11. pow-prod-down 0.0%

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

      12. sqrt-prod 0.0%

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

   4. Applied egg-rr 56.7%

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

      1. associate-*l/ 56.7%

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

      2. *-lft-identity 56.7%

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

      3. unpow2 56.7%

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

      4. rem-sqrt-square 56.7%

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

      5. unpow2 56.7%

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

      6. rem-sqrt-square 89.7%

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

   6. Simplified 89.7%

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

      1. div-inv 89.7%

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

      2. *-commutative 89.7%

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

      3. add-sqr-sqrt 53.3%

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

      4. fabs-sqr 53.3%

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

      5. add-sqr-sqrt 49.8%

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

      6. associate-*r* 49.9%

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

      7. add-sqr-sqrt 40.8%

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

      8. fabs-sqr 40.8%

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

      9. add-sqr-sqrt 90.0%

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

      10. associate-*r* 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 72.9%

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

Alternative 2: 97.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ s_m = \left|s\right| \\ [x_m, c, s_m] = \mathsf{sort}([x_m, c, s_m])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x\_m \cdot s\_m\right)\\ t_1 := x\_m \cdot \left(c \cdot s\_m\right)\\ \mathbf{if}\;c \leq 6 \cdot 10^{-245}:\\ \;\;\;\;\frac{\cos \left(x\_m \cdot -2\right)}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\_m\right)}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
s_m = (fabs.f64 s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c s_m)
 :precision binary64
 (let* ((t_0 (* c (* x_m s_m))) (t_1 (* x_m (* c s_m))))
   (if (<= c 6e-245)
     (/ (cos (* x_m -2.0)) (* t_1 t_1))
     (/ (/ (cos (* 2.0 x_m)) t_0) t_0))))

C

x_m = fabs(x);
s_m = fabs(s);
assert(x_m < c && c < s_m);
double code(double x_m, double c, double s_m) {
	double t_0 = c * (x_m * s_m);
	double t_1 = x_m * (c * s_m);
	double tmp;
	if (c <= 6e-245) {
		tmp = cos((x_m * -2.0)) / (t_1 * t_1);
	} else {
		tmp = (cos((2.0 * x_m)) / t_0) / t_0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
s_m = abs(s)
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x_m * s_m)
    t_1 = x_m * (c * s_m)
    if (c <= 6d-245) then
        tmp = cos((x_m * (-2.0d0))) / (t_1 * t_1)
    else
        tmp = (cos((2.0d0 * x_m)) / t_0) / t_0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
s_m = Math.abs(s);
assert x_m < c && c < s_m;
public static double code(double x_m, double c, double s_m) {
	double t_0 = c * (x_m * s_m);
	double t_1 = x_m * (c * s_m);
	double tmp;
	if (c <= 6e-245) {
		tmp = Math.cos((x_m * -2.0)) / (t_1 * t_1);
	} else {
		tmp = (Math.cos((2.0 * x_m)) / t_0) / t_0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
s_m = math.fabs(s)
[x_m, c, s_m] = sort([x_m, c, s_m])
def code(x_m, c, s_m):
	t_0 = c * (x_m * s_m)
	t_1 = x_m * (c * s_m)
	tmp = 0
	if c <= 6e-245:
		tmp = math.cos((x_m * -2.0)) / (t_1 * t_1)
	else:
		tmp = (math.cos((2.0 * x_m)) / t_0) / t_0
	return tmp

Julia

x_m = abs(x)
s_m = abs(s)
x_m, c, s_m = sort([x_m, c, s_m])
function code(x_m, c, s_m)
	t_0 = Float64(c * Float64(x_m * s_m))
	t_1 = Float64(x_m * Float64(c * s_m))
	tmp = 0.0
	if (c <= 6e-245)
		tmp = Float64(cos(Float64(x_m * -2.0)) / Float64(t_1 * t_1));
	else
		tmp = Float64(Float64(cos(Float64(2.0 * x_m)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
s_m = abs(s);
x_m, c, s_m = num2cell(sort([x_m, c, s_m])){:}
function tmp_2 = code(x_m, c, s_m)
	t_0 = c * (x_m * s_m);
	t_1 = x_m * (c * s_m);
	tmp = 0.0;
	if (c <= 6e-245)
		tmp = cos((x_m * -2.0)) / (t_1 * t_1);
	else
		tmp = (cos((2.0 * x_m)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x_m, c, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c_, s$95$m_] := Block[{t$95$0 = N[(c * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(c * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, 6e-245], N[(N[Cos[N[(x$95$m * -2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 6.0000000000000004e-245

   1. Initial program 67.0%

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

      1. associate-/l/ 67.0%

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

      2. remove-double-neg 67.0%

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

      3. distribute-frac-neg 67.0%

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

      4. distribute-neg-frac 67.0%

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

      5. remove-double-neg 67.0%

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

      6. *-commutative 67.0%

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

      7. associate-*r* 60.2%

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

      8. unpow2 60.2%

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

      9. associate-/r* 59.5%

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

      10. cos-neg 59.5%

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

      11. *-commutative 59.5%

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

      12. distribute-rgt-neg-in 59.5%

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

      13. metadata-eval 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in x around inf 60.2%

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

      1. associate-/r* 60.2%

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

      2. unpow2 60.2%

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

      3. unpow2 60.2%

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

      4. swap-sqr 75.1%

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

      5. unpow2 75.1%

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

      6. associate-/r* 75.1%

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

      7. *-commutative 75.1%

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

      8. unpow2 75.1%

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

      9. rem-square-sqrt 75.1%

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

      10. swap-sqr 84.8%

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

      11. unpow2 84.8%

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

      12. unpow2 84.8%

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

      13. rem-sqrt-square 96.8%

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

   7. Simplified 96.8%

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

      1. unpow-prod-down 75.1%

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

      2. unpow2 75.1%

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

      3. pow2 75.1%

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

      4. sqr-abs 75.1%

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

      5. swap-sqr 96.8%

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

      6. associate-*r* 95.5%

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

      7. associate-*r* 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 6.0000000000000004e-245 < c

   1. Initial program 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. add-sqr-sqrt 65.4%

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

      3. times-frac 65.4%

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

      4. sqrt-prod 65.4%

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

      5. sqrt-pow1 65.4%

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

      6. metadata-eval 65.4%

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

      7. pow1 65.4%

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

      8. *-commutative 65.4%

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

      9. associate-*r* 61.6%

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

      10. unpow2 61.6%

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

      11. pow-prod-down 65.4%

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

      12. sqrt-prod 65.5%

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

   4. Applied egg-rr 91.9%

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

      1. associate-*l/ 91.9%

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

      2. *-lft-identity 91.9%

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

      3. unpow2 91.9%

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

      4. rem-sqrt-square 91.9%

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

      5. unpow2 91.9%

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

      6. rem-sqrt-square 98.9%

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

   6. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. *-commutative 98.9%

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

      3. add-sqr-sqrt 48.3%

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

      4. fabs-sqr 48.3%

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

      5. add-sqr-sqrt 59.9%

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

      6. associate-*r* 59.2%

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

      7. add-sqr-sqrt 38.6%

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

      8. fabs-sqr 38.6%

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

      9. add-sqr-sqrt 95.3%

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

      10. associate-*r* 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. un-div-inv 96.0%

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

      2. associate-*l* 95.3%

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

      3. associate-*l* 98.9%

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

   10. Applied egg-rr 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 6 \cdot 10^{-245}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. *-un-lft-identity 66.2%

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

   2. add-sqr-sqrt 66.2%

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

   3. times-frac 66.2%

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

   4. sqrt-prod 66.2%

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

   5. sqrt-pow1 49.9%

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

   6. metadata-eval 49.9%

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

   7. pow1 49.9%

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

   8. *-commutative 49.9%

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

   9. associate-*r* 46.4%

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

   10. unpow2 46.4%

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

   11. pow-prod-down 49.9%

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

   12. sqrt-prod 49.9%

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

4. Applied egg-rr 88.1%

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

   1. associate-*l/ 88.2%

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

   2. *-lft-identity 88.2%

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

   3. unpow2 88.2%

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

   4. rem-sqrt-square 88.2%

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

   5. unpow2 88.2%

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

   6. rem-sqrt-square 97.8%

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

6. Simplified 97.8%

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

   1. div-inv 97.8%

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

   2. *-commutative 97.8%

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

   3. add-sqr-sqrt 55.0%

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

   4. fabs-sqr 55.0%

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

   5. add-sqr-sqrt 63.4%

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

   6. associate-*r* 62.4%

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

   7. add-sqr-sqrt 46.5%

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

   8. fabs-sqr 46.5%

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

   9. add-sqr-sqrt 95.4%

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

   10. associate-*r* 97.1%

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

8. Applied egg-rr 97.1%

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

   1. associate-/r* 97.2%

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

   2. div-inv 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-/r* 97.2%

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

10. Applied egg-rr 97.2%

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

11. Final simplification 97.2%

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

Alternative 4: 97.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. *-un-lft-identity 66.2%

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

   2. add-sqr-sqrt 66.2%

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

   3. times-frac 66.2%

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

   4. sqrt-prod 66.2%

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

   5. sqrt-pow1 49.9%

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

   6. metadata-eval 49.9%

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

   7. pow1 49.9%

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

   8. *-commutative 49.9%

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

   9. associate-*r* 46.4%

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

   10. unpow2 46.4%

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

   11. pow-prod-down 49.9%

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

   12. sqrt-prod 49.9%

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

4. Applied egg-rr 88.1%

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

   1. associate-*l/ 88.2%

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

   2. *-lft-identity 88.2%

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

   3. unpow2 88.2%

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

   4. rem-sqrt-square 88.2%

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

   5. unpow2 88.2%

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

   6. rem-sqrt-square 97.8%

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

6. Simplified 97.8%

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

   1. div-inv 97.8%

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

   2. *-commutative 97.8%

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

   3. add-sqr-sqrt 55.0%

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

   4. fabs-sqr 55.0%

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

   5. add-sqr-sqrt 63.4%

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

   6. associate-*r* 62.4%

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

   7. add-sqr-sqrt 46.5%

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

   8. fabs-sqr 46.5%

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

   9. add-sqr-sqrt 95.4%

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

   10. associate-*r* 97.1%

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

8. Applied egg-rr 97.1%

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

9. Final simplification 97.1%

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

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. associate-/l/ 66.3%

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

   2. remove-double-neg 66.3%

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

   3. distribute-frac-neg 66.3%

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

   4. distribute-neg-frac 66.3%

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

   5. remove-double-neg 66.3%

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

   6. *-commutative 66.3%

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

   7. associate-*r* 60.9%

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

   8. unpow2 60.9%

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

   9. associate-/r* 60.2%

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

   10. cos-neg 60.2%

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

   11. *-commutative 60.2%

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

   12. distribute-rgt-neg-in 60.2%

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

   13. metadata-eval 60.2%

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

3. Simplified 60.2%

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

5. Taylor expanded in x around inf 60.9%

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

   1. associate-/r* 60.9%

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

   2. unpow2 60.9%

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

   3. unpow2 60.9%

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

   4. swap-sqr 75.6%

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

   5. unpow2 75.6%

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

   6. associate-/r* 75.6%

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

   7. *-commutative 75.6%

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

   8. unpow2 75.6%

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

   9. rem-square-sqrt 75.6%

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

   10. swap-sqr 88.0%

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

   11. unpow2 88.0%

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

   12. unpow2 88.0%

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

   13. rem-sqrt-square 97.5%

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

7. Simplified 97.5%

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

   1. unpow-prod-down 75.6%

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

   2. unpow2 75.6%

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

   3. pow2 75.6%

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

   4. sqr-abs 75.6%

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

   5. swap-sqr 97.5%

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

   6. associate-*r* 95.1%

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

   7. associate-*r* 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 6: 80.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in x around 0 55.7%

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

   1. associate-/r* 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. swap-sqr 65.4%

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

   5. unpow2 65.4%

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

   6. associate-/r* 65.5%

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

   7. unpow2 65.5%

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

   8. rem-square-sqrt 65.5%

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

   9. swap-sqr 75.5%

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

   10. unpow2 75.5%

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

   11. unpow2 75.5%

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

   12. rem-sqrt-square 81.6%

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

5. Simplified 81.6%

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

   1. *-un-lft-identity 81.6%

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

   2. pow-flip 81.6%

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

   3. add-sqr-sqrt 45.4%

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

   4. fabs-sqr 45.4%

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

   5. add-sqr-sqrt 81.6%

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

   6. associate-*r* 81.5%

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

   7. metadata-eval 81.5%

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

7. Applied egg-rr 81.5%

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

   1. *-lft-identity 81.5%

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

   2. associate-*l* 81.6%

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

9. Simplified 81.6%

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

10. Final simplification 81.6%

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

Alternative 7: 79.3% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in x around 0 55.7%

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

   1. associate-/r* 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. swap-sqr 65.4%

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

   5. unpow2 65.4%

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

   6. associate-/r* 65.5%

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

   7. unpow2 65.5%

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

   8. rem-square-sqrt 65.5%

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

   9. swap-sqr 75.5%

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

   10. unpow2 75.5%

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

   11. unpow2 75.5%

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

   12. rem-sqrt-square 81.6%

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

5. Simplified 81.6%

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

   1. unpow-prod-down 75.6%

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

   2. unpow2 75.6%

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

   3. pow2 75.6%

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

   4. sqr-abs 75.6%

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

   5. swap-sqr 97.5%

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

   6. associate-*r* 95.1%

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

   7. associate-*r* 96.9%

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

7. Applied egg-rr 81.5%

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

   1. associate-/r* 81.5%

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

   2. *-un-lft-identity 81.5%

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

   3. associate-*l* 80.5%

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

   4. times-frac 79.1%

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

   5. *-commutative 79.1%

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

   6. *-commutative 79.1%

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

   7. associate-*r* 80.2%

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

   8. *-commutative 80.2%

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

   9. associate-/r* 80.2%

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

    \[\leadsto \frac{1}{c} \cdot \frac{\frac{\frac{1}{x \cdot s}}{c}}{x \cdot s} \]
11. Add Preprocessing

Alternative 8: 79.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in x around 0 55.7%

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

   1. associate-/r* 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. swap-sqr 65.4%

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

   5. unpow2 65.4%

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

   6. associate-/r* 65.5%

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

   7. unpow2 65.5%

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

   8. rem-square-sqrt 65.5%

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

   9. swap-sqr 75.5%

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

   10. unpow2 75.5%

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

   11. unpow2 75.5%

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

   12. rem-sqrt-square 81.6%

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

5. Simplified 81.6%

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

   1. /-rgt-identity 81.6%

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

   2. clear-num 81.6%

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

   3. pow-flip 81.6%

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

   4. add-sqr-sqrt 45.4%

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

   5. fabs-sqr 45.4%

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

   6. add-sqr-sqrt 81.6%

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

   7. associate-*r* 81.5%

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

   8. metadata-eval 81.5%

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

7. Applied egg-rr 81.5%

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

   1. pow-flip 81.5%

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

   2. metadata-eval 81.5%

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

   3. pow2 81.5%

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

   4. *-commutative 81.5%

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

   5. *-commutative 81.5%

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

   6. associate-*r* 80.5%

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

   7. *-commutative 80.5%

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

   8. associate-*r* 79.0%

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

   9. associate-*l* 80.1%

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

9. Applied egg-rr 80.1%

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

10. Final simplification 80.1%

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

Alternative 9: 79.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

3. Taylor expanded in x around 0 55.7%

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

   1. associate-/r* 55.7%

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

   2. unpow2 55.7%

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

   3. unpow2 55.7%

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

   4. swap-sqr 65.4%

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

   5. unpow2 65.4%

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

   6. associate-/r* 65.5%

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

   7. unpow2 65.5%

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

   8. rem-square-sqrt 65.5%

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

   9. swap-sqr 75.5%

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

   10. unpow2 75.5%

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

   11. unpow2 75.5%

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

   12. rem-sqrt-square 81.6%

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

5. Simplified 81.6%

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

   1. unpow-prod-down 75.6%

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

   2. unpow2 75.6%

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

   3. pow2 75.6%

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

   4. sqr-abs 75.6%

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

   5. swap-sqr 97.5%

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

   6. associate-*r* 95.1%

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

   7. associate-*r* 96.9%

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

7. Applied egg-rr 81.5%

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

8. Final simplification 81.5%

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

Reproduce

herbie shell --seed 2024091 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))