mixedcos

Percentage Accurate: 65.4% → 99.1%

Time: 12.0s

Alternatives: 7

Speedup: 24.1×

• 30.0% 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: 65.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x -2.0))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))
        INFINITY)
     (* (/ (* (/ 1.0 x) (/ t_0 s)) c) (/ (/ 1.0 (* x s)) c))
     (* (/ (/ t_0 x) (* c s)) (/ (/ 1.0 x) (* c s))))))

C

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

Java

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

Python

def code(x, c, s):
	t_0 = math.cos((x * -2.0))
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (((1.0 / x) * (t_0 / s)) / c) * ((1.0 / (x * s)) / c)
	else:
		tmp = ((t_0 / x) / (c * s)) * ((1.0 / x) / (c * s))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x * -2.0))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(Float64(Float64(1.0 / x) * Float64(t_0 / s)) / c) * Float64(Float64(1.0 / Float64(x * s)) / c));
	else
		tmp = Float64(Float64(Float64(t_0 / x) / Float64(c * s)) * Float64(Float64(1.0 / x) / Float64(c * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * -2.0));
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (((1.0 / x) * (t_0 / s)) / c) * ((1.0 / (x * s)) / c);
	else
		tmp = ((t_0 / x) / (c * s)) * ((1.0 / x) / (c * s));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(c * s), $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 78.3%

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

      1. associate-/l/ 78.3%

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

      2. remove-double-neg 78.3%

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

      3. distribute-frac-neg 78.3%

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

      4. distribute-neg-frac 78.3%

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

      5. remove-double-neg 78.3%

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

      6. *-commutative 78.3%

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

      7. associate-*r* 72.4%

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

      8. unpow2 72.4%

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

      9. associate-/r* 72.5%

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

      10. cos-neg 72.5%

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

      11. *-commutative 72.5%

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

      12. distribute-rgt-neg-in 72.5%

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

      13. metadata-eval 72.5%

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

   3. Simplified 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. unpow2 72.5%

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

      3. times-frac 72.5%

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

      4. add-sqr-sqrt 38.7%

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

      5. sqrt-unprod 57.1%

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

      6. *-commutative 57.1%

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

      7. *-commutative 57.1%

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

      8. swap-sqr 57.1%

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

      9. metadata-eval 57.1%

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

      10. metadata-eval 57.1%

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

      11. swap-sqr 57.1%

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

      12. sqrt-unprod 30.9%

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

      13. add-sqr-sqrt 72.5%

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

   6. Applied egg-rr 72.5%

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

      1. *-commutative 72.5%

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

      2. unpow2 72.5%

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

      3. times-frac 89.7%

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

      4. add-sqr-sqrt 37.4%

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

      5. sqrt-unprod 67.0%

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

      6. *-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. swap-sqr 67.0%

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

      9. metadata-eval 67.0%

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

      10. metadata-eval 67.0%

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

      11. swap-sqr 67.0%

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

      12. sqrt-unprod 47.6%

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

      13. add-sqr-sqrt 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. unpow2 89.7%

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

      2. times-frac 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/l/ 99.6%

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

      6. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. *-un-lft-identity 99.6%

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

       2. times-frac 99.6%

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

   12. Applied egg-rr 99.6%

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

   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. Step-by-step derivation

      1. associate-/l/ 0.0%

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

      2. remove-double-neg 0.0%

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

      3. distribute-frac-neg 0.0%

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

      4. distribute-neg-frac 0.0%

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

      5. remove-double-neg 0.0%

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

      6. *-commutative 0.0%

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

      7. associate-*r* 0.0%

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

      8. unpow2 0.0%

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

      9. associate-/r* 0.0%

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

      10. cos-neg 0.0%

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

      11. *-commutative 0.0%

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

      12. distribute-rgt-neg-in 0.0%

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

      13. metadata-eval 0.0%

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

   3. Simplified 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. unpow2 0.0%

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

      3. times-frac 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. swap-sqr 0.0%

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

      9. metadata-eval 0.0%

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

      10. metadata-eval 0.0%

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

      11. swap-sqr 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. associate-/l/ 0.5%

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

      2. *-commutative 0.5%

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

      3. unpow-prod-down 71.7%

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

      4. unpow2 71.7%

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

      5. times-frac 99.3%

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

      6. add-sqr-sqrt 45.1%

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

      7. sqrt-unprod 67.4%

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

      8. *-commutative 67.4%

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

      9. *-commutative 67.4%

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

      10. swap-sqr 67.4%

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

      11. metadata-eval 67.4%

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

      12. metadata-eval 67.4%

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

      13. swap-sqr 67.4%

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

      14. sqrt-unprod 43.3%

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

      15. add-sqr-sqrt 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

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

Alternative 2: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\cos \left(x \cdot -2\right)}{x}}{c \cdot s} \cdot \frac{\frac{1}{x}}{c \cdot s} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (* (/ (/ (cos (* x -2.0)) x) (* c s)) (/ (/ 1.0 x) (* c s))))

C

double code(double x, double c, double s) {
	return ((cos((x * -2.0)) / x) / (c * s)) * ((1.0 / x) / (c * s));
}

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((x * (-2.0d0))) / x) / (c * s)) * ((1.0d0 / x) / (c * s))
end function

Java

public static double code(double x, double c, double s) {
	return ((Math.cos((x * -2.0)) / x) / (c * s)) * ((1.0 / x) / (c * s));
}

Python

def code(x, c, s):
	return ((math.cos((x * -2.0)) / x) / (c * s)) * ((1.0 / x) / (c * s))

Julia

function code(x, c, s)
	return Float64(Float64(Float64(cos(Float64(x * -2.0)) / x) / Float64(c * s)) * Float64(Float64(1.0 / x) / Float64(c * s)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = ((cos((x * -2.0)) / x) / (c * s)) * ((1.0 / x) / (c * s));
end

Wolfram

code[x_, c_, s_] := N[(N[(N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. associate-/l/ 62.1%

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

   2. remove-double-neg 62.1%

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

   3. distribute-frac-neg 62.1%

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

   4. distribute-neg-frac 62.1%

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

   5. remove-double-neg 62.1%

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

   6. *-commutative 62.1%

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

   7. associate-*r* 57.4%

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

   8. unpow2 57.4%

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

   9. associate-/r* 57.5%

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

   10. cos-neg 57.5%

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

   11. *-commutative 57.5%

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

   12. distribute-rgt-neg-in 57.5%

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

   13. metadata-eval 57.5%

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

3. Simplified 57.5%

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

   1. *-un-lft-identity 57.5%

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

   2. unpow2 57.5%

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

   3. times-frac 57.5%

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

   4. add-sqr-sqrt 30.7%

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

   5. sqrt-unprod 45.3%

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

   6. *-commutative 45.3%

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

   7. *-commutative 45.3%

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

   8. swap-sqr 45.3%

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

   9. metadata-eval 45.3%

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

   10. metadata-eval 45.3%

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

   11. swap-sqr 45.3%

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

   12. sqrt-unprod 24.5%

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

   13. add-sqr-sqrt 57.5%

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

6. Applied egg-rr 57.5%

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

   1. associate-/l/ 57.5%

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

   2. *-commutative 57.5%

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

   3. unpow-prod-down 77.3%

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

   4. unpow2 77.3%

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

   5. times-frac 97.3%

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

   6. add-sqr-sqrt 41.5%

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

   7. sqrt-unprod 72.2%

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

   8. *-commutative 72.2%

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

   9. *-commutative 72.2%

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

   10. swap-sqr 72.2%

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

   11. metadata-eval 72.2%

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

   12. metadata-eval 72.2%

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

   13. swap-sqr 72.2%

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

   14. sqrt-unprod 49.1%

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

   15. add-sqr-sqrt 97.3%

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

8. Applied egg-rr 97.3%

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

Alternative 3: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{1}{t\_0} \cdot \frac{\cos \left(x \cdot -2\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (* (/ 1.0 t_0) (/ (cos (* x -2.0)) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) * (cos((x * -2.0)) / t_0);
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = (1.0d0 / t_0) * (cos((x * (-2.0d0))) / t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) * (Math.cos((x * -2.0)) / t_0);
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	return (1.0 / t_0) * (math.cos((x * -2.0)) / t_0)

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(Float64(1.0 / t_0) * Float64(cos(Float64(x * -2.0)) / t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = (1.0 / t_0) * (cos((x * -2.0)) / t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.1%

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

   1. associate-/l/ 62.1%

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

   2. remove-double-neg 62.1%

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

   3. distribute-frac-neg 62.1%

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

   4. distribute-neg-frac 62.1%

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

   5. remove-double-neg 62.1%

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

   6. *-commutative 62.1%

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

   7. associate-*r* 57.4%

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

   8. unpow2 57.4%

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

   9. associate-/r* 57.5%

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

   10. cos-neg 57.5%

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

   11. *-commutative 57.5%

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

   12. distribute-rgt-neg-in 57.5%

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

   13. metadata-eval 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in x around inf 57.4%

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

   1. associate-/r* 57.0%

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

   2. unpow2 57.0%

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

   3. unpow2 57.0%

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

   4. swap-sqr 73.0%

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

   5. unpow2 73.0%

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

   6. associate-/r* 73.3%

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

   7. *-commutative 73.3%

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

   8. unpow2 73.3%

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

   9. rem-square-sqrt 73.3%

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

   10. swap-sqr 85.3%

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

   11. unpow2 85.3%

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

   12. unpow2 85.3%

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

   13. rem-sqrt-square 96.2%

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

7. Simplified 96.2%

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

   1. *-un-lft-identity 96.2%

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

   2. unpow2 96.2%

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

   3. times-frac 96.6%

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

   4. add-sqr-sqrt 59.9%

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

   5. fabs-sqr 59.9%

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

   6. add-sqr-sqrt 69.6%

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

   7. add-sqr-sqrt 56.4%

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

   8. fabs-sqr 56.4%

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

   9. add-sqr-sqrt 96.6%

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

9. Applied egg-rr 96.6%

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

10. Final simplification 96.6%

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

Alternative 4: 97.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\cos \left(x \cdot -2\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ (cos (* x -2.0)) (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return cos((x * -2.0)) / (t_0 * t_0);
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = cos((x * (-2.0d0))) / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return Math.cos((x * -2.0)) / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	return math.cos((x * -2.0)) / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(cos(Float64(x * -2.0)) / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = cos((x * -2.0)) / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.1%

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

   1. associate-/l/ 62.1%

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

   2. remove-double-neg 62.1%

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

   3. distribute-frac-neg 62.1%

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

   4. distribute-neg-frac 62.1%

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

   5. remove-double-neg 62.1%

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

   6. *-commutative 62.1%

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

   7. associate-*r* 57.4%

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

   8. unpow2 57.4%

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

   9. associate-/r* 57.5%

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

   10. cos-neg 57.5%

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

   11. *-commutative 57.5%

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

   12. distribute-rgt-neg-in 57.5%

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

   13. metadata-eval 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in x around inf 57.4%

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

   1. associate-/r* 57.0%

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

   2. unpow2 57.0%

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

   3. unpow2 57.0%

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

   4. swap-sqr 73.0%

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

   5. unpow2 73.0%

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

   6. associate-/r* 73.3%

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

   7. *-commutative 73.3%

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

   8. unpow2 73.3%

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

   9. rem-square-sqrt 73.3%

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

   10. swap-sqr 85.3%

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

   11. unpow2 85.3%

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

   12. unpow2 85.3%

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

   13. rem-sqrt-square 96.2%

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

7. Simplified 96.2%

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

   1. unpow-prod-down 73.3%

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

   2. unpow2 73.3%

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

   3. pow2 73.3%

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

   4. sqr-abs 73.3%

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

   5. swap-sqr 96.2%

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

9. Applied egg-rr 96.2%

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

10. Final simplification 96.2%

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

Alternative 5: 78.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. associate-/l/ 62.1%

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

   2. remove-double-neg 62.1%

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

   3. distribute-frac-neg 62.1%

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

   4. distribute-neg-frac 62.1%

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

   5. remove-double-neg 62.1%

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

   6. *-commutative 62.1%

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

   7. associate-*r* 57.4%

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

   8. unpow2 57.4%

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

   9. associate-/r* 57.5%

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

   10. cos-neg 57.5%

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

   11. *-commutative 57.5%

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

   12. distribute-rgt-neg-in 57.5%

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

   13. metadata-eval 57.5%

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

3. Simplified 57.5%

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

5. Taylor expanded in x around inf 57.4%

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

   1. associate-/r* 57.0%

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

   2. unpow2 57.0%

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

   3. unpow2 57.0%

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

   4. swap-sqr 73.0%

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

   5. unpow2 73.0%

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

   6. associate-/r* 73.3%

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

   7. *-commutative 73.3%

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

   8. unpow2 73.3%

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

   9. rem-square-sqrt 73.3%

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

   10. swap-sqr 85.3%

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

   11. unpow2 85.3%

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

   12. unpow2 85.3%

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

   13. rem-sqrt-square 96.2%

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

7. Simplified 96.2%

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

8. Taylor expanded in x around 0 63.7%

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

   1. associate-/r* 63.5%

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

   2. unpow2 63.5%

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

   3. sqr-abs 63.5%

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

   4. unpow2 63.5%

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

   5. associate-/r* 63.7%

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

   6. unpow2 63.7%

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

   7. unpow2 63.7%

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

   8. swap-sqr 79.8%

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

   9. unpow2 79.8%

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

   10. unpow-1 79.8%

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

   11. exp-to-pow 47.3%

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

   12. *-commutative 47.3%

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

   13. exp-prod 47.5%

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

   14. *-commutative 47.5%

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

   15. associate-*l* 47.5%

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

   16. metadata-eval 47.5%

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

   17. exp-to-pow 80.1%

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

10. Simplified 80.1%

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

11. Final simplification 80.1%

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

Alternative 6: 78.5% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ (/ 1.0 t_0) t_0)))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) / t_0;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = (1.0d0 / t_0) / t_0
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (1.0 / t_0) / t_0;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	return (1.0 / t_0) / t_0

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in x around 0 52.8%

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

   1. associate-/r* 52.4%

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

   2. unpow2 52.4%

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

   3. unpow2 52.4%

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

   4. swap-sqr 63.5%

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

   5. unpow2 63.5%

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

   6. associate-/r* 63.7%

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

   7. unpow2 63.7%

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

   8. rem-square-sqrt 63.7%

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

   9. swap-sqr 73.2%

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

   10. unpow2 73.2%

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

   11. unpow2 73.2%

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

   12. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. unpow-prod-down 73.3%

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

   2. unpow2 73.3%

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

   3. pow2 73.3%

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

   4. sqr-abs 73.3%

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

   5. swap-sqr 96.2%

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

7. Applied egg-rr 79.8%

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

   1. metadata-eval 79.8%

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

   2. frac-times 80.0%

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

   3. associate-*r* 78.7%

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

   4. associate-/l/ 78.7%

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

   5. associate-*r* 80.2%

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

   6. associate-/l/ 80.2%

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

   7. associate-*r/ 77.6%

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

   8. times-frac 75.2%

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

   9. associate-/l/ 75.2%

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

   10. associate-*r* 76.2%

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

   11. *-commutative 76.2%

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

   12. associate-/l/ 76.2%

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

   13. *-commutative 76.2%

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

9. Applied egg-rr 76.2%

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

    1. *-commutative 76.2%

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

    2. frac-times 80.1%

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

    3. *-un-lft-identity 80.1%

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

    4. *-commutative 80.1%

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

11. Applied egg-rr 80.1%

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

12. Final simplification 80.1%

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

Alternative 7: 78.3% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	return 1.0 / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in x around 0 52.8%

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

   1. associate-/r* 52.4%

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

   2. unpow2 52.4%

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

   3. unpow2 52.4%

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

   4. swap-sqr 63.5%

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

   5. unpow2 63.5%

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

   6. associate-/r* 63.7%

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

   7. unpow2 63.7%

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

   8. rem-square-sqrt 63.7%

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

   9. swap-sqr 73.2%

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

   10. unpow2 73.2%

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

   11. unpow2 73.2%

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

   12. rem-sqrt-square 79.8%

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

5. Simplified 79.8%

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

   1. unpow-prod-down 73.3%

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

   2. unpow2 73.3%

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

   3. pow2 73.3%

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

   4. sqr-abs 73.3%

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

   5. swap-sqr 96.2%

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

7. Applied egg-rr 79.8%

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

8. Final simplification 79.8%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))