mixedcos

Percentage Accurate: 67.1% → 99.2%

Time: 16.5s

Alternatives: 14

Speedup: 24.1×

• 32.3% 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: 67.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;\frac{t\_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos x, \cos x, -{\sin x}^{2}\right)}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* 2.0 x))) (t_2 (* x (* c s))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ (/ t_1 t_0) t_0)
     (/ (fma (cos x) (cos x) (- (pow (sin x) 2.0))) (* t_2 t_2)))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = fma(cos(x), cos(x), -pow(sin(x), 2.0)) / (t_2 * t_2);
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	else
		tmp = Float64(fma(cos(x), cos(x), Float64(-(sin(x) ^ 2.0))) / Float64(t_2 * t_2));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[x], $MachinePrecision] + (-N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

   4. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.4%

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

      3. div-inv 99.4%

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

      4. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) 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. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-2neg 85.3%

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

      3. frac-2neg 85.3%

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

      4. metadata-eval 85.3%

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

      5. frac-times 85.3%

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

      6. *-commutative 85.3%

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

      7. *-commutative 85.3%

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

      8. associate-*r* 85.6%

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

      9. distribute-rgt-neg-in 85.6%

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

      10. *-commutative 85.6%

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

      11. associate-*r* 97.1%

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

      12. distribute-rgt-neg-in 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. cos-2 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. fma-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in x around inf 99.2%

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

       1. neg-mul-1 99.2%

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

   13. Simplified 99.2%

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

4. Final simplification 99.3%

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

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;\frac{t\_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\cos x}^{2} - {\sin x}^{2}}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* 2.0 x))) (t_2 (* x (* c s))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ (/ t_1 t_0) t_0)
     (/ (- (pow (cos x) 2.0) (pow (sin x) 2.0)) (* t_2 t_2)))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = (pow(cos(x), 2.0) - pow(sin(x), 2.0)) / (t_2 * t_2);
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = (Math.pow(Math.cos(x), 2.0) - Math.pow(Math.sin(x), 2.0)) / (t_2 * t_2);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((2.0 * x))
	t_2 = x * (c * s)
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (t_1 / t_0) / t_0
	else:
		tmp = (math.pow(math.cos(x), 2.0) - math.pow(math.sin(x), 2.0)) / (t_2 * t_2)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	else
		tmp = Float64(Float64((cos(x) ^ 2.0) - (sin(x) ^ 2.0)) / Float64(t_2 * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((2.0 * x));
	t_2 = x * (c * s);
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (t_1 / t_0) / t_0;
	else
		tmp = ((cos(x) ^ 2.0) - (sin(x) ^ 2.0)) / (t_2 * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

   4. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.4%

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

      3. div-inv 99.4%

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

      4. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) 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. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-2neg 85.3%

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

      3. frac-2neg 85.3%

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

      4. metadata-eval 85.3%

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

      5. frac-times 85.3%

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

      6. *-commutative 85.3%

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

      7. *-commutative 85.3%

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

      8. associate-*r* 85.6%

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

      9. distribute-rgt-neg-in 85.6%

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

      10. *-commutative 85.6%

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

      11. associate-*r* 97.1%

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

      12. distribute-rgt-neg-in 97.1%

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

   6. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. cos-2 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. fma-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in x around inf 99.1%

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

       1. +-commutative 99.1%

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

       2. neg-mul-1 99.1%

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

       3. unsub-neg 99.1%

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

   13. Simplified 99.1%

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

4. Final simplification 99.3%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;\frac{t\_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* 2.0 x))) (t_2 (* x (* c s))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ (/ t_1 t_0) t_0)
     (/ t_1 (* t_2 t_2)))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = t_1 / (t_2 * t_2);
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = t_1 / (t_2 * t_2);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((2.0 * x))
	t_2 = x * (c * s)
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (t_1 / t_0) / t_0
	else:
		tmp = t_1 / (t_2 * t_2)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	else
		tmp = Float64(t_1 / Float64(t_2 * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((2.0 * x));
	t_2 = x * (c * s);
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (t_1 / t_0) / t_0;
	else
		tmp = t_1 / (t_2 * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(t$95$1 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

   4. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l/ 99.4%

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

      3. div-inv 99.4%

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

      4. *-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) 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. Applied egg-rr 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-2neg 85.3%

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

      3. frac-2neg 85.3%

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

      4. metadata-eval 85.3%

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

      5. frac-times 85.3%

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

      6. *-commutative 85.3%

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

      7. *-commutative 85.3%

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

      8. associate-*r* 85.6%

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

      9. distribute-rgt-neg-in 85.6%

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

      10. *-commutative 85.6%

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

      11. associate-*r* 97.1%

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

      12. distribute-rgt-neg-in 97.1%

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

   6. Applied egg-rr 97.1%

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

4. Final simplification 99.0%

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

Alternative 4: 95.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;c \leq 5.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{t\_1 \cdot \frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* 2.0 x))))
   (if (<= c 5.2e-240)
     (/ (* t_1 (/ (/ 1.0 (* x (* c s))) (* c s))) x)
     (/ (/ t_1 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double tmp;
	if (c <= 5.2e-240) {
		tmp = (t_1 * ((1.0 / (x * (c * s))) / (c * s))) / x;
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((2.0d0 * x))
    if (c <= 5.2d-240) then
        tmp = (t_1 * ((1.0d0 / (x * (c * s))) / (c * s))) / x
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((2.0 * x));
	double tmp;
	if (c <= 5.2e-240) {
		tmp = (t_1 * ((1.0 / (x * (c * s))) / (c * s))) / x;
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((2.0 * x))
	tmp = 0
	if c <= 5.2e-240:
		tmp = (t_1 * ((1.0 / (x * (c * s))) / (c * s))) / x
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (c <= 5.2e-240)
		tmp = Float64(Float64(t_1 * Float64(Float64(1.0 / Float64(x * Float64(c * s))) / Float64(c * s))) / x);
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((2.0 * x));
	tmp = 0.0;
	if (c <= 5.2e-240)
		tmp = (t_1 * ((1.0 / (x * (c * s))) / (c * s))) / x;
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, 5.2e-240], N[(N[(t$95$1 * N[(N[(1.0 / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 5.19999999999999984e-240

   1. Initial program 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. add-sqr-sqrt 64.6%

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

      3. times-frac 64.6%

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

   4. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. frac-2neg 96.2%

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

      3. frac-2neg 96.2%

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

      4. metadata-eval 96.2%

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

      5. frac-times 96.2%

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

      6. *-commutative 96.2%

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

      7. *-commutative 96.2%

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

      8. associate-*r* 94.5%

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

      9. distribute-rgt-neg-in 94.5%

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

      10. *-commutative 94.5%

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

      11. associate-*r* 96.6%

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

      12. distribute-rgt-neg-in 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. *-un-lft-identity 96.6%

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

      2. associate-*r* 90.5%

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

      3. times-frac 91.0%

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

      4. associate-*l* 88.3%

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

      5. add-sqr-sqrt 41.9%

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

      6. sqrt-unprod 51.4%

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

      7. sqr-neg 51.4%

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

      8. sqrt-prod 18.4%

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

      9. add-sqr-sqrt 28.5%

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

   8. Applied egg-rr 88.3%

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

      1. associate-*r/ 88.3%

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

      2. frac-2neg 88.3%

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

   10. Applied egg-rr 91.4%

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

   if 5.19999999999999984e-240 < c

   1. Initial program 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. add-sqr-sqrt 66.4%

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

      3. times-frac 66.4%

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

   4. Applied egg-rr 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l/ 97.8%

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

      3. div-inv 97.9%

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

      4. *-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right) \cdot \frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{c \cdot s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;c \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{t\_1}{x \cdot \left(x \cdot \left(c \cdot s\right)\right)}}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* 2.0 x))))
   (if (<= c 2e-276)
     (/ (/ t_1 (* x (* x (* c s)))) (* c s))
     (/ (/ t_1 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double tmp;
	if (c <= 2e-276) {
		tmp = (t_1 / (x * (x * (c * s)))) / (c * s);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((2.0d0 * x))
    if (c <= 2d-276) then
        tmp = (t_1 / (x * (x * (c * s)))) / (c * s)
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((2.0 * x));
	double tmp;
	if (c <= 2e-276) {
		tmp = (t_1 / (x * (x * (c * s)))) / (c * s);
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((2.0 * x))
	tmp = 0
	if c <= 2e-276:
		tmp = (t_1 / (x * (x * (c * s)))) / (c * s)
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (c <= 2e-276)
		tmp = Float64(Float64(t_1 / Float64(x * Float64(x * Float64(c * s)))) / Float64(c * s));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((2.0 * x));
	tmp = 0.0;
	if (c <= 2e-276)
		tmp = (t_1 / (x * (x * (c * s)))) / (c * s);
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, 2e-276], N[(N[(t$95$1 / N[(x * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 2e-276

   1. Initial program 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. add-sqr-sqrt 64.7%

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

      3. times-frac 64.7%

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

   4. Applied egg-rr 96.1%

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

      1. *-commutative 96.1%

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

      2. frac-2neg 96.1%

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

      3. frac-2neg 96.1%

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

      4. metadata-eval 96.1%

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

      5. frac-times 96.0%

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

      6. *-commutative 96.0%

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

      7. *-commutative 96.0%

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

      8. associate-*r* 94.3%

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

      9. distribute-rgt-neg-in 94.3%

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

      10. *-commutative 94.3%

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

      11. associate-*r* 96.5%

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

      12. distribute-rgt-neg-in 96.5%

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

   6. Applied egg-rr 96.5%

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

      1. *-un-lft-identity 96.5%

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

      2. associate-*l* 93.2%

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

      3. times-frac 92.3%

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

   8. Applied egg-rr 90.0%

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

      1. associate-*l/ 90.5%

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

      2. *-lft-identity 90.5%

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

      3. associate-*r* 92.8%

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

      4. *-commutative 92.8%

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

   10. Simplified 92.8%

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

   if 2e-276 < c

   1. Initial program 66.3%

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

      1. *-un-lft-identity 66.3%

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

      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. Applied egg-rr 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l/ 97.9%

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

      3. div-inv 97.9%

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

      4. *-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2 \cdot 10^{-276}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(c \cdot s\right)\right)}}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return (cos((2.0 * x)) / c) / ((c * (x * s)) * (x * 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((2.0d0 * x)) / c) / ((c * (x * s)) * (x * s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.0%

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

   1. associate-*l/ 97.0%

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

   2. *-un-lft-identity 97.0%

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

   3. associate-/r* 97.0%

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

   4. associate-/l/ 93.5%

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

   5. *-commutative 93.5%

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

6. Applied egg-rr 93.5%

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

7. Final simplification 93.5%

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

Alternative 7: 97.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (cos((2.0 * x)) / 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((2.0d0 * x)) / 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((2.0 * x)) / t_0) / t_0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.5%

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

4. Applied egg-rr 97.0%

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

   1. *-commutative 97.0%

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

   2. associate-*l/ 97.0%

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

   3. div-inv 97.0%

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

   4. *-commutative 97.0%

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

6. Applied egg-rr 97.0%

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

7. Final simplification 97.0%

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

Alternative 8: 79.1% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ t_1 := \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s))) (t_1 (/ 1.0 (* c (* x s)))))
   (if (<= x 2.6e+44) (* t_1 t_1) (/ (/ -1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double t_1 = 1.0 / (c * (x * s));
	double tmp;
	if (x <= 2.6e+44) {
		tmp = t_1 * t_1;
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (c * s)
    t_1 = 1.0d0 / (c * (x * s))
    if (x <= 2.6d+44) then
        tmp = t_1 * t_1
    else
        tmp = ((-1.0d0) / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double t_1 = 1.0 / (c * (x * s));
	double tmp;
	if (x <= 2.6e+44) {
		tmp = t_1 * t_1;
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	t_1 = 1.0 / (c * (x * s))
	tmp = 0
	if x <= 2.6e+44:
		tmp = t_1 * t_1
	else:
		tmp = (-1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	t_1 = Float64(1.0 / Float64(c * Float64(x * s)))
	tmp = 0.0
	if (x <= 2.6e+44)
		tmp = Float64(t_1 * t_1);
	else
		tmp = Float64(Float64(-1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (c * s);
	t_1 = 1.0 / (c * (x * s));
	tmp = 0.0;
	if (x <= 2.6e+44)
		tmp = t_1 * t_1;
	else
		tmp = (-1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.6e+44], N[(t$95$1 * t$95$1), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999999e44

   1. Initial program 67.4%

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

      1. *-un-lft-identity 67.4%

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

      2. add-sqr-sqrt 67.3%

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

      3. times-frac 67.3%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in x around 0 85.4%

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

   if 2.5999999999999999e44 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around 0 53.2%

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

      1. associate-/r* 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

      4. unpow2 53.2%

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

      5. swap-sqr 56.8%

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

      6. unpow2 56.8%

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

      7. associate-/r* 56.8%

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

      8. unpow2 56.8%

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

      9. unpow2 56.8%

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

      10. swap-sqr 61.2%

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

      11. unpow2 61.2%

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

      12. *-commutative 61.2%

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

   5. Simplified 61.2%

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

      1. unpow2 61.2%

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

      2. associate-*r* 61.2%

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

      3. *-commutative 61.2%

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

      4. associate-*l* 61.3%

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

   7. Applied egg-rr 61.3%

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

   9. Applied egg-rr 62.1%

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

       1. associate-/r* 62.1%

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

       2. *-commutative 62.1%

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

       3. associate-/l/ 62.1%

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

   11. Simplified 62.1%

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

   13. Applied egg-rr 63.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s))))
   (if (<= x 2.6e+44)
     (* (/ 1.0 (* c (* x s))) (/ (/ 1.0 c) (* x s)))
     (/ (/ -1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if (x <= 2.6e+44) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) / (x * s));
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = x * (c * s)
    if (x <= 2.6d+44) then
        tmp = (1.0d0 / (c * (x * s))) * ((1.0d0 / c) / (x * s))
    else
        tmp = ((-1.0d0) / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if (x <= 2.6e+44) {
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) / (x * s));
	} else {
		tmp = (-1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	tmp = 0
	if x <= 2.6e+44:
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) / (x * s))
	else:
		tmp = (-1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (x <= 2.6e+44)
		tmp = Float64(Float64(1.0 / Float64(c * Float64(x * s))) * Float64(Float64(1.0 / c) / Float64(x * s)));
	else
		tmp = Float64(Float64(-1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (c * s);
	tmp = 0.0;
	if (x <= 2.6e+44)
		tmp = (1.0 / (c * (x * s))) * ((1.0 / c) / (x * s));
	else
		tmp = (-1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.6e+44], N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999999e44

   1. Initial program 67.4%

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

      1. *-un-lft-identity 67.4%

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

      2. add-sqr-sqrt 67.3%

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

      3. times-frac 67.3%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in x around 0 85.4%

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

      1. associate-/r* 85.4%

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

   7. Simplified 85.4%

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

   if 2.5999999999999999e44 < x

   1. Initial program 59.8%

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

   3. Taylor expanded in x around 0 53.2%

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

      1. associate-/r* 53.2%

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

      2. *-commutative 53.2%

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

      3. unpow2 53.2%

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

      4. unpow2 53.2%

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

      5. swap-sqr 56.8%

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

      6. unpow2 56.8%

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

      7. associate-/r* 56.8%

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

      8. unpow2 56.8%

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

      9. unpow2 56.8%

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

      10. swap-sqr 61.2%

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

      11. unpow2 61.2%

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

      12. *-commutative 61.2%

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

   5. Simplified 61.2%

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

      1. unpow2 61.2%

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

      2. associate-*r* 61.2%

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

      3. *-commutative 61.2%

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

      4. associate-*l* 61.3%

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

   7. Applied egg-rr 61.3%

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

   9. Applied egg-rr 62.1%

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

       1. associate-/r* 62.1%

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

       2. *-commutative 62.1%

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

       3. associate-/l/ 62.1%

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

   11. Simplified 62.1%

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

   13. Applied egg-rr 63.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return 1.0 / ((c * s) * (x * (c * (x * 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 = 1.0d0 / ((c * s) * (x * (c * (x * s))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0 54.3%

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

   1. associate-/r* 54.1%

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

   2. *-commutative 54.1%

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

   3. unpow2 54.1%

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

   4. unpow2 54.1%

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

   5. swap-sqr 65.7%

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

   6. unpow2 65.7%

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

   7. associate-/r* 65.8%

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

   8. unpow2 65.8%

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

   9. unpow2 65.8%

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

   10. swap-sqr 79.3%

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

   11. unpow2 79.3%

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

   12. *-commutative 79.3%

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

5. Simplified 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*r* 79.0%

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

   3. *-commutative 79.0%

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

   4. associate-*l* 77.2%

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

7. Applied egg-rr 77.2%

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

8. Final simplification 77.2%

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

Alternative 11: 76.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return 1.0 / ((c * s) * (x * (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 = 1.0d0 / ((c * s) * (x * (x * (c * s))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0 54.3%

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

   1. associate-/r* 54.1%

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

   2. *-commutative 54.1%

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

   3. unpow2 54.1%

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

   4. unpow2 54.1%

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

   5. swap-sqr 65.7%

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

   6. unpow2 65.7%

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

   7. associate-/r* 65.8%

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

   8. unpow2 65.8%

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

   9. unpow2 65.8%

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

   10. swap-sqr 79.3%

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

   11. unpow2 79.3%

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

   12. *-commutative 79.3%

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

5. Simplified 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*r* 79.0%

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

   3. *-commutative 79.0%

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

   4. associate-*l* 77.2%

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

7. Applied egg-rr 77.2%

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

   1. add-sqr-sqrt 41.7%

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

   2. sqrt-prod 53.5%

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

   3. sqr-neg 53.5%

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

   4. sqrt-unprod 16.6%

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

   5. add-sqr-sqrt 38.8%

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

   6. *-commutative 38.8%

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

   7. add-sqr-sqrt 22.2%

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

   8. sqrt-prod 50.6%

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

   9. sqr-neg 50.6%

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

   10. sqrt-unprod 35.5%

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

   11. add-sqr-sqrt 77.2%

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

   12. associate-*l* 77.7%

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

   13. pow1 77.7%

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

9. Applied egg-rr 77.2%

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

    1. unpow1 77.2%

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

    2. associate-*r* 77.7%

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

    3. *-commutative 77.7%

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

11. Simplified 77.7%

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

12. Final simplification 77.7%

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

Alternative 12: 76.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return 1.0 / (c * ((c * (x * s)) * (x * 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 = 1.0d0 / (c * ((c * (x * s)) * (x * s)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0 54.3%

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

   1. associate-/r* 54.1%

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

   2. *-commutative 54.1%

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

   3. unpow2 54.1%

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

   4. unpow2 54.1%

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

   5. swap-sqr 65.7%

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

   6. unpow2 65.7%

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

   7. associate-/r* 65.8%

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

   8. unpow2 65.8%

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

   9. unpow2 65.8%

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

   10. swap-sqr 79.3%

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

   11. unpow2 79.3%

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

   12. *-commutative 79.3%

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

5. Simplified 79.3%

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

   1. *-commutative 79.3%

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

   2. unpow2 79.3%

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

   3. *-commutative 79.3%

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

   4. associate-*r* 76.9%

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

7. Applied egg-rr 76.9%

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

8. Final simplification 76.9%

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

Alternative 13: 77.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return (1.0 / c) / ((c * (x * s)) * (x * 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 = (1.0d0 / c) / ((c * (x * s)) * (x * s))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0 54.3%

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

   1. associate-/r* 54.1%

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

   2. *-commutative 54.1%

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

   3. unpow2 54.1%

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

   4. unpow2 54.1%

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

   5. swap-sqr 65.7%

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

   6. unpow2 65.7%

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

   7. associate-/r* 65.8%

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

   8. unpow2 65.8%

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

   9. unpow2 65.8%

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

   10. swap-sqr 79.3%

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

   11. unpow2 79.3%

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

   12. *-commutative 79.3%

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

5. Simplified 79.3%

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

   1. pow-flip 79.5%

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

   2. *-commutative 79.5%

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

   3. pow-flip 79.3%

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

   4. unpow2 79.3%

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

   5. metadata-eval 79.3%

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

   6. frac-times 79.5%

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

   7. associate-/r* 79.5%

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

   8. associate-/r* 79.4%

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

7. Applied egg-rr 79.4%

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

   1. associate-/l/ 79.5%

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

   2. *-commutative 79.5%

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

   3. frac-times 77.0%

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

   4. *-un-lft-identity 77.0%

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

   5. *-commutative 77.0%

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

   6. *-commutative 77.0%

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

9. Applied egg-rr 77.0%

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

10. Final simplification 77.0%

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

Alternative 14: 79.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = x * (c * 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 = x * (c * s)
    code = (1.0d0 / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in x around 0 54.3%

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

   1. associate-/r* 54.1%

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

   2. *-commutative 54.1%

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

   3. unpow2 54.1%

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

   4. unpow2 54.1%

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

   5. swap-sqr 65.7%

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

   6. unpow2 65.7%

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

   7. associate-/r* 65.8%

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

   8. unpow2 65.8%

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

   9. unpow2 65.8%

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

   10. swap-sqr 79.3%

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

   11. unpow2 79.3%

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

   12. *-commutative 79.3%

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

5. Simplified 79.3%

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

   1. unpow2 79.3%

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

   2. associate-*r* 79.0%

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

   3. *-commutative 79.0%

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

   4. associate-*l* 77.2%

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

7. Applied egg-rr 77.2%

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

9. Applied egg-rr 38.8%

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

    1. associate-/r* 38.8%

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

    2. *-commutative 38.8%

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

    3. associate-/l/ 38.8%

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

11. Simplified 38.8%

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

13. Applied egg-rr 79.6%

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

14. Final simplification 79.6%

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

Reproduce

herbie shell --seed 2024061 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))