mixedcos

Percentage Accurate: 66.8% → 99.1%

Time: 14.8s

Alternatives: 8

Speedup: 24.1×

• 31.3% 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: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.6%

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

      1. associate-/r* 81.6%

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

      2. *-commutative 81.6%

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

      3. unpow2 81.6%

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

      4. sqr-neg 81.6%

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

      5. unpow2 81.6%

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

      6. cos-neg 81.6%

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

      7. *-commutative 81.6%

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

      8. distribute-rgt-neg-in 81.6%

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

      9. metadata-eval 81.6%

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

      10. unpow2 81.6%

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

      11. sqr-neg 81.6%

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

      12. unpow2 81.6%

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

      13. associate-*r* 72.6%

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

      14. unpow2 72.6%

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

      15. *-commutative 72.6%

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

   3. Simplified 72.6%

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

      1. associate-/l/ 72.6%

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

      2. add-sqr-sqrt 57.2%

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

      3. add-sqr-sqrt 72.6%

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

      4. add-sqr-sqrt 26.6%

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

      5. sqrt-unprod 57.8%

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

      6. swap-sqr 57.8%

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

      7. metadata-eval 57.8%

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

      8. metadata-eval 57.8%

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

      9. swap-sqr 57.8%

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

      10. *-commutative 57.8%

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

      11. *-commutative 57.8%

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

      12. sqrt-unprod 40.6%

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

      13. add-sqr-sqrt 72.6%

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

      14. *-un-lft-identity 72.6%

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

      15. *-commutative 72.6%

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

      16. unpow2 72.6%

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

      17. associate-*r* 81.6%

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

      18. associate-*l* 83.3%

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

      19. *-commutative 83.3%

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

      20. *-commutative 83.3%

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

   6. Applied egg-rr 83.8%

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

   7. Taylor expanded in c around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r* 71.5%

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

      3. unpow2 71.5%

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

      4. unpow2 71.5%

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

      5. swap-sqr 74.8%

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

      6. unpow2 74.8%

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

      7. *-commutative 74.8%

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

      8. unpow2 74.8%

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

      9. associate-*r* 83.8%

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

      10. *-commutative 83.8%

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

      11. *-lft-identity 83.8%

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

      12. associate-*l/ 83.8%

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

   9. Simplified 99.7%

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

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

   1. Initial program 0.0%

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

      1. associate-/r* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. sqr-neg 0.0%

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

      5. unpow2 0.0%

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

      6. cos-neg 0.0%

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

      7. *-commutative 0.0%

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

      8. distribute-rgt-neg-in 0.0%

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

      9. metadata-eval 0.0%

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

      10. unpow2 0.0%

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

      11. sqr-neg 0.0%

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

      12. unpow2 0.0%

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

      13. associate-*r* 0.0%

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

      14. unpow2 0.0%

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

      15. *-commutative 0.0%

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

   3. Simplified 0.0%

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

      1. associate-/l/ 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. swap-sqr 0.0%

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

      7. metadata-eval 0.0%

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

      8. metadata-eval 0.0%

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

      9. swap-sqr 0.0%

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

      10. *-commutative 0.0%

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

      11. *-commutative 0.0%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. *-un-lft-identity 0.0%

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

      15. *-commutative 0.0%

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

      16. unpow2 0.0%

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

      17. associate-*r* 0.0%

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

      18. associate-*l* 0.2%

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

      19. *-commutative 0.2%

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

      20. *-commutative 0.2%

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

   6. Applied egg-rr 80.6%

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

   7. Taylor expanded in c around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.3%

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

      3. unpow2 0.3%

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

      4. unpow2 0.3%

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

      5. swap-sqr 67.5%

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

      6. unpow2 67.5%

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

      7. *-commutative 67.5%

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

      8. unpow2 67.5%

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

      9. associate-*r* 80.2%

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

      10. *-commutative 80.2%

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

      11. *-lft-identity 80.2%

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

      12. associate-*l/ 80.2%

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

   9. Simplified 79.6%

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

       1. *-un-lft-identity 79.6%

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

   11. Applied egg-rr 79.6%

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

       1. *-lft-identity 79.6%

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

       2. associate-*r* 97.5%

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

       3. *-commutative 97.5%

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

   13. Simplified 97.5%

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

4. Final simplification 99.3%

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

Alternative 2: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot s\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2e-8)
   (pow (* c (* x s)) -2.0)
   (/ (/ (cos (* 2.0 x)) c) (* (* x (* c s)) (* x s)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2e-8) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = (cos((2.0 * x)) / c) / ((x * (c * s)) * (x * s));
	}
	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) :: tmp
    if (x <= 2d-8) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (cos((2.0d0 * x)) / c) / ((x * (c * s)) * (x * s))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2e-8) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = (Math.cos((2.0 * x)) / c) / ((x * (c * s)) * (x * s));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2e-8:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = (math.cos((2.0 * x)) / c) / ((x * (c * s)) * (x * s))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2e-8)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(2.0 * x)) / c) / Float64(Float64(x * Float64(c * s)) * Float64(x * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2e-8)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = (cos((2.0 * x)) / c) / ((x * (c * s)) * (x * s));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2e-8], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / c), $MachinePrecision] / N[(N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-8

   1. Initial program 63.9%

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

      1. associate-/r* 63.9%

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

      2. *-commutative 63.9%

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

      3. unpow2 63.9%

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

      4. sqr-neg 63.9%

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

      5. unpow2 63.9%

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

      6. cos-neg 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. metadata-eval 63.9%

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

      10. unpow2 63.9%

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

      11. sqr-neg 63.9%

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

      12. unpow2 63.9%

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

      13. associate-*r* 56.7%

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

      14. unpow2 56.7%

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

      15. *-commutative 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in x around 0 52.4%

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

      1. associate-/r* 52.4%

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

      2. *-commutative 52.4%

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

      3. unpow2 52.4%

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

      4. unpow2 52.4%

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

      5. swap-sqr 67.7%

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

      6. unpow2 67.7%

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

      7. associate-/r* 67.7%

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

      8. unpow2 67.7%

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

      9. unpow2 67.7%

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

      10. swap-sqr 82.4%

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

      11. unpow2 82.4%

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

   7. Simplified 82.4%

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

   8. Taylor expanded in c around 0 52.4%

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

      1. unpow2 52.4%

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

      2. *-commutative 52.4%

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

      3. unpow2 52.4%

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

      4. unpow2 52.4%

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

      5. swap-sqr 67.7%

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

      6. swap-sqr 82.4%

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

      7. associate-*l* 79.8%

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

      8. associate-*l* 82.2%

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

      9. associate-/l/ 82.4%

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

      10. *-lft-identity 82.4%

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

      11. associate-*l/ 82.4%

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

      12. unpow-1 82.4%

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

      13. unpow-1 82.4%

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

      14. pow-sqr 82.4%

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

      15. associate-*l* 82.7%

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

      16. *-commutative 82.7%

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

      17. metadata-eval 82.7%

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

   10. Simplified 82.7%

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

   if 2e-8 < x

   1. Initial program 67.9%

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

      1. associate-/r* 68.0%

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

      2. *-commutative 68.0%

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

      3. unpow2 68.0%

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

      4. sqr-neg 68.0%

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

      5. unpow2 68.0%

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

      6. cos-neg 68.0%

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

      7. *-commutative 68.0%

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

      8. distribute-rgt-neg-in 68.0%

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

      9. metadata-eval 68.0%

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

      10. unpow2 68.0%

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

      11. sqr-neg 68.0%

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

      12. unpow2 68.0%

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

      13. associate-*r* 60.8%

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

      14. unpow2 60.8%

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

      15. *-commutative 60.8%

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

   3. Simplified 60.8%

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

      1. associate-/l/ 60.8%

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

      2. add-sqr-sqrt 32.3%

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

      3. add-sqr-sqrt 60.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 31.2%

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

      6. swap-sqr 31.2%

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

      7. metadata-eval 31.2%

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

      8. metadata-eval 31.2%

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

      9. swap-sqr 31.2%

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

      10. *-commutative 31.2%

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

      11. *-commutative 31.2%

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

      12. sqrt-unprod 57.2%

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

      13. add-sqr-sqrt 60.8%

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

      14. *-un-lft-identity 60.8%

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

      15. *-commutative 60.8%

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

      16. unpow2 60.8%

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

      17. associate-*r* 67.9%

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

      18. associate-*l* 69.4%

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

      19. *-commutative 69.4%

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

      20. *-commutative 69.4%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in c around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-*r* 58.1%

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

      3. unpow2 58.1%

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

      4. unpow2 58.1%

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

      5. swap-sqr 71.7%

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

      6. unpow2 71.7%

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

      7. *-commutative 71.7%

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

      8. unpow2 71.7%

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

      9. associate-*r* 80.5%

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

      10. *-commutative 80.5%

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

      11. *-lft-identity 80.5%

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

      12. associate-*l/ 80.5%

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

   9. Simplified 95.6%

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

       1. *-un-lft-identity 95.6%

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

   11. Applied egg-rr 95.6%

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

       1. *-lft-identity 95.6%

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

       2. associate-*r* 99.7%

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

       3. *-commutative 99.7%

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

   13. Simplified 99.7%

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

       1. *-commutative 99.7%

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

       2. associate-*r* 95.6%

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

       3. *-commutative 95.6%

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

       4. associate-*r* 98.1%

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

       5. metadata-eval 98.1%

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

       6. pow-prod-up 98.1%

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

       7. inv-pow 98.1%

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

       8. inv-pow 98.1%

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

       9. associate-*l* 98.0%

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

       10. associate-*l/ 98.0%

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

       11. associate-*r* 94.0%

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

       12. frac-times 94.0%

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

       13. associate-*r/ 94.0%

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

       14. frac-times 91.4%

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

   15. Applied egg-rr 92.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * 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 65.0%

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

   1. associate-/r* 65.0%

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

   2. *-commutative 65.0%

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

   3. unpow2 65.0%

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

   4. sqr-neg 65.0%

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

   5. unpow2 65.0%

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

   6. cos-neg 65.0%

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

   7. *-commutative 65.0%

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

   8. distribute-rgt-neg-in 65.0%

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

   9. metadata-eval 65.0%

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

   10. unpow2 65.0%

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

   11. sqr-neg 65.0%

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

   12. unpow2 65.0%

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

   13. associate-*r* 57.8%

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

   14. unpow2 57.8%

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

   15. *-commutative 57.8%

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

3. Simplified 57.8%

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

   1. associate-/l/ 57.8%

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

   2. add-sqr-sqrt 45.6%

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

   3. add-sqr-sqrt 57.8%

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

   4. add-sqr-sqrt 21.2%

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

   5. sqrt-unprod 46.1%

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

   6. swap-sqr 46.1%

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

   7. metadata-eval 46.1%

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

   8. metadata-eval 46.1%

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

   9. swap-sqr 46.1%

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

   10. *-commutative 46.1%

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

   11. *-commutative 46.1%

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

   12. sqrt-unprod 32.3%

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

   13. add-sqr-sqrt 57.8%

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

   14. *-un-lft-identity 57.8%

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

   15. *-commutative 57.8%

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

   16. unpow2 57.8%

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

   17. associate-*r* 65.0%

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

   18. associate-*l* 66.4%

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

   19. *-commutative 66.4%

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

   20. *-commutative 66.4%

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

6. Applied egg-rr 83.2%

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

7. Taylor expanded in c around 0 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*r* 57.0%

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

   3. unpow2 57.0%

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

   4. unpow2 57.0%

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

   5. swap-sqr 73.3%

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

   6. unpow2 73.3%

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

   7. *-commutative 73.3%

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

   8. unpow2 73.3%

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

   9. associate-*r* 83.1%

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

   10. *-commutative 83.1%

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

   11. *-lft-identity 83.1%

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

   12. associate-*l/ 83.0%

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

9. Simplified 95.6%

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

    1. *-un-lft-identity 95.6%

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

11. Applied egg-rr 95.6%

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

    1. *-lft-identity 95.6%

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

    2. associate-*r* 97.6%

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

    3. *-commutative 97.6%

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

13. Simplified 97.6%

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

    1. *-commutative 97.6%

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

    2. associate-*r* 95.6%

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

    3. *-commutative 95.6%

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

    4. associate-*r* 96.6%

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

    5. metadata-eval 96.6%

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

    6. pow-prod-up 96.5%

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

    7. inv-pow 96.5%

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

    8. inv-pow 96.5%

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

    9. un-div-inv 96.6%

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

    10. associate-*l/ 96.6%

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

    11. associate-*r* 93.0%

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

    12. *-commutative 93.0%

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

15. Applied egg-rr 97.6%

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

16. Final simplification 97.6%

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

Alternative 4: 78.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 65.0%

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

   2. *-commutative 65.0%

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

   3. unpow2 65.0%

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

   4. sqr-neg 65.0%

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

   5. unpow2 65.0%

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

   6. cos-neg 65.0%

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

   7. *-commutative 65.0%

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

   8. distribute-rgt-neg-in 65.0%

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

   9. metadata-eval 65.0%

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

   10. unpow2 65.0%

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

   11. sqr-neg 65.0%

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

   12. unpow2 65.0%

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

   13. associate-*r* 57.8%

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

   14. unpow2 57.8%

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

   15. *-commutative 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in x around 0 52.3%

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

   1. associate-/r* 52.3%

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

   2. *-commutative 52.3%

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

   3. unpow2 52.3%

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

   4. unpow2 52.3%

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

   5. swap-sqr 65.8%

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

   6. unpow2 65.8%

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

   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 77.8%

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

   11. unpow2 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in c around 0 52.3%

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

   1. unpow2 52.3%

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

   2. *-commutative 52.3%

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

   3. unpow2 52.3%

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

   4. unpow2 52.3%

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

   5. swap-sqr 65.8%

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

   6. swap-sqr 77.8%

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

   7. associate-*l* 75.8%

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

   8. associate-*l* 77.7%

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

   9. associate-/l/ 77.8%

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

   10. *-lft-identity 77.8%

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

   11. associate-*l/ 77.8%

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

   12. unpow-1 77.8%

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

   13. unpow-1 77.8%

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

   14. pow-sqr 77.9%

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

   15. associate-*l* 77.9%

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

   16. *-commutative 77.9%

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

   17. metadata-eval 77.9%

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

10. Simplified 77.9%

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

11. Final simplification 77.9%

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

Alternative 5: 78.1% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 65.0%

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

   2. *-commutative 65.0%

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

   3. unpow2 65.0%

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

   4. sqr-neg 65.0%

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

   5. unpow2 65.0%

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

   6. cos-neg 65.0%

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

   7. *-commutative 65.0%

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

   8. distribute-rgt-neg-in 65.0%

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

   9. metadata-eval 65.0%

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

   10. unpow2 65.0%

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

   11. sqr-neg 65.0%

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

   12. unpow2 65.0%

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

   13. associate-*r* 57.8%

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

   14. unpow2 57.8%

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

   15. *-commutative 57.8%

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

3. Simplified 57.8%

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

6. Applied egg-rr 95.5%

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

   1. *-un-lft-identity 95.5%

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

   2. times-frac 95.6%

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

   3. *-commutative 95.6%

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

8. Applied egg-rr 95.6%

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

9. Taylor expanded in x around 0 77.9%

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

    1. associate-/r* 77.9%

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

11. Simplified 77.9%

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

Alternative 6: 78.2% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 65.0%

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

   2. *-commutative 65.0%

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

   3. unpow2 65.0%

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

   4. sqr-neg 65.0%

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

   5. unpow2 65.0%

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

   6. cos-neg 65.0%

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

   7. *-commutative 65.0%

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

   8. distribute-rgt-neg-in 65.0%

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

   9. metadata-eval 65.0%

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

   10. unpow2 65.0%

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

   11. sqr-neg 65.0%

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

   12. unpow2 65.0%

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

   13. associate-*r* 57.8%

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

   14. unpow2 57.8%

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

   15. *-commutative 57.8%

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

3. Simplified 57.8%

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

6. Applied egg-rr 95.5%

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

7. Taylor expanded in x around 0 77.9%

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

Alternative 7: 78.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 65.0%

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

   2. *-commutative 65.0%

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

   3. unpow2 65.0%

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

   4. sqr-neg 65.0%

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

   5. unpow2 65.0%

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

   6. cos-neg 65.0%

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

   7. *-commutative 65.0%

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

   8. distribute-rgt-neg-in 65.0%

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

   9. metadata-eval 65.0%

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

   10. unpow2 65.0%

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

   11. sqr-neg 65.0%

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

   12. unpow2 65.0%

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

   13. associate-*r* 57.8%

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

   14. unpow2 57.8%

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

   15. *-commutative 57.8%

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

3. Simplified 57.8%

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

   1. associate-/l/ 57.8%

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

   2. add-sqr-sqrt 45.6%

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

   3. add-sqr-sqrt 57.8%

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

   4. add-sqr-sqrt 21.2%

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

   5. sqrt-unprod 46.1%

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

   6. swap-sqr 46.1%

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

   7. metadata-eval 46.1%

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

   8. metadata-eval 46.1%

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

   9. swap-sqr 46.1%

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

   10. *-commutative 46.1%

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

   11. *-commutative 46.1%

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

   12. sqrt-unprod 32.3%

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

   13. add-sqr-sqrt 57.8%

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

   14. *-un-lft-identity 57.8%

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

   15. *-commutative 57.8%

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

   16. unpow2 57.8%

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

   17. associate-*r* 65.0%

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

   18. associate-*l* 66.4%

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

   19. *-commutative 66.4%

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

   20. *-commutative 66.4%

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

6. Applied egg-rr 83.2%

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

7. Taylor expanded in c around 0 57.8%

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

   1. *-commutative 57.8%

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

   2. associate-*r* 57.0%

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

   3. unpow2 57.0%

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

   4. unpow2 57.0%

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

   5. swap-sqr 73.3%

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

   6. unpow2 73.3%

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

   7. *-commutative 73.3%

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

   8. unpow2 73.3%

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

   9. associate-*r* 83.1%

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

   10. *-commutative 83.1%

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

   11. *-lft-identity 83.1%

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

   12. associate-*l/ 83.0%

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

9. Simplified 95.6%

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

    1. *-commutative 95.6%

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

    2. associate-*l* 96.6%

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

    3. metadata-eval 96.6%

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

    4. pow-flip 96.3%

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

    5. pow2 96.3%

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

    6. associate-/r* 96.6%

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

    7. associate-*l* 93.0%

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

    8. *-commutative 93.0%

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

    9. associate-*l* 95.6%

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

    10. *-commutative 95.6%

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

11. Applied egg-rr 95.6%

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

12. Taylor expanded in x around 0 77.9%

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

13. Final simplification 77.9%

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

Alternative 8: 78.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. associate-/r* 65.0%

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

   2. *-commutative 65.0%

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

   3. unpow2 65.0%

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

   4. sqr-neg 65.0%

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

   5. unpow2 65.0%

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

   6. cos-neg 65.0%

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

   7. *-commutative 65.0%

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

   8. distribute-rgt-neg-in 65.0%

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

   9. metadata-eval 65.0%

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

   10. unpow2 65.0%

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

   11. sqr-neg 65.0%

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

   12. unpow2 65.0%

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

   13. associate-*r* 57.8%

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

   14. unpow2 57.8%

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

   15. *-commutative 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in x around 0 52.3%

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

   1. associate-/r* 52.3%

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

   2. *-commutative 52.3%

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

   3. unpow2 52.3%

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

   4. unpow2 52.3%

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

   5. swap-sqr 65.8%

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

   6. unpow2 65.8%

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

   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 77.8%

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

   11. unpow2 77.8%

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

7. Simplified 77.8%

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

   1. unpow2 77.8%

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

   2. associate-*r* 75.8%

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

   3. associate-*r* 77.7%

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

9. Applied egg-rr 77.7%

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

10. Taylor expanded in c around 0 75.8%

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

    1. *-commutative 75.8%

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

12. Simplified 75.8%

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

13. Taylor expanded in c around 0 77.8%

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

    1. *-commutative 75.8%

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

15. Simplified 77.8%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))