mixedcos

Percentage Accurate: 66.3% → 97.5%

Time: 15.3s

Alternatives: 9

Speedup: 24.1×

• 30.8% 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: 66.3% 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: 97.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (c * s);
	tmp = (cos((x * 2.0)) / 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[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

6. Applied egg-rr 96.9%

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

   1. associate-*l/ 96.9%

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

   2. *-commutative 96.9%

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

   3. unpow2 96.9%

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

   4. frac-times 97.2%

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

   5. associate-*l/ 97.2%

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

   6. div-inv 97.3%

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

   7. *-commutative 97.3%

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

   8. *-commutative 97.3%

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

   9. associate-*l* 96.0%

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

   10. *-commutative 96.0%

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

   11. *-commutative 96.0%

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

   12. associate-*l* 98.3%

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

   13. *-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

Alternative 2: 86.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 1.3e-8)
   (pow (* c (* x s)) -2.0)
   (/ (/ (cos (* x 2.0)) x) (* (* c s) (* s (* x c))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.3e-8) {
		tmp = pow((c * (x * s)), -2.0);
	} else {
		tmp = (cos((x * 2.0)) / x) / ((c * s) * (s * (x * c)));
	}
	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 <= 1.3d-8) then
        tmp = (c * (x * s)) ** (-2.0d0)
    else
        tmp = (cos((x * 2.0d0)) / x) / ((c * s) * (s * (x * c)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.3e-8) {
		tmp = Math.pow((c * (x * s)), -2.0);
	} else {
		tmp = (Math.cos((x * 2.0)) / x) / ((c * s) * (s * (x * c)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 1.3e-8:
		tmp = math.pow((c * (x * s)), -2.0)
	else:
		tmp = (math.cos((x * 2.0)) / x) / ((c * s) * (s * (x * c)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 1.3e-8)
		tmp = Float64(c * Float64(x * s)) ^ -2.0;
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / x) / Float64(Float64(c * s) * Float64(s * Float64(x * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.3e-8)
		tmp = (c * (x * s)) ^ -2.0;
	else
		tmp = (cos((x * 2.0)) / x) / ((c * s) * (s * (x * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 1.3e-8], N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[(N[(c * s), $MachinePrecision] * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3000000000000001e-8

   1. Initial program 61.5%

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

      1. associate-/r* 60.8%

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

      2. cos-neg 60.8%

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

      3. distribute-rgt-neg-out 60.8%

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

      4. distribute-rgt-neg-out 60.8%

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

      5. *-commutative 60.8%

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

      6. distribute-rgt-neg-in 60.8%

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

      7. metadata-eval 60.8%

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

      8. *-commutative 60.8%

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

      9. associate-*l* 57.1%

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

      10. unpow2 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around 0 52.5%

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

      1. associate-/r* 52.1%

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

      2. *-commutative 52.1%

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

      3. unpow2 52.1%

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

      4. unpow2 52.1%

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

      5. swap-sqr 66.0%

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

      6. unpow2 66.0%

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

      7. associate-/r* 66.4%

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

      8. unpow2 66.4%

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

      9. unpow2 66.4%

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

      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 52.5%

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

      1. associate-*r* 54.1%

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

      2. unpow2 54.1%

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

      3. unpow2 54.1%

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

      4. swap-sqr 68.7%

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

      5. unpow2 68.7%

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

      6. swap-sqr 82.8%

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

      7. *-commutative 82.8%

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

      8. *-commutative 82.8%

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

      9. associate-/l/ 82.8%

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

      10. *-lft-identity 82.8%

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

      11. associate-*l/ 82.7%

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

      12. unpow-1 82.7%

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

      13. unpow-1 82.7%

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

      14. pow-sqr 82.8%

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

      15. metadata-eval 82.8%

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

      16. *-commutative 82.8%

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

      17. associate-*r* 82.7%

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

   10. Simplified 82.7%

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

   if 1.3000000000000001e-8 < x

   1. Initial program 62.2%

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

      1. associate-/r* 61.1%

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

      2. cos-neg 61.1%

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

      3. distribute-rgt-neg-out 61.1%

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

      4. distribute-rgt-neg-out 61.1%

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

      5. *-commutative 61.1%

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

      6. distribute-rgt-neg-in 61.1%

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

      7. metadata-eval 61.1%

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

      8. *-commutative 61.1%

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

      9. associate-*l* 54.6%

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

      10. unpow2 54.6%

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

   3. Simplified 54.6%

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

   6. Applied egg-rr 97.6%

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

      1. unpow2 97.6%

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

      2. associate-*r* 90.4%

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

      3. associate-*r* 90.3%

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

      4. *-commutative 90.3%

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

      5. associate-*l* 91.8%

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

      6. *-commutative 91.8%

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

   8. Applied egg-rr 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-*l/ 91.8%

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

      3. *-un-lft-identity 91.8%

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

      4. associate-*l* 91.8%

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

      5. *-commutative 91.8%

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

      6. associate-*r* 99.0%

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

      7. associate-/l/ 99.6%

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

      8. associate-/r* 99.6%

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

      9. associate-/l/ 90.6%

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

      10. *-commutative 90.6%

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

      11. *-commutative 90.6%

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

      12. associate-*l* 83.3%

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

      13. *-commutative 83.3%

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

   10. Applied egg-rr 83.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

6. Applied egg-rr 96.9%

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

   1. associate-*l/ 96.9%

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

   2. unpow2 96.9%

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

   3. *-un-lft-identity 96.9%

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

   4. associate-/r* 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/l/ 92.6%

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

   7. *-commutative 92.6%

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

   8. *-commutative 92.6%

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

   9. associate-*l* 91.3%

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

   10. *-commutative 91.3%

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

8. Applied egg-rr 91.3%

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

Alternative 4: 78.4% 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 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

5. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

7. Simplified 76.5%

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

8. Taylor expanded in c around 0 51.3%

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

   1. associate-*r* 52.6%

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

   2. unpow2 52.6%

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

   3. unpow2 52.6%

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

   4. swap-sqr 65.3%

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

   5. unpow2 65.3%

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

   6. swap-sqr 76.8%

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

   7. *-commutative 76.8%

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

   8. *-commutative 76.8%

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

   9. associate-/l/ 76.8%

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

   10. *-lft-identity 76.8%

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

   11. associate-*l/ 76.8%

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

   12. unpow-1 76.8%

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

   13. unpow-1 76.8%

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

   14. pow-sqr 76.8%

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

   15. metadata-eval 76.8%

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

   16. *-commutative 76.8%

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

   17. associate-*r* 76.7%

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

10. Simplified 76.7%

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

11. Final simplification 76.7%

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

Alternative 5: 78.4% 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 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

6. Applied egg-rr 97.2%

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

7. Taylor expanded in x around 0 76.7%

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

Alternative 6: 78.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

6. Applied egg-rr 96.9%

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

   1. associate-*l/ 96.9%

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

   2. *-commutative 96.9%

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

   3. unpow2 96.9%

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

   4. frac-times 97.2%

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

   5. associate-*l/ 97.2%

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

   6. div-inv 97.3%

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

   7. *-commutative 97.3%

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

   8. *-commutative 97.3%

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

   9. associate-*l* 96.0%

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

   10. *-commutative 96.0%

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

   11. *-commutative 96.0%

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

   12. associate-*l* 98.3%

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

   13. *-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

9. Taylor expanded in x around 0 75.9%

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

    1. associate-*r* 76.8%

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

    2. remove-double-neg 76.8%

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

    3. distribute-lft-neg-in 76.8%

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

    4. *-commutative 76.8%

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

    5. distribute-rgt-neg-out 76.8%

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

    6. remove-double-neg 76.8%

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

11. Simplified 76.8%

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

Alternative 7: 78.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

5. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

7. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. *-commutative 76.5%

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

   3. *-commutative 76.5%

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

   4. associate-*l* 75.9%

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

   5. *-commutative 75.9%

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

   6. associate-*l* 76.8%

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

   7. *-commutative 76.8%

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

9. Applied egg-rr 76.8%

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

Alternative 8: 75.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

5. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

7. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*r* 74.9%

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

   3. associate-*l* 72.8%

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

   4. *-commutative 72.8%

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

   5. associate-*l* 73.4%

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

   6. *-commutative 73.4%

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

9. Applied egg-rr 73.4%

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

10. Final simplification 73.4%

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

Alternative 9: 75.1% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. associate-/r* 60.9%

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

   2. cos-neg 60.9%

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

   3. distribute-rgt-neg-out 60.9%

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

   4. distribute-rgt-neg-out 60.9%

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

   5. *-commutative 60.9%

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

   6. distribute-rgt-neg-in 60.9%

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

   7. metadata-eval 60.9%

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

   8. *-commutative 60.9%

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

   9. associate-*l* 56.5%

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

   10. unpow2 56.5%

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

3. Simplified 56.5%

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

5. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

7. Simplified 76.5%

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

   1. unpow2 76.5%

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

   2. associate-*r* 74.9%

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

   3. associate-*l* 72.8%

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

   4. *-commutative 72.8%

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

   5. associate-*l* 73.4%

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

   6. *-commutative 73.4%

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

9. Applied egg-rr 73.4%

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

10. Taylor expanded in x around 0 72.8%

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

11. Final simplification 72.8%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))