mixedcos

Percentage Accurate: 67.1% → 97.1%

Time: 11.9s

Alternatives: 10

Speedup: 9.0×

• 31.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 70.9

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. *-commutative N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-down N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot c\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s c) x)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) 4e+191)
     (/ (cos (+ x x)) (* (* (* (* c c) x) (* s x)) s))
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= 4e+191) {
		tmp = cos((x + x)) / ((((c * c) * x) * (s * x)) * s);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s * c) * x
    if ((cos((2.0d0 * x)) / ((((s ** 2.0d0) * x) * x) * (c ** 2.0d0))) <= 4d+191) then
        tmp = cos((x + x)) / ((((c * c) * x) * (s * x)) * s)
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c, 2.0))) <= 4e+191) {
		tmp = Math.cos((x + x)) / ((((c * c) * x) * (s * x)) * s);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (s * c) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s, 2.0) * x) * x) * math.pow(c, 2.0))) <= 4e+191:
		tmp = math.cos((x + x)) / ((((c * c) * x) * (s * x)) * s)
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * c) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= 4e+191)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(Float64(c * c) * x) * Float64(s * x)) * s));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (s * c) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= 4e+191)
		tmp = cos((x + x)) / ((((c * c) * x) * (s * x)) * s);
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+191], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$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))) < 4.00000000000000029e191

   1. Initial program 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 94.0

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

   4. Applied rewrites 94.0%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 94.0

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

   6. Applied rewrites 94.0%

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

   if 4.00000000000000029e191 < (/.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 53.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 62.6

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 86.8

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 86.8

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 86.8

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

   9. Applied rewrites 86.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq 4 \cdot 10^{+191}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot c\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s \cdot s}}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s c) x)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -4e-214)
     (/ (/ (/ -2.0 (* s s)) c) c)
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -4e-214) {
		tmp = ((-2.0 / (s * s)) / c) / c;
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (s * c) * x
    if ((cos((2.0d0 * x)) / ((((s ** 2.0d0) * x) * x) * (c ** 2.0d0))) <= (-4d-214)) then
        tmp = (((-2.0d0) / (s * s)) / c) / c
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c, 2.0))) <= -4e-214) {
		tmp = ((-2.0 / (s * s)) / c) / c;
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (s * c) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s, 2.0) * x) * x) * math.pow(c, 2.0))) <= -4e-214:
		tmp = ((-2.0 / (s * s)) / c) / c
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * c) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -4e-214)
		tmp = Float64(Float64(Float64(-2.0 / Float64(s * s)) / c) / c);
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (s * c) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -4e-214)
		tmp = ((-2.0 / (s * s)) / c) / c;
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-214], N[(N[(N[(-2.0 / N[(s * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$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))) < -3.99999999999999965e-214

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.6%

      \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]

   if -3.99999999999999965e-214 < (/.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 70.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 88.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 88.5

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 88.5

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

   9. Applied rewrites 88.5%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s \cdot s}}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot c\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c x) s)))
   (if (<= x 5.8e-17)
     (/ (/ 1.0 (* (* s x) c)) (* (* s c) x))
     (/ (cos (* 2.0 x)) (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if (x <= 5.8e-17) {
		tmp = (1.0 / ((s * x) * c)) / ((s * c) * x);
	} else {
		tmp = cos((2.0 * x)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * x) * s
    if (x <= 5.8d-17) then
        tmp = (1.0d0 / ((s * x) * c)) / ((s * c) * x)
    else
        tmp = cos((2.0d0 * x)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if (x <= 5.8e-17) {
		tmp = (1.0 / ((s * x) * c)) / ((s * c) * x);
	} else {
		tmp = Math.cos((2.0 * x)) / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * x) * s
	tmp = 0
	if x <= 5.8e-17:
		tmp = (1.0 / ((s * x) * c)) / ((s * c) * x)
	else:
		tmp = math.cos((2.0 * x)) / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * x) * s)
	tmp = 0.0
	if (x <= 5.8e-17)
		tmp = Float64(Float64(1.0 / Float64(Float64(s * x) * c)) / Float64(Float64(s * c) * x));
	else
		tmp = Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * x) * s;
	tmp = 0.0;
	if (x <= 5.8e-17)
		tmp = (1.0 / ((s * x) * c)) / ((s * c) * x);
	else
		tmp = cos((2.0 * x)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[x, 5.8e-17], N[(N[(1.0 / N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8000000000000006e-17

   1. Initial program 70.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.8

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

   9. Applied rewrites 86.8%

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

   if 5.8000000000000006e-17 < x

   1. Initial program 71.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot c\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 70.9

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. *-commutative N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-down N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.6

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.6

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 99.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.6

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 6: 97.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 70.9

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. *-commutative N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-down N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 99.2

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

4. Applied rewrites 99.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 99.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 99.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 99.2

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

6. Applied rewrites 99.2%

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

7. Final simplification 99.2%

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

Alternative 7: 67.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot s\right) \cdot x\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 6.4e-105)
   (/ 1.0 (* (* (* (* c c) s) x) (* s x)))
   (/ 1.0 (* (* (* s c) s) (* (* x x) c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 6.4e-105) {
		tmp = 1.0 / ((((c * c) * s) * x) * (s * x));
	} else {
		tmp = 1.0 / (((s * c) * s) * ((x * 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 <= 6.4d-105) then
        tmp = 1.0d0 / ((((c * c) * s) * x) * (s * x))
    else
        tmp = 1.0d0 / (((s * c) * s) * ((x * x) * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 6.4e-105) {
		tmp = 1.0 / ((((c * c) * s) * x) * (s * x));
	} else {
		tmp = 1.0 / (((s * c) * s) * ((x * x) * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 6.4e-105:
		tmp = 1.0 / ((((c * c) * s) * x) * (s * x))
	else:
		tmp = 1.0 / (((s * c) * s) * ((x * x) * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 6.4e-105)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * c) * s) * x) * Float64(s * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s * c) * s) * Float64(Float64(x * x) * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 6.4e-105)
		tmp = 1.0 / ((((c * c) * s) * x) * (s * x));
	else
		tmp = 1.0 / (((s * c) * s) * ((x * x) * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 6.4e-105], N[(1.0 / N[(N[(N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision] * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s * c), $MachinePrecision] * s), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.39999999999999962e-105

   1. Initial program 71.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 77.0

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

   10. Applied rewrites 77.0%

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

   12. Applied rewrites 73.4%

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

   if 6.39999999999999962e-105 < x

   1. Initial program 68.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 71.2

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

   10. Applied rewrites 71.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot s\right) \cdot x\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot s\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 78.3

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

4. Applied rewrites 78.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 84.0

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 84.0

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 84.0

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

9. Applied rewrites 84.0%

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

10. Final simplification 84.0%

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

Alternative 9: 76.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 78.3

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

4. Applied rewrites 78.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-*l* N/A

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

   20. lower-*.f64 N/A

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

   21. lower-*.f64 82.5

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 82.5

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

9. Applied rewrites 82.5%

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

10. Final simplification 82.5%

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

Alternative 10: 65.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

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 / (((s * c) * s) * ((x * x) * c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 78.3

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

4. Applied rewrites 78.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 69.5%

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

8. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 75.1

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

10. Applied rewrites 75.1%

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

11. Final simplification 75.1%

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

Reproduce

herbie shell --seed 2024272 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))