mixedcos

Percentage Accurate: 67.4% → 97.3%

Time: 9.0s

Alternatives: 6

Speedup: 9.0×

• 32.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 2.3× speedup

Math

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

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(s * Float64(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[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 66.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 66.0

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. pow-prod-down N/A

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

   16. lower-pow.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 76.1

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

   19. lift-pow.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 76.1

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

4. Applied rewrites 76.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow-prod-down N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 98.5

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 98.5

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

6. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2 N/A

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

   4. lower-+.f64 98.5

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 2: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -4e-231)
     (/
      (fma
       (fma (fma -0.08888888888888889 (* x x) 0.6666666666666666) (* x x) -2.0)
       (* x x)
       1.0)
      (* (* (* s x) (* (* c c) x)) s))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -4e-231) {
		tmp = fma(fma(fma(-0.08888888888888889, (x * x), 0.6666666666666666), (x * x), -2.0), (x * x), 1.0) / (((s * x) * ((c * c) * x)) * s);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -4e-231)
		tmp = Float64(fma(fma(fma(-0.08888888888888889, Float64(x * x), 0.6666666666666666), Float64(x * x), -2.0), Float64(x * x), 1.0) / Float64(Float64(Float64(s * x) * Float64(Float64(c * c) * x)) * s));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $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-231], N[(N[(N[(N[(-0.08888888888888889 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(s * x), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $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))) < -4e-231

   1. Initial program 72.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right)}}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right) + 1}}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2\right) \cdot {x}^{2}} + 1}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) - 2, {x}^{2}, 1\right)}}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(2\right)\right)}, {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(2\right)\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{-2}, {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{-4}{45} \cdot {x}^{2}, {x}^{2}, -2\right)}, {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{45} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-4}{45}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, -2\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, -2\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, -2\right), {x}^{2}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      14. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, -2\right), \color{blue}{x \cdot x}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

      15. lower-*.f64 51.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), \color{blue}{x \cdot x}, 1\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

   7. Applied rewrites 51.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)\right) \cdot s} \]

   if -4e-231 < (/.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 65.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 65.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.6

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 75.2

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.4

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 84.5

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

   9. Applied rewrites 84.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 85.1

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 85.1

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

   11. Applied rewrites 85.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -4 \cdot 10^{-231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := s \cdot \left(c \cdot x\right)\\ t_1 := \left(s \cdot x\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-249}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* c x))) (t_1 (* (* s x) c)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -1e-249)
     (/ (fma -2.0 (* x x) 1.0) (* t_0 t_0))
     (/ (/ 1.0 t_1) t_1))))

C

double code(double x, double c, double s) {
	double t_0 = s * (c * x);
	double t_1 = (s * x) * c;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -1e-249) {
		tmp = fma(-2.0, (x * x), 1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(s * Float64(c * x))
	t_1 = Float64(Float64(s * x) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -1e-249)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * x), $MachinePrecision] * c), $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], -1e-249], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $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))) < -1.00000000000000005e-249

   1. Initial program 73.9%

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

   3. 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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 48.8

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

   8. Applied rewrites 48.8%

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

   if -1.00000000000000005e-249 < (/.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 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 65.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.4

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 75.1

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

      19. lift-pow.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 75.1

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

   4. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow-prod-down N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 98.4

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 84.9

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

   9. Applied rewrites 84.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 85.5

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 85.5

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

   11. Applied rewrites 85.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-249}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(s \cdot x\right) \cdot c}}{\left(s \cdot x\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* c x)))) (/ (cos (+ 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(s * Float64(c * x))
	return Float64(cos(Float64(x + x)) / Float64(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[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
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 98.2

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

5. Applied rewrites 98.2%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 98.2

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Alternative 5: 79.0% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 66.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 66.0

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. pow-prod-down N/A

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

   16. lower-pow.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 76.1

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

   19. lift-pow.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 76.1

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

4. Applied rewrites 76.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow-prod-down N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. associate-/r* N/A

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

   17. lower-/.f64 N/A

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

   18. lower-/.f64 98.5

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f64 98.5

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 79.0

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

9. Applied rewrites 79.0%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. lift-*.f64 79.5

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 79.5

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

11. Applied rewrites 79.5%

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

12. Final simplification 79.5%

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

Alternative 6: 78.7% accurate, 9.0× speedup

Math

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

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(s * Float64(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[(s * N[(c * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
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 98.2

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.8%

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

9. Final simplification 79.8%

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

Reproduce

herbie shell --seed 2024243 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))