mixedcos

Percentage Accurate: 67.0% → 97.4%

Time: 10.6s

Alternatives: 8

Speedup: 2.4×

• 32.0% 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: 67.0% 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: 97.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.4%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow2 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 97.2

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

4. Applied rewrites 97.2%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 97.5

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

   7. lift-*.f64 N/A

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

   8. count-2 N/A

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

   9. lift-+.f64 97.5

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 96.5

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 98.6

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

6. Applied rewrites 98.6%

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

7. Final simplification 98.6%

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

Alternative 2: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -2e+21)
     (/ -2.0 (* c (* c (* s s))))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if ((cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -2e+21) {
		tmp = -2.0 / (c * (c * (s * 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 = c * (x * s)
    if ((cos((x * 2.0d0)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-2d+21)) then
        tmp = (-2.0d0) / (c * (c * (s * 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 = c * (x * s);
	double tmp;
	if ((Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -2e+21) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if (math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -2e+21:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -2e+21)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -2e+21)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e+21], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $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))) < -2e21

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 37.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 52.2%

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

   if -2e21 < (/.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 67.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))