mixedcos

Percentage Accurate: 67.3% → 96.9%

Time: 11.3s

Alternatives: 9

Speedup: 9.0×

• 32.4% 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: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\right)\\ \frac{\cos \left(x + x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x)))) (/ (cos (+ x x)) (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return cos((x + x)) / (t_0 * t_0);
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x)
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return Math.cos((x + x)) / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	return math.cos((x + x)) / (t_0 * t_0)

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. 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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 67.8

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 67.8

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

4. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 67.8

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

6. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. swap-sqr N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 97.2

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 97.2

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 97.2

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

8. Applied rewrites 97.2%

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

9. Final simplification 97.2%

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

Alternative 2: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s\_m}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(s\_m \cdot x\right) \cdot \left(s\_m \cdot x\right)\right) \cdot c\_m\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (* (* (pow s_m 2.0) x) x) (pow c_m 2.0)))
        -2e-69)
     (/ (fma -2.0 (* x x) 1.0) (* (* (* (* s_m x) (* s_m x)) c_m) c_m))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s_m, 2.0) * x) * x) * pow(c_m, 2.0))) <= -2e-69) {
		tmp = fma(-2.0, (x * x), 1.0) / ((((s_m * x) * (s_m * x)) * c_m) * c_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(Float64(s_m * x) * Float64(s_m * x)) * c_m) * c_m));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-69], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(s$95$m * x), $MachinePrecision] * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $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))) < -1.9999999999999999e-69

   1. Initial program 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 96.8

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

   4. Applied rewrites 96.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 43.0

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

   9. Applied rewrites 43.0%

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

   if -1.9999999999999999e-69 < (/.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 66.7%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.7

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 84.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s\_m}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s\_m}}{c\_m}}{c\_m \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (* (* (pow s_m 2.0) x) x) (pow c_m 2.0)))
        -2e-69)
     (/ (/ (/ -2.0 s_m) c_m) (* c_m s_m))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s_m, 2.0) * x) * x) * pow(c_m, 2.0))) <= -2e-69) {
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x)
    if ((cos((2.0d0 * x)) / ((((s_m ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= (-2d-69)) then
        tmp = (((-2.0d0) / s_m) / c_m) / (c_m * s_m)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s_m, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= -2e-69) {
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s_m, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= -2e-69:
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = Float64(Float64(Float64(-2.0 / s_m) / c_m) / Float64(c_m * s_m));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-69], N[(N[(N[(-2.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(c$95$m * s$95$m), $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))) < -1.9999999999999999e-69

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      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. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.1

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. pow-prod-down N/A

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

      21. lower-pow.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

   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 28.7%

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

   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 43.0%

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

   if -1.9999999999999999e-69 < (/.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 66.7%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.7

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

   9. Applied rewrites 84.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s}}{c}}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s\_m}^{2} \cdot x\right) \cdot x\right) \cdot {c\_m}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s\_m}}{c\_m}}{c\_m \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (* (* (pow s_m 2.0) x) x) (pow c_m 2.0)))
        -2e-69)
     (/ (/ (/ -2.0 s_m) c_m) (* c_m s_m))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s_m, 2.0) * x) * x) * pow(c_m, 2.0))) <= -2e-69) {
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (s_m * x)
    if ((cos((2.0d0 * x)) / ((((s_m ** 2.0d0) * x) * x) * (c_m ** 2.0d0))) <= (-2d-69)) then
        tmp = (((-2.0d0) / s_m) / c_m) / (c_m * s_m)
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s_m, 2.0) * x) * x) * Math.pow(c_m, 2.0))) <= -2e-69) {
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s_m, 2.0) * x) * x) * math.pow(c_m, 2.0))) <= -2e-69:
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m)
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = Float64(Float64(Float64(-2.0 / s_m) / c_m) / Float64(c_m * s_m));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s_m ^ 2.0) * x) * x) * (c_m ^ 2.0))) <= -2e-69)
		tmp = ((-2.0 / s_m) / c_m) / (c_m * s_m);
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-69], N[(N[(N[(-2.0 / s$95$m), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1.9999999999999999e-69

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      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. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 72.1

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. pow-prod-down N/A

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

      21. lower-pow.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

   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 28.7%

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

   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 43.0%

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

   if -1.9999999999999999e-69 < (/.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 66.7%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 68.7

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 65.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. swap-sqr N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 84.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 84.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 84.9

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

   9. Applied rewrites 84.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{-2}{s}}{c}}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\right)\\ \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s\_m \cdot s\_m\right) \cdot c\_m\right) \cdot \left(\left(c\_m \cdot x\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x))))
   (if (<= x 0.78)
     (/
      (/
       (fma
        (fma
         (fma -0.08888888888888889 (* x x) 0.6666666666666666)
         (* x x)
         -2.0)
        (* x x)
        1.0)
       t_0)
      t_0)
     (/ (cos (+ x x)) (* (* (* s_m s_m) c_m) (* (* c_m x) x))))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	double tmp;
	if (x <= 0.78) {
		tmp = (fma(fma(fma(-0.08888888888888889, (x * x), 0.6666666666666666), (x * x), -2.0), (x * x), 1.0) / t_0) / t_0;
	} else {
		tmp = cos((x + x)) / (((s_m * s_m) * c_m) * ((c_m * x) * x));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	tmp = 0.0
	if (x <= 0.78)
		tmp = Float64(Float64(fma(fma(fma(-0.08888888888888889, Float64(x * x), 0.6666666666666666), Float64(x * x), -2.0), Float64(x * x), 1.0) / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(s_m * s_m) * c_m) * Float64(Float64(c_m * x) * x)));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.78], N[(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] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(N[(c$95$m * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow-prod-down N/A

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

      16. lower-pow.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.7

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

   6. Applied rewrites 85.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 64.9

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

   9. Applied rewrites 64.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*l* N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/r* N/A

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

       12. lower-/.f64 N/A

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

   11. Applied rewrites 71.1%

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

   if 0.78000000000000003 < x

   1. Initial program 59.6%

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

      1. lift-*.f64 N/A

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

      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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 61.2

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 61.2

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 61.2

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

   6. Applied rewrites 61.2%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.1

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

   9. Applied rewrites 78.1%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, -2\right), x \cdot x, 1\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot \left(\left(c \cdot x\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s\_m \cdot x\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* s_m x)))) (/ 1.0 (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return 1.0 / (t_0 * t_0);
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (s_m * x)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (s_m * x);
	return 1.0 / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (s_m * x)
	return 1.0 / (t_0 * t_0)

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(s_m * x))
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (s_m * x);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.0%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. 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. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 67.8

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 67.8

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

4. Applied rewrites 67.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 60.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. swap-sqr N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 77.3

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 77.3

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 77.3

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

9. Applied rewrites 77.3%

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

Alternative 7: 78.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{\left(\left(\left(c\_m \cdot \left(s\_m \cdot x\right)\right) \cdot x\right) \cdot s\_m\right) \cdot c\_m} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* (* (* (* c_m (* s_m x)) x) s_m) c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / ((((c_m * (s_m * x)) * x) * s_m) * c_m);
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((((c_m * (s_m * x)) * x) * s_m) * c_m)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / ((((c_m * (s_m * x)) * x) * s_m) * c_m);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / ((((c_m * (s_m * x)) * x) * s_m) * c_m)

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(Float64(Float64(Float64(c_m * Float64(s_m * x)) * x) * s_m) * c_m))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / ((((c_m * (s_m * x)) * x) * s_m) * c_m);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(N[(N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 65.5

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

5. Applied rewrites 65.5%

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

7. Applied rewrites 63.9%

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

9. Applied rewrites 69.5%

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

11. Applied rewrites 74.8%

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

12. Final simplification 74.8%

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

Alternative 8: 76.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{\left(c\_m \cdot s\_m\right) \cdot \left(\left(\left(s\_m \cdot x\right) \cdot x\right) \cdot c\_m\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* (* c_m s_m) (* (* (* s_m x) x) c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / ((c_m * s_m) * (((s_m * x) * x) * c_m));
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / ((c_m * s_m) * (((s_m * x) * x) * c_m))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / ((c_m * s_m) * (((s_m * x) * x) * c_m));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / ((c_m * s_m) * (((s_m * x) * x) * c_m))

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(Float64(Float64(s_m * x) * x) * c_m)))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / ((c_m * s_m) * (((s_m * x) * x) * c_m));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(N[(N[(s$95$m * x), $MachinePrecision] * x), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 65.5

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

5. Applied rewrites 65.5%

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

7. Applied rewrites 63.9%

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

9. Applied rewrites 69.5%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 71.6%

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

13. Final simplification 71.6%

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

Alternative 9: 62.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{\left(\left(s\_m \cdot s\_m\right) \cdot c\_m\right) \cdot \left(\left(c\_m \cdot x\right) \cdot x\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ 1.0 (* (* (* s_m s_m) c_m) (* (* c_m x) x))))

C

s_m = fabs(s);
c_m = fabs(c);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return 1.0 / (((s_m * s_m) * c_m) * ((c_m * x) * x));
}

Fortran

s_m = abs(s)
c_m = abs(c)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 1.0d0 / (((s_m * s_m) * c_m) * ((c_m * x) * x))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return 1.0 / (((s_m * s_m) * c_m) * ((c_m * x) * x));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return 1.0 / (((s_m * s_m) * c_m) * ((c_m * x) * x))

Julia

s_m = abs(s)
c_m = abs(c)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(1.0 / Float64(Float64(Float64(s_m * s_m) * c_m) * Float64(Float64(c_m * x) * x)))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = 1.0 / (((s_m * s_m) * c_m) * ((c_m * x) * x));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(1.0 / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * N[(N[(c$95$m * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 65.5

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

5. Applied rewrites 65.5%

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

7. Applied rewrites 63.9%

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

9. Applied rewrites 69.5%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 65.0%

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

13. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2024241 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))