mixedcos

Percentage Accurate: 67.5% → 95.9%

Time: 8.0s

Alternatives: 10

Speedup: 2.4×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% 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: 95.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{c\_m}}{\left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(\left(c\_m \cdot s\_m\right) \cdot x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 4.9e-89)
   (/ (/ (fma -2.0 (* x_m x_m) 1.0) c_m) (* (* c_m (* s_m x_m)) (* s_m x_m)))
   (/ (cos (+ x_m x_m)) (* (* x_m c_m) (* s_m (* (* c_m s_m) x_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4.9e-89) {
		tmp = (fma(-2.0, (x_m * x_m), 1.0) / c_m) / ((c_m * (s_m * x_m)) * (s_m * x_m));
	} else {
		tmp = cos((x_m + x_m)) / ((x_m * c_m) * (s_m * ((c_m * s_m) * x_m)));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 4.9e-89)
		tmp = Float64(Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / c_m) / Float64(Float64(c_m * Float64(s_m * x_m)) * Float64(s_m * x_m)));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(Float64(c_m * s_m) * x_m))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 4.9e-89], N[(N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(N[(c$95$m * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9e-89

   1. Initial program 59.1%

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

      1. lift-/.f64 N/A

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

      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-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 80.9%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-/r* N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 69.3%

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

   if 4.9e-89 < x

   1. Initial program 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.3

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 96.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.2

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

   8. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 96.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.2

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

   10. Applied rewrites 96.2%

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

Alternative 2: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-2}{\left(\left(c\_m \cdot s\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m))))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
        -2e-175)
     (/ -2.0 (* (* (* c_m s_m) c_m) s_m))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-175) {
		tmp = -2.0 / (((c_m * s_m) * c_m) * s_m);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = s_m * (c_m * x_m)
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-2d-175)) then
        tmp = (-2.0d0) / (((c_m * s_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);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -2e-175) {
		tmp = -2.0 / (((c_m * s_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_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (c_m * x_m)
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -2e-175:
		tmp = -2.0 / (((c_m * s_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_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-175)
		tmp = Float64(-2.0 / Float64(Float64(Float64(c_m * s_m) * 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_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = s_m * (c_m * x_m);
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -2e-175)
		tmp = -2.0 / (((c_m * s_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]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-175], N[(-2.0 / N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * 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))) < -2e-175

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.3%

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

   10. Applied rewrites 39.3%

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

   12. Applied rewrites 39.3%

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

   if -2e-175 < (/.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 62.0%

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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.0

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 93.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.6

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

   6. Applied rewrites 93.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 98.1

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

      13. lift-cos.f64 N/A

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

      14. cos-neg-rev N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-*.f64 98.1

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

   8. Applied rewrites 98.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.8%

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

Alternative 3: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-2}{\left(\left(c\_m \cdot s\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(s\_m \cdot \left(c\_m \cdot x\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -2e-175)
   (/ -2.0 (* (* (* c_m s_m) c_m) s_m))
   (/ 1.0 (* (* x_m c_m) (* s_m (* s_m (* c_m x_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -2e-175) {
		tmp = -2.0 / (((c_m * s_m) * c_m) * s_m);
	} else {
		tmp = 1.0 / ((x_m * c_m) * (s_m * (s_m * (c_m * x_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-2d-175)) then
        tmp = (-2.0d0) / (((c_m * s_m) * c_m) * s_m)
    else
        tmp = 1.0d0 / ((x_m * c_m) * (s_m * (s_m * (c_m * x_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -2e-175) {
		tmp = -2.0 / (((c_m * s_m) * c_m) * s_m);
	} else {
		tmp = 1.0 / ((x_m * c_m) * (s_m * (s_m * (c_m * x_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -2e-175:
		tmp = -2.0 / (((c_m * s_m) * c_m) * s_m)
	else:
		tmp = 1.0 / ((x_m * c_m) * (s_m * (s_m * (c_m * x_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -2e-175)
		tmp = Float64(-2.0 / Float64(Float64(Float64(c_m * s_m) * c_m) * s_m));
	else
		tmp = Float64(1.0 / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(s_m * Float64(c_m * x_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -2e-175)
		tmp = -2.0 / (((c_m * s_m) * c_m) * s_m);
	else
		tmp = 1.0 / ((x_m * c_m) * (s_m * (s_m * (c_m * x_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-175], N[(-2.0 / N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -2e-175

   1. Initial program 64.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 27.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.3%

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

   10. Applied rewrites 39.3%

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

   12. Applied rewrites 39.3%

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

   if -2e-175 < (/.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 62.0%

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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.0

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 76.0

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

   4. Applied rewrites 76.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 93.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.6

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 83.0%

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

Alternative 4: 93.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ \mathbf{if}\;x\_m \leq 6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{c\_m}}{\left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot x\_m\right)}\\ \mathbf{elif}\;x\_m \leq 4.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(x\_m \cdot \left(\left(\left(c\_m \cdot s\_m\right) \cdot c\_m\right) \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x\_m \cdot \left(\left(c\_m \cdot x\_m\right) \cdot \left(\left(s\_m \cdot s\_m\right) \cdot c\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))))
   (if (<= x_m 6e-17)
     (/ (/ (fma -2.0 (* x_m x_m) 1.0) c_m) (* (* c_m (* s_m x_m)) (* s_m x_m)))
     (if (<= x_m 4.8e+120)
       (/ t_0 (* x_m (* x_m (* (* (* c_m s_m) c_m) s_m))))
       (/ t_0 (* x_m (* (* c_m x_m) (* (* s_m s_m) c_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double tmp;
	if (x_m <= 6e-17) {
		tmp = (fma(-2.0, (x_m * x_m), 1.0) / c_m) / ((c_m * (s_m * x_m)) * (s_m * x_m));
	} else if (x_m <= 4.8e+120) {
		tmp = t_0 / (x_m * (x_m * (((c_m * s_m) * c_m) * s_m)));
	} else {
		tmp = t_0 / (x_m * ((c_m * x_m) * ((s_m * s_m) * c_m)));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	tmp = 0.0
	if (x_m <= 6e-17)
		tmp = Float64(Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / c_m) / Float64(Float64(c_m * Float64(s_m * x_m)) * Float64(s_m * x_m)));
	elseif (x_m <= 4.8e+120)
		tmp = Float64(t_0 / Float64(x_m * Float64(x_m * Float64(Float64(Float64(c_m * s_m) * c_m) * s_m))));
	else
		tmp = Float64(t_0 / Float64(x_m * Float64(Float64(c_m * x_m) * Float64(Float64(s_m * s_m) * c_m))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 6e-17], N[(N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.8e+120], N[(t$95$0 / N[(x$95$m * N[(x$95$m * N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x$95$m * N[(N[(c$95$m * x$95$m), $MachinePrecision] * N[(N[(s$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.00000000000000012e-17

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 82.2%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-/r* N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 71.3%

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

   if 6.00000000000000012e-17 < x < 4.80000000000000002e120

   1. Initial program 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 85.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 85.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 99.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lift-*.f64 N/A

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

   10. Applied rewrites 90.5%

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

   if 4.80000000000000002e120 < x

   1. Initial program 63.5%

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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 77.0

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 93.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.4

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

   6. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.4

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

   8. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

   10. Applied rewrites 74.5%

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

Alternative 5: 96.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{c\_m}}{\left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(s\_m \cdot \left(c\_m \cdot x\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 4.9e-89)
   (/ (/ (fma -2.0 (* x_m x_m) 1.0) c_m) (* (* c_m (* s_m x_m)) (* s_m x_m)))
   (/ (cos (+ x_m x_m)) (* (* x_m c_m) (* s_m (* s_m (* c_m x_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4.9e-89) {
		tmp = (fma(-2.0, (x_m * x_m), 1.0) / c_m) / ((c_m * (s_m * x_m)) * (s_m * x_m));
	} else {
		tmp = cos((x_m + x_m)) / ((x_m * c_m) * (s_m * (s_m * (c_m * x_m))));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 4.9e-89)
		tmp = Float64(Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / c_m) / Float64(Float64(c_m * Float64(s_m * x_m)) * Float64(s_m * x_m)));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(x_m * c_m) * Float64(s_m * Float64(s_m * Float64(c_m * x_m)))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 4.9e-89], N[(N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(s$95$m * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9e-89

   1. Initial program 59.1%

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

      1. lift-/.f64 N/A

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

      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-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 80.9%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-/r* N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 69.3%

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

   if 4.9e-89 < x

   1. Initial program 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.3

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.3

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 78.3

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

   4. Applied rewrites 78.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 96.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.2

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

   8. Applied rewrites 96.2%

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

Alternative 6: 89.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{c\_m}}{\left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right) \cdot \left(s\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{x\_m \cdot \left(x\_m \cdot \left(\left(\left(c\_m \cdot s\_m\right) \cdot c\_m\right) \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 6e-17)
   (/ (/ (fma -2.0 (* x_m x_m) 1.0) c_m) (* (* c_m (* s_m x_m)) (* s_m x_m)))
   (/ (cos (+ x_m x_m)) (* x_m (* x_m (* (* (* c_m s_m) c_m) s_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 6e-17) {
		tmp = (fma(-2.0, (x_m * x_m), 1.0) / c_m) / ((c_m * (s_m * x_m)) * (s_m * x_m));
	} else {
		tmp = cos((x_m + x_m)) / (x_m * (x_m * (((c_m * s_m) * c_m) * s_m)));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 6e-17)
		tmp = Float64(Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / c_m) / Float64(Float64(c_m * Float64(s_m * x_m)) * Float64(s_m * x_m)));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(x_m * Float64(x_m * Float64(Float64(Float64(c_m * s_m) * c_m) * s_m))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 6e-17], N[(N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / c$95$m), $MachinePrecision] / N[(N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(x$95$m * N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.00000000000000012e-17

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   4. Applied rewrites 82.2%

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

   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. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/r* N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-/r* N/A

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

      9. div-add N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 71.3%

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

   if 6.00000000000000012e-17 < x

   1. Initial program 69.2%

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

      1. lift-*.f64 N/A

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

      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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.9

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 79.9

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

   4. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.4

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

   8. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*l* N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lift-*.f64 N/A

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

   10. Applied rewrites 84.8%

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

Alternative 7: 97.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 75.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 75.8

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 93.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 93.7

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

6. Applied rewrites 93.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 97.8

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

   13. lift-cos.f64 N/A

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

   14. cos-neg-rev N/A

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

   15. lift-*.f64 N/A

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

   16. distribute-lft-neg-in N/A

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

   17. metadata-eval N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-*.f64 97.8

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

8. Applied rewrites 97.8%

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

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. count-2-rev N/A

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

   5. flip-+ N/A

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

   6. distribute-neg-frac N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{\frac{\cos \left(\frac{\mathsf{neg}\left(\color{blue}{0}\right)}{x - x}\right)}{s \cdot \left(c \cdot x\right)}}{s \cdot \left(c \cdot x\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\cos \left(\frac{\color{blue}{0}}{x - x}\right)}{s \cdot \left(c \cdot x\right)}}{s \cdot \left(c \cdot x\right)} \]

   11. +-inverses N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. flip-+ N/A

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

   15. lift-+.f64 97.8

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

10. Applied rewrites 97.8%

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

Alternative 8: 97.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\_m\right)\\ \frac{\cos \left(2 \cdot x\_m\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x_m)))) (/ (cos (* 2.0 x_m)) (* t_0 t_0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	return cos((2.0 * x_m)) / (t_0 * t_0);
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = s_m * (c_m * x_m)
    code = cos((2.0d0 * x_m)) / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x_m);
	return Math.cos((2.0 * x_m)) / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = s_m * (c_m * x_m)
	return math.cos((2.0 * x_m)) / (t_0 * t_0)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x_m))
	return Float64(cos(Float64(2.0 * x_m)) / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	t_0 = s_m * (c_m * x_m);
	tmp = cos((2.0 * x_m)) / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 62.2%

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

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 75.8

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 75.8

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 93.7

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 93.7

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

6. Applied rewrites 93.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 97.5

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

8. Applied rewrites 97.5%

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

Alternative 9: 25.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{\frac{-2}{s\_m \cdot c\_m}}{s\_m \cdot c\_m} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m) :precision binary64 (/ (/ -2.0 (* s_m c_m)) (* s_m c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return (-2.0 / (s_m * c_m)) / (s_m * c_m);
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = ((-2.0d0) / (s_m * c_m)) / (s_m * c_m)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return (-2.0 / (s_m * c_m)) / (s_m * c_m);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return (-2.0 / (s_m * c_m)) / (s_m * c_m)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(Float64(-2.0 / Float64(s_m * c_m)) / Float64(s_m * c_m))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = (-2.0 / (s_m * c_m)) / (s_m * c_m);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(N[(-2.0 / N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] / N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

5. Applied rewrites 48.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 23.8%

   \[\leadsto \frac{\frac{-2}{s \cdot c}}{\color{blue}{s \cdot c}} \]
9. Add Preprocessing

Alternative 10: 25.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{-2}{\left(\left(c\_m \cdot s\_m\right) \cdot c\_m\right) \cdot s\_m} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m) :precision binary64 (/ -2.0 (* (* (* c_m s_m) c_m) s_m)))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((c_m * s_m) * c_m) * s_m);
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (-2.0d0) / (((c_m * s_m) * c_m) * s_m)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (((c_m * s_m) * c_m) * s_m);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (((c_m * s_m) * c_m) * s_m)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(-2.0 / Float64(Float64(Float64(c_m * s_m) * c_m) * s_m))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / (((c_m * s_m) * c_m) * s_m);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(-2.0 / N[(N[(N[(c$95$m * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

5. Applied rewrites 48.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 23.8%

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

10. Applied rewrites 24.9%

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

12. Applied rewrites 24.9%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))