mixedcos

Percentage Accurate: 66.6% → 99.2%

Time: 8.9s

Alternatives: 8

Speedup: 2.3×

• 31.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% 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: 99.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 10^{-31}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)) (t_1 (* (* s_m x_m) c_m)))
   (if (<= x_m 1e-31) (/ 1.0 (* t_1 t_1)) (/ (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 = (c_m * x_m) * s_m;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1e-31) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = cos((x_m + x_m)) / (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) :: t_1
    real(8) :: tmp
    t_0 = (c_m * x_m) * s_m
    t_1 = (s_m * x_m) * c_m
    if (x_m <= 1d-31) then
        tmp = 1.0d0 / (t_1 * t_1)
    else
        tmp = cos((x_m + x_m)) / (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 = (c_m * x_m) * s_m;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1e-31) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = Math.cos((x_m + x_m)) / (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 = (c_m * x_m) * s_m
	t_1 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 1e-31:
		tmp = 1.0 / (t_1 * t_1)
	else:
		tmp = math.cos((x_m + x_m)) / (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(Float64(c_m * x_m) * s_m)
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1e-31)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(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 = (c_m * x_m) * s_m;
	t_1 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1e-31)
		tmp = 1.0 / (t_1 * t_1);
	else
		tmp = cos((x_m + x_m)) / (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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1e-31], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-31

   1. Initial program 65.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 65.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 96.1

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if 1e-31 < x

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-31}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(c\_m \cdot x\_m\right) \cdot s\_m\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

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 (* (* c_m x_m) s_m)) (t_1 (* (* s_m x_m) c_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -1e-135)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ 1.0 (* t_1 t_1)))))

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 = (c_m * x_m) * s_m;
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -1e-135) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = 1.0 / (t_1 * t_1);
	}
	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(Float64(c_m * x_m) * s_m)
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	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[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-135], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 * t$95$1), $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))) < -1e-135

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 49.9

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

   8. Applied rewrites 49.9%

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

   if -1e-135 < (/.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 64.8%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 64.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 96.7

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.3

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

   7. Applied rewrites 85.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{-2}{\left(s\_m \cdot c\_m\right) \cdot c\_m}}{s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 x_m) c_m)))
   (if (<=
        (/ (cos (* 2.0 x_m)) (* (* (* (pow s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -1e-135)
     (/ (/ -2.0 (* (* s_m c_m) c_m)) s_m)
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
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 * x_m) * c_m;
	double tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -1e-135) {
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
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 * x_m) * c_m
    if ((cos((2.0d0 * x_m)) / ((((s_m ** 2.0d0) * x_m) * x_m) * (c_m ** 2.0d0))) <= (-1d-135)) then
        tmp = ((-2.0d0) / ((s_m * c_m) * c_m)) / s_m
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
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 * x_m) * c_m;
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (((Math.pow(s_m, 2.0) * x_m) * x_m) * Math.pow(c_m, 2.0))) <= -1e-135) {
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_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 * x_m) * c_m
	tmp = 0
	if (math.cos((2.0 * x_m)) / (((math.pow(s_m, 2.0) * x_m) * x_m) * math.pow(c_m, 2.0))) <= -1e-135:
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_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(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = Float64(Float64(-2.0 / Float64(Float64(s_m * c_m) * c_m)) / s_m);
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_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 * x_m) * c_m;
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
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[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-135], N[(N[(-2.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-135

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.0%

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

   if -1e-135 < (/.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 64.8%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 64.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 96.7

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.3

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

   7. Applied rewrites 85.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{-2}{\left(s \cdot c\right) \cdot c}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot c\_m\right) \cdot x\_m\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{-2}{\left(s\_m \cdot c\_m\right) \cdot c\_m}}{s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
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 s_m 2.0) x_m) x_m) (pow c_m 2.0)))
        -1e-135)
     (/ (/ -2.0 (* (* s_m c_m) c_m)) s_m)
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
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(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -1e-135) {
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
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)) / ((((s_m ** 2.0d0) * x_m) * x_m) * (c_m ** 2.0d0))) <= (-1d-135)) then
        tmp = ((-2.0d0) / ((s_m * c_m) * c_m)) / s_m
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
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(s_m, 2.0) * x_m) * x_m) * Math.pow(c_m, 2.0))) <= -1e-135) {
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_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(s_m, 2.0) * x_m) * x_m) * math.pow(c_m, 2.0))) <= -1e-135:
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_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(Float64(s_m * c_m) * x_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = Float64(Float64(-2.0 / Float64(Float64(s_m * c_m) * c_m)) / s_m);
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_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)) / ((((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
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[(N[(s$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-135], N[(N[(-2.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-135

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.0%

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

   if -1e-135 < (/.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 64.8%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 64.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 96.7

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.7

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

   4. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 96.3

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 96.3

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

   6. Applied rewrites 96.3%

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

   7. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.3

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

   9. Applied rewrites 84.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{-2}{\left(s \cdot c\right) \cdot c}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(\left({s\_m}^{2} \cdot x\_m\right) \cdot x\_m\right) \cdot {c\_m}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{-2}{\left(s\_m \cdot c\_m\right) \cdot c\_m}}{s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c\_m \cdot x\_m\right) \cdot s\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

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 s_m 2.0) x_m) x_m) (pow c_m 2.0)))
      -1e-135)
   (/ (/ -2.0 (* (* s_m c_m) c_m)) s_m)
   (/ 1.0 (* (* (* (* c_m x_m) s_m) (* s_m x_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 tmp;
	if ((cos((2.0 * x_m)) / (((pow(s_m, 2.0) * x_m) * x_m) * pow(c_m, 2.0))) <= -1e-135) {
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	} else {
		tmp = 1.0 / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_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)) / ((((s_m ** 2.0d0) * x_m) * x_m) * (c_m ** 2.0d0))) <= (-1d-135)) then
        tmp = ((-2.0d0) / ((s_m * c_m) * c_m)) / s_m
    else
        tmp = 1.0d0 / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_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(s_m, 2.0) * x_m) * x_m) * Math.pow(c_m, 2.0))) <= -1e-135) {
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	} else {
		tmp = 1.0 / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_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(s_m, 2.0) * x_m) * x_m) * math.pow(c_m, 2.0))) <= -1e-135:
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m
	else:
		tmp = 1.0 / ((((c_m * x_m) * s_m) * (s_m * x_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)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(Float64((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = Float64(Float64(-2.0 / Float64(Float64(s_m * c_m) * c_m)) / s_m);
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c_m * x_m) * s_m) * Float64(s_m * x_m)) * c_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)) / ((((s_m ^ 2.0) * x_m) * x_m) * (c_m ^ 2.0))) <= -1e-135)
		tmp = (-2.0 / ((s_m * c_m) * c_m)) / s_m;
	else
		tmp = 1.0 / ((((c_m * x_m) * s_m) * (s_m * x_m)) * c_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[(N[(N[Power[s$95$m, 2.0], $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[Power[c$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-135], N[(N[(-2.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $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))) < -1e-135

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 4.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 50.0%

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

   if -1e-135 < (/.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 64.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 68.5%

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

   9. Applied rewrites 68.3%

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

   11. Applied rewrites 81.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{-2}{\left(s \cdot c\right) \cdot c}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;x\_m \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(\left(s\_m \cdot c\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right) \cdot \left(c\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

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))) (t_1 (* (* s_m x_m) c_m)))
   (if (<= x_m 1.1e-44)
     (/ 1.0 (* t_1 t_1))
     (if (<= x_m 4.2e+108)
       (/ t_0 (* (* (* (* (* s_m c_m) s_m) x_m) c_m) x_m))
       (/ t_0 (* (* (* s_m s_m) (* c_m x_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 t_0 = cos((x_m + x_m));
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.1e-44) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x_m <= 4.2e+108) {
		tmp = t_0 / (((((s_m * c_m) * s_m) * x_m) * c_m) * x_m);
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = (s_m * x_m) * c_m
    if (x_m <= 1.1d-44) then
        tmp = 1.0d0 / (t_1 * t_1)
    else if (x_m <= 4.2d+108) then
        tmp = t_0 / (((((s_m * c_m) * s_m) * x_m) * c_m) * x_m)
    else
        tmp = t_0 / (((s_m * s_m) * (c_m * x_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 t_0 = Math.cos((x_m + x_m));
	double t_1 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.1e-44) {
		tmp = 1.0 / (t_1 * t_1);
	} else if (x_m <= 4.2e+108) {
		tmp = t_0 / (((((s_m * c_m) * s_m) * x_m) * c_m) * x_m);
	} else {
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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):
	t_0 = math.cos((x_m + x_m))
	t_1 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 1.1e-44:
		tmp = 1.0 / (t_1 * t_1)
	elif x_m <= 4.2e+108:
		tmp = t_0 / (((((s_m * c_m) * s_m) * x_m) * c_m) * x_m)
	else:
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.1e-44)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	elseif (x_m <= 4.2e+108)
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(Float64(s_m * c_m) * s_m) * x_m) * c_m) * x_m));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(s_m * s_m) * Float64(c_m * x_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)
	t_0 = cos((x_m + x_m));
	t_1 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1.1e-44)
		tmp = 1.0 / (t_1 * t_1);
	elseif (x_m <= 4.2e+108)
		tmp = t_0 / (((((s_m * c_m) * s_m) * x_m) * c_m) * x_m);
	else
		tmp = t_0 / (((s_m * s_m) * (c_m * x_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_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.1e-44], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4.2e+108], N[(t$95$0 / N[(N[(N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.10000000000000006e-44

   1. Initial program 64.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 64.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 96.0

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if 1.10000000000000006e-44 < x < 4.20000000000000019e108

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.1

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

   5. Applied rewrites 96.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   9. Applied rewrites 98.2%

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

   if 4.20000000000000019e108 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 99.6

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 89.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(\left(s \cdot c\right) \cdot s\right) \cdot x\right) \cdot c\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot s\right) \cdot \left(c \cdot x\right)\right) \cdot \left(c \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(\left(s\_m \cdot c\_m\right) \cdot s\_m\right) \cdot x\_m\right) \cdot c\_m\right) \cdot x\_m}\\ \end{array} \end{array} \]

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 x_m) c_m)))
   (if (<= x_m 1.1e-44)
     (/ 1.0 (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* (* (* (* (* s_m c_m) s_m) x_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 t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.1e-44) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / (((((s_m * c_m) * s_m) * x_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) :: t_0
    real(8) :: tmp
    t_0 = (s_m * x_m) * c_m
    if (x_m <= 1.1d-44) then
        tmp = 1.0d0 / (t_0 * t_0)
    else
        tmp = cos((x_m + x_m)) / (((((s_m * c_m) * s_m) * x_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 t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.1e-44) {
		tmp = 1.0 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x_m + x_m)) / (((((s_m * c_m) * s_m) * x_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):
	t_0 = (s_m * x_m) * c_m
	tmp = 0
	if x_m <= 1.1e-44:
		tmp = 1.0 / (t_0 * t_0)
	else:
		tmp = math.cos((x_m + x_m)) / (((((s_m * c_m) * s_m) * x_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)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.1e-44)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(Float64(s_m * c_m) * s_m) * x_m) * 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)
	t_0 = (s_m * x_m) * c_m;
	tmp = 0.0;
	if (x_m <= 1.1e-44)
		tmp = 1.0 / (t_0 * t_0);
	else
		tmp = cos((x_m + x_m)) / (((((s_m * c_m) * s_m) * x_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_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.1e-44], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.10000000000000006e-44

   1. Initial program 64.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 64.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. pow-prod-down N/A

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

      17. pow-prod-down N/A

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

      18. lower-pow.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 96.0

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unswap-sqr N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if 1.10000000000000006e-44 < x

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lower-+.f64 98.3

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 88.5%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(\left(\left(s \cdot c\right) \cdot s\right) \cdot x\right) \cdot c\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.2% accurate, 10.1× speedup

Math

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

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 (* (* s_m c_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 / ((s_m * c_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) / ((s_m * c_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 / ((s_m * c_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 / ((s_m * c_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(Float64(-2.0 / Float64(Float64(s_m * c_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 / ((s_m * c_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[(N[(-2.0 / N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision] / s$95$m), $MachinePrecision]

Derivation

1. Initial program 65.4%

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

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. associate-/l/ N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

5. Applied rewrites 73.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.4%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 25.7%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))