mixedcos

Percentage Accurate: 66.9% → 95.2%

Time: 8.8s

Alternatives: 9

Speedup: 2.4×

• 31.5% 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: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 8 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(c\_m \cdot s\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 8e-52)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (+ x_m x_m)) (* (* c_m s_m) (* s_m (* x_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 8e-52) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (x_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) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 8d-52) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (x_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 t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 8e-52) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (x_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):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 8e-52:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (x_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)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 8e-52)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(c_m * s_m) * Float64(s_m * Float64(x_m * Float64(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)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 8e-52)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (x_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_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 8e-52], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.0000000000000001e-52

   1. Initial program 63.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 71.0

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

   5. Simplified 71.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 84.1

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

   7. Applied egg-rr 84.1%

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

   if 8.0000000000000001e-52 < x

   1. Initial program 78.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. associate-*l* N/A

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

      2. unpow-prod-down N/A

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

      3. pow2 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. count-2 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. swap-sqr N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 N/A

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

      21. *-lowering-*.f64 89.8

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

   6. Applied egg-rr 89.8%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 96.9

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

   8. Applied egg-rr 96.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<=
        (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
        -1e-160)
     (/ -2.0 (* c_m (* c_m (* s_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 = c_m * (x_m * s_m);
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_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 = c_m * (x_m * s_m)
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-1d-160)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_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 = c_m * (x_m * s_m);
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_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 = c_m * (x_m * s_m)
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -1e-160:
		tmp = -2.0 / (c_m * (c_m * (s_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(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = -2.0 / (c_m * (c_m * (s_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-160], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999999e-161

   1. Initial program 68.3%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 59.7

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

   4. Applied egg-rr 59.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 28.7

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

   7. Simplified 28.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 43.5

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

   10. Simplified 43.5%

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

   if -9.9999999999999999e-161 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 76.2

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

   5. Simplified 76.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 87.3

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

   7. Applied egg-rr 87.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

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 (* c_m (* x_m s_m))))
   (if (<=
        (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
        -1e-160)
     (/ -2.0 (* c_m (* c_m (* s_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 = c_m * (x_m * s_m);
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_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 = c_m * (x_m * s_m)
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-1d-160)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_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 = c_m * (x_m * s_m);
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_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 = c_m * (x_m * s_m)
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -1e-160:
		tmp = -2.0 / (c_m * (c_m * (s_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(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_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 = c_m * (x_m * s_m);
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = -2.0 / (c_m * (c_m * (s_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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-160], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999999e-161

   1. Initial program 68.3%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 59.7

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

   4. Applied egg-rr 59.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 28.7

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

   7. Simplified 28.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 43.5

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

   10. Simplified 43.5%

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

   if -9.9999999999999999e-161 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 76.2

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

   5. Simplified 76.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 87.0

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

   7. Applied egg-rr 87.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(\left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right) \cdot \left(x\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
      -1e-160)
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))
   (/ 1.0 (* c_m (* (* x_m (* c_m s_m)) (* x_m s_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * ((x_m * (c_m * s_m)) * (x_m * s_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((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-1d-160)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c_m * ((x_m * (c_m * s_m)) * (x_m * s_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((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * ((x_m * (c_m * s_m)) * (x_m * s_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((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -1e-160:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (c_m * ((x_m * (c_m * s_m)) * (x_m * s_m)))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(Float64(x_m * Float64(c_m * s_m)) * Float64(x_m * s_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((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (c_m * ((x_m * (c_m * s_m)) * (x_m * s_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[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-160], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999999e-161

   1. Initial program 68.3%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 59.7

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

   4. Applied egg-rr 59.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 28.7

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

   7. Simplified 28.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 43.5

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

   10. Simplified 43.5%

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

   if -9.9999999999999999e-161 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 67.8

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

   7. Simplified 67.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. swap-sqr N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 83.7

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

   9. Applied egg-rr 83.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(s\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
      -1e-160)
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))
   (/ 1.0 (* c_m (* c_m (* x_m (* s_m (* x_m s_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * (c_m * (x_m * (s_m * (x_m * s_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((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-1d-160)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c_m * (c_m * (x_m * (s_m * (x_m * s_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((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -1e-160) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * (c_m * (x_m * (s_m * (x_m * s_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((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -1e-160:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (c_m * (c_m * (x_m * (s_m * (x_m * s_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(x_m * Float64(s_m * Float64(x_m * s_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((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -1e-160)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (c_m * (c_m * (x_m * (s_m * (x_m * s_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[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-160], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(x$95$m * N[(s$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999999e-161

   1. Initial program 68.3%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 59.7

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

   4. Applied egg-rr 59.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 28.7

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

   7. Simplified 28.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 43.5

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

   10. Simplified 43.5%

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

   if -9.9999999999999999e-161 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 97.7

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 67.8

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

   7. Simplified 67.8%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 78.1

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

   9. Applied egg-rr 78.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{elif}\;x\_m \leq 1.9 \cdot 10^{+159}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(c\_m \cdot s\_m\right) \cdot \left(s\_m \cdot \left(c\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot t\_0\right)}\\ \end{array} \end{array} \]

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 (* c_m (* x_m s_m))))
   (if (<= x_m 5e-23)
     (/ (/ 1.0 t_0) t_0)
     (if (<= x_m 1.9e+159)
       (/ (cos (+ x_m x_m)) (* (* c_m s_m) (* s_m (* c_m (* x_m x_m)))))
       (/ 1.0 (* (* c_m s_m) (* x_m 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 tmp;
	if (x_m <= 5e-23) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x_m <= 1.9e+159) {
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (c_m * (x_m * x_m))));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * 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 = c_m * (x_m * s_m)
    if (x_m <= 5d-23) then
        tmp = (1.0d0 / t_0) / t_0
    else if (x_m <= 1.9d+159) then
        tmp = cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (c_m * (x_m * x_m))))
    else
        tmp = 1.0d0 / ((c_m * s_m) * (x_m * 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 tmp;
	if (x_m <= 5e-23) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x_m <= 1.9e+159) {
		tmp = Math.cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (c_m * (x_m * x_m))));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * 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)
	tmp = 0
	if x_m <= 5e-23:
		tmp = (1.0 / t_0) / t_0
	elif x_m <= 1.9e+159:
		tmp = math.cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (c_m * (x_m * x_m))))
	else:
		tmp = 1.0 / ((c_m * s_m) * (x_m * 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(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 5e-23)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	elseif (x_m <= 1.9e+159)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(c_m * s_m) * Float64(s_m * Float64(c_m * Float64(x_m * x_m)))));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(x_m * 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);
	tmp = 0.0;
	if (x_m <= 5e-23)
		tmp = (1.0 / t_0) / t_0;
	elseif (x_m <= 1.9e+159)
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (s_m * (c_m * (x_m * x_m))));
	else
		tmp = 1.0 / ((c_m * s_m) * (x_m * 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[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 5e-23], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x$95$m, 1.9e+159], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(s$95$m * N[(c$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.0000000000000002e-23

   1. Initial program 62.7%

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

   3. 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. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. associate-/r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 84.5

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

   7. Applied egg-rr 84.5%

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

   if 5.0000000000000002e-23 < x < 1.89999999999999983e159

   1. Initial program 77.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 99.3

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

   4. Applied egg-rr 99.3%

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

      1. associate-*l* N/A

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

      2. unpow-prod-down N/A

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

      3. pow2 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. count-2 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. swap-sqr N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. associate-*r* N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 N/A

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

      21. *-lowering-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   if 1.89999999999999983e159 < x

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 86.3

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

   5. Simplified 86.3%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. swap-sqr N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 87.0

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

   7. Applied egg-rr 87.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+159}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;{c\_m}^{2} \leq 2 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

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
 (if (<= (pow c_m 2.0) 2e+231)
   (/ 1.0 (* c_m (* c_m (* s_m (* s_m (* x_m x_m))))))
   (/ -2.0 (* (* s_m s_m) (* c_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 (pow(c_m, 2.0) <= 2e+231) {
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))));
	} else {
		tmp = -2.0 / ((s_m * s_m) * (c_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 ((c_m ** 2.0d0) <= 2d+231) then
        tmp = 1.0d0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))))
    else
        tmp = (-2.0d0) / ((s_m * s_m) * (c_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.pow(c_m, 2.0) <= 2e+231) {
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))));
	} else {
		tmp = -2.0 / ((s_m * s_m) * (c_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.pow(c_m, 2.0) <= 2e+231:
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))))
	else:
		tmp = -2.0 / ((s_m * s_m) * (c_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 ((c_m ^ 2.0) <= 2e+231)
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(s_m * Float64(s_m * Float64(x_m * x_m))))));
	else
		tmp = Float64(-2.0 / Float64(Float64(s_m * s_m) * Float64(c_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 ((c_m ^ 2.0) <= 2e+231)
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))));
	else
		tmp = -2.0 / ((s_m * s_m) * (c_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[Power[c$95$m, 2.0], $MachinePrecision], 2e+231], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 2.0000000000000001e231

   1. Initial program 70.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 63.5

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

   7. Simplified 63.5%

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

   if 2.0000000000000001e231 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 61.2%

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 68.0

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 42.2

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

   7. Simplified 42.2%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 43.6

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

   10. Simplified 43.6%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 49.1

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

   12. Applied egg-rr 49.1%

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

Alternative 8: 96.7% accurate, 2.4× speedup

Math

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

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)))) (/ (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);
	return cos((x_m + x_m)) / (t_0 * t_0);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. pow-lowering-pow.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 97.5

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

4. Applied egg-rr 97.5%

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 98.1

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

6. Applied egg-rr 98.1%

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

   1. count-2 N/A

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

   2. +-lowering-+.f64 98.1

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

8. Applied egg-rr 98.1%

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

9. Final simplification 98.1%

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

Alternative 9: 31.6% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 68.8

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

4. Applied egg-rr 68.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 40.1

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

7. Simplified 40.1%

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

8. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 28.1

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

10. Simplified 28.1%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))