mixedcos

Percentage Accurate: 65.8% → 97.4%

Time: 17.7s

Alternatives: 9

Speedup: 24.1×

• 30.7% 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: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x \cdot s_m\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c_m \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{\frac{t_1}{s_m}}{c_m \cdot x}}{x \cdot \left(c_m \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))) (t_1 (cos (* x 2.0))))
   (if (<= c_m 1.25e-203)
     (/ (/ (/ t_1 s_m) (* c_m x)) (* x (* c_m s_m)))
     (/ (/ t_1 t_0) t_0))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (c_m <= 1.25e-203) {
		tmp = ((t_1 / s_m) / (c_m * x)) / (x * (c_m * s_m));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    t_1 = cos((x * 2.0d0))
    if (c_m <= 1.25d-203) then
        tmp = ((t_1 / s_m) / (c_m * x)) / (x * (c_m * s_m))
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (c_m <= 1.25e-203) {
		tmp = ((t_1 / s_m) / (c_m * x)) / (x * (c_m * s_m));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if c_m <= 1.25e-203:
		tmp = ((t_1 / s_m) / (c_m * x)) / (x * (c_m * s_m))
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (c_m <= 1.25e-203)
		tmp = Float64(Float64(Float64(t_1 / s_m) / Float64(c_m * x)) / Float64(x * Float64(c_m * s_m)));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (c_m <= 1.25e-203)
		tmp = ((t_1 / s_m) / (c_m * x)) / (x * (c_m * s_m));
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c$95$m, 1.25e-203], N[(N[(N[(t$95$1 / s$95$m), $MachinePrecision] / N[(c$95$m * x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.25e-203

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-*r* 61.7%

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

      3. unpow2 61.7%

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

      4. pow-prod-down 77.1%

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

   4. Applied egg-rr 77.1%

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

      1. *-un-lft-identity 77.1%

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

      2. pow-prod-down 96.4%

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

      3. pow2 96.4%

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

      4. frac-times 96.4%

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

      5. associate-*r/ 96.4%

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

      6. associate-*r* 95.8%

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

      7. times-frac 94.4%

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

      8. associate-/r* 94.5%

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

      9. *-commutative 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. *-commutative 94.5%

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

      2. associate-/l/ 93.9%

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

      3. *-commutative 93.9%

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

      4. associate-*r* 93.8%

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

      5. *-commutative 93.8%

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

      6. associate-*r* 93.1%

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

   8. Simplified 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-/l/ 93.1%

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

      3. frac-times 93.1%

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

      4. *-un-lft-identity 93.1%

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

      5. associate-*r* 95.0%

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

      6. *-commutative 95.0%

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

      7. associate-*l* 93.1%

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

      8. *-commutative 93.1%

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

      9. associate-*r* 93.7%

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

      10. *-commutative 93.7%

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

      11. associate-*l* 96.4%

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

      12. *-commutative 96.4%

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

      13. associate-/l/ 96.4%

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

   10. Applied egg-rr 96.9%

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

       1. *-un-lft-identity 96.9%

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

       2. associate-*r* 96.3%

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

       3. times-frac 96.3%

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

   12. Applied egg-rr 96.3%

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

       1. associate-*l/ 96.4%

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

       2. *-lft-identity 96.4%

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

       3. *-commutative 96.4%

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

   14. Simplified 96.4%

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

   if 1.25e-203 < c

   1. Initial program 62.4%

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

      1. *-un-lft-identity 62.4%

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

      2. add-sqr-sqrt 62.4%

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

      3. times-frac 62.3%

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

   4. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l/ 98.6%

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

      3. div-inv 98.7%

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

      4. *-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{s}}{c \cdot x}}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c_m \cdot \left(x \cdot s_m\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ t_2 := x \cdot \left(c_m \cdot s_m\right)\\ \mathbf{if}\;c_m \leq 1.02 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{t_1}{t_2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m))) (t_1 (cos (* x 2.0))) (t_2 (* x (* c_m s_m))))
   (if (<= c_m 1.02e-232) (/ (/ t_1 t_2) t_2) (/ (/ t_1 t_0) t_0))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double t_1 = cos((x * 2.0));
	double t_2 = x * (c_m * s_m);
	double tmp;
	if (c_m <= 1.02e-232) {
		tmp = (t_1 / t_2) / t_2;
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c_m * (x * s_m)
    t_1 = cos((x * 2.0d0))
    t_2 = x * (c_m * s_m)
    if (c_m <= 1.02d-232) then
        tmp = (t_1 / t_2) / t_2
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	double t_1 = Math.cos((x * 2.0));
	double t_2 = x * (c_m * s_m);
	double tmp;
	if (c_m <= 1.02e-232) {
		tmp = (t_1 / t_2) / t_2;
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	t_1 = math.cos((x * 2.0))
	t_2 = x * (c_m * s_m)
	tmp = 0
	if c_m <= 1.02e-232:
		tmp = (t_1 / t_2) / t_2
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	t_1 = cos(Float64(x * 2.0))
	t_2 = Float64(x * Float64(c_m * s_m))
	tmp = 0.0
	if (c_m <= 1.02e-232)
		tmp = Float64(Float64(t_1 / t_2) / t_2);
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp_2 = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	t_1 = cos((x * 2.0));
	t_2 = x * (c_m * s_m);
	tmp = 0.0;
	if (c_m <= 1.02e-232)
		tmp = (t_1 / t_2) / t_2;
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c$95$m, 1.02e-232], N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.0200000000000001e-232

   1. Initial program 65.8%

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

      1. *-commutative 65.8%

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

      2. associate-*r* 61.2%

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

      3. unpow2 61.2%

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

      4. pow-prod-down 77.4%

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

   4. Applied egg-rr 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. pow-prod-down 96.2%

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

      3. pow2 96.2%

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

      4. frac-times 96.2%

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

      5. associate-*r/ 96.2%

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

      6. associate-*r* 95.6%

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

      7. times-frac 94.9%

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

      8. associate-/r* 95.0%

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

      9. *-commutative 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/l/ 94.2%

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

      3. *-commutative 94.2%

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

      4. associate-*r* 94.2%

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

      5. *-commutative 94.2%

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

      6. associate-*r* 93.4%

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

   8. Simplified 93.4%

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

      1. *-commutative 93.4%

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

      2. associate-/l/ 93.4%

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

      3. frac-times 93.4%

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

      4. *-un-lft-identity 93.4%

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

      5. associate-*r* 94.8%

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

      6. *-commutative 94.8%

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

      7. associate-*l* 92.7%

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

      8. *-commutative 92.7%

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

      9. associate-*r* 93.4%

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

      10. *-commutative 93.4%

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

      11. associate-*l* 96.2%

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

      12. *-commutative 96.2%

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

      13. associate-/l/ 96.2%

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

   10. Applied egg-rr 96.8%

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

   if 1.0200000000000001e-232 < c

   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. Step-by-step derivation

      1. *-un-lft-identity 63.0%

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

      2. add-sqr-sqrt 62.9%

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

      3. times-frac 62.9%

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

   4. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l/ 98.7%

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

      3. div-inv 98.7%

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

      4. *-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.02 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. add-sqr-sqrt 64.4%

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

   3. times-frac 64.4%

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

4. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-/r* 97.5%

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

   3. frac-times 92.9%

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

   4. div-inv 92.9%

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

   5. *-commutative 92.9%

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

6. Applied egg-rr 92.9%

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

7. Final simplification 92.9%

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

Alternative 4: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. add-sqr-sqrt 64.4%

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

   3. times-frac 64.4%

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

4. Applied egg-rr 97.4%

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

   1. *-commutative 97.4%

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

   2. associate-*l/ 97.4%

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

   3. div-inv 97.4%

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

   4. *-commutative 97.4%

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

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

Alternative 5: 79.7% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. add-sqr-sqrt 64.4%

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

   3. times-frac 64.4%

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

4. Applied egg-rr 97.4%

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

5. Taylor expanded in x around 0 77.3%

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

   1. associate-/r* 77.3%

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

   2. *-commutative 77.3%

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

   3. *-rgt-identity 77.3%

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

   4. associate-*r/ 77.3%

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

   5. associate-*l/ 77.3%

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

   6. *-lft-identity 77.3%

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

   7. *-commutative 77.3%

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

7. Simplified 77.3%

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

8. Final simplification 77.3%

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

Alternative 6: 76.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.5%

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

   2. *-commutative 51.5%

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

   3. unpow2 51.5%

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

   4. unpow2 51.5%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 77.1%

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

   11. unpow2 77.1%

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

   12. *-commutative 77.1%

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

5. Simplified 77.1%

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

   1. unpow2 77.1%

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

   2. associate-*r* 75.1%

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

   3. *-commutative 75.1%

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

   4. associate-*l* 74.3%

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

7. Applied egg-rr 74.3%

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

8. Final simplification 74.3%

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

Alternative 7: 76.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.5%

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

   2. *-commutative 51.5%

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

   3. unpow2 51.5%

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

   4. unpow2 51.5%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 77.1%

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

   11. unpow2 77.1%

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

   12. *-commutative 77.1%

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

5. Simplified 77.1%

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

   1. unpow2 77.1%

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

   2. associate-*r* 75.1%

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

   3. *-commutative 75.1%

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

   4. associate-*l* 74.3%

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

7. Applied egg-rr 74.3%

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

8. Taylor expanded in c around 0 74.3%

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

   1. associate-*r* 74.8%

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

10. Simplified 74.8%

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

11. Final simplification 74.8%

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

Alternative 8: 79.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.5%

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

   2. *-commutative 51.5%

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

   3. unpow2 51.5%

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

   4. unpow2 51.5%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 77.1%

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

   11. unpow2 77.1%

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

   12. *-commutative 77.1%

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

5. Simplified 77.1%

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

   1. *-commutative 77.1%

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

   2. pow2 77.1%

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

7. Applied egg-rr 77.1%

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

8. Final simplification 77.1%

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

Alternative 9: 78.7% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

3. Taylor expanded in x around 0 51.5%

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

   1. associate-/r* 51.5%

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

   2. *-commutative 51.5%

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

   3. unpow2 51.5%

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

   4. unpow2 51.5%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 77.1%

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

   11. unpow2 77.1%

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

   12. *-commutative 77.1%

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

5. Simplified 77.1%

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

   1. *-commutative 77.1%

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

   2. pow2 77.1%

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

   3. *-commutative 77.1%

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

   4. associate-*r* 74.7%

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

7. Applied egg-rr 74.7%

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

8. Final simplification 74.7%

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))