mixedcos

Percentage Accurate: 67.5% → 99.3%

Time: 9.9s

Alternatives: 4

Speedup: 24.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{{\left(c \cdot \left|x \cdot s\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0 \cdot t_0}{t_1}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))) (t_1 (cos (* x 2.0))))
   (if (<= x 5e-6)
     (/ t_1 (pow (* c (fabs (* x s))) 2.0))
     (/ 1.0 (/ (* t_0 t_0) t_1)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (x <= 5e-6) {
		tmp = t_1 / pow((c * fabs((x * s))), 2.0);
	} else {
		tmp = 1.0 / ((t_0 * t_0) / t_1);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = cos((x * 2.0d0))
    if (x <= 5d-6) then
        tmp = t_1 / ((c * abs((x * s))) ** 2.0d0)
    else
        tmp = 1.0d0 / ((t_0 * t_0) / t_1)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (x <= 5e-6) {
		tmp = t_1 / Math.pow((c * Math.abs((x * s))), 2.0);
	} else {
		tmp = 1.0 / ((t_0 * t_0) / t_1);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if x <= 5e-6:
		tmp = t_1 / math.pow((c * math.fabs((x * s))), 2.0)
	else:
		tmp = 1.0 / ((t_0 * t_0) / t_1)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (x <= 5e-6)
		tmp = Float64(t_1 / (Float64(c * abs(Float64(x * s))) ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 * t_0) / t_1));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (x <= 5e-6)
		tmp = t_1 / ((c * abs((x * s))) ^ 2.0);
	else
		tmp = 1.0 / ((t_0 * t_0) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 5e-6], N[(t$95$1 / N[Power[N[(c * N[Abs[N[(x * s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000041e-6

   1. Initial program 62.3%

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

   2. Taylor expanded in c around 0 55.6%

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

      1. *-commutative 55.6%

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

      2. unpow2 55.6%

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

      3. unpow2 55.6%

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

      4. swap-sqr 75.8%

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

      5. unpow2 75.8%

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

      6. unpow2 75.8%

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

      7. rem-square-sqrt 75.8%

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

      8. swap-sqr 82.8%

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

      9. unpow2 82.8%

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

      10. unpow2 82.8%

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

      11. rem-sqrt-square 96.8%

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

      12. *-commutative 96.8%

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

   4. Simplified 96.8%

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

   if 5.00000000000000041e-6 < x

   1. Initial program 62.5%

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

      1. clear-num 62.5%

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

      2. associate-/r/ 62.5%

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

      3. associate-/r* 62.4%

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

      4. pow-flip 62.5%

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

      5. metadata-eval 62.5%

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

      6. *-commutative 62.5%

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

      7. associate-*r* 53.7%

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

      8. unpow2 53.7%

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

      9. pow-prod-down 77.7%

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

   3. Applied egg-rr 77.7%

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

   5. Applied egg-rr 96.4%

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

      1. pow2 96.4%

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot \left|x \cdot s\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}{\cos \left(x \cdot 2\right)}}\\ \end{array} \]

Alternative 2: 99.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0 \cdot t_0}{\cos \left(x \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))) (t_1 (* c (* x s))))
   (if (<= x 2e-35)
     (/ 1.0 (* t_1 t_1))
     (/ 1.0 (/ (* t_0 t_0) (cos (* x 2.0)))))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 2e-35) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = 1.0 / ((t_0 * t_0) / cos((x * 2.0)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = c * (x * s)
    if (x <= 2d-35) then
        tmp = 1.0d0 / (t_1 * t_1)
    else
        tmp = 1.0d0 / ((t_0 * t_0) / cos((x * 2.0d0)))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 2e-35) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = 1.0 / ((t_0 * t_0) / Math.cos((x * 2.0)));
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	t_1 = c * (x * s)
	tmp = 0
	if x <= 2e-35:
		tmp = 1.0 / (t_1 * t_1)
	else:
		tmp = 1.0 / ((t_0 * t_0) / math.cos((x * 2.0)))
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 2e-35)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(1.0 / Float64(Float64(t_0 * t_0) / cos(Float64(x * 2.0))));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 2e-35)
		tmp = 1.0 / (t_1 * t_1);
	else
		tmp = 1.0 / ((t_0 * t_0) / cos((x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e-35], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000002e-35

   1. Initial program 61.8%

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

   2. Taylor expanded in x around 0 52.5%

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

      1. associate-/r* 52.5%

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

      2. *-commutative 52.5%

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

      3. unpow2 52.5%

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

      4. unpow2 52.5%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. associate-/r* 69.6%

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

      8. rem-square-sqrt 69.6%

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

      9. unpow2 69.6%

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

      10. swap-sqr 75.1%

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

      11. unpow2 75.1%

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

      12. unpow2 75.1%

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

      13. rem-sqrt-square 85.4%

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

      14. *-commutative 85.4%

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

   4. Simplified 85.4%

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

      1. *-commutative 85.4%

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

      2. unpow-prod-down 69.6%

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

      3. add-sqr-sqrt 44.1%

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

      4. fabs-sqr 44.1%

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

      5. add-sqr-sqrt 69.6%

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

      6. pow1 69.6%

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

      7. metadata-eval 69.6%

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

      8. sqrt-pow1 69.6%

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

      9. pow2 69.6%

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

      10. pow2 69.6%

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

      11. swap-sqr 75.1%

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

      12. sqrt-pow1 55.3%

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

      13. metadata-eval 55.3%

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

      14. pow1 55.3%

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

      15. associate-*l* 55.3%

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

      16. sqrt-pow1 84.6%

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

      17. metadata-eval 84.6%

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

      18. pow1 84.6%

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

      19. associate-*l* 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in s around 0 84.6%

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

   8. Taylor expanded in s around 0 85.4%

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

   if 2.00000000000000002e-35 < x

   1. Initial program 63.9%

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

      1. clear-num 63.9%

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

      2. associate-/r/ 63.9%

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

      3. associate-/r* 63.8%

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

      4. pow-flip 63.9%

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

      5. metadata-eval 63.9%

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

      6. *-commutative 63.9%

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

      7. associate-*r* 56.7%

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

      8. unpow2 56.7%

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

      9. pow-prod-down 76.3%

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

   3. Applied egg-rr 76.3%

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

   5. Applied egg-rr 96.9%

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

      1. pow2 96.9%

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

   7. Applied egg-rr 96.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}{\cos \left(x \cdot 2\right)}}\\ \end{array} \]

Alternative 3: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ c = |c|\\ s = |s|\\ [c, s] = \mathsf{sort}([c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))) (t_1 (* c (* x s))))
   (if (<= x 2e-35) (/ 1.0 (* t_1 t_1)) (/ (cos (* x 2.0)) (* t_0 t_0)))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 2e-35) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = s * (x * c)
    t_1 = c * (x * s)
    if (x <= 2d-35) then
        tmp = 1.0d0 / (t_1 * t_1)
    else
        tmp = cos((x * 2.0d0)) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
c = Math.abs(c);
s = Math.abs(s);
assert c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 2e-35) {
		tmp = 1.0 / (t_1 * t_1);
	} else {
		tmp = Math.cos((x * 2.0)) / (t_0 * t_0);
	}
	return tmp;
}

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	t_1 = c * (x * s)
	tmp = 0
	if x <= 2e-35:
		tmp = 1.0 / (t_1 * t_1)
	else:
		tmp = math.cos((x * 2.0)) / (t_0 * t_0)
	return tmp

Julia

x = abs(x)
c = abs(c)
s = abs(s)
c, s = sort([c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 2e-35)
		tmp = Float64(1.0 / Float64(t_1 * t_1));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
c = abs(c)
s = abs(s)
c, s = num2cell(sort([c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 2e-35)
		tmp = 1.0 / (t_1 * t_1);
	else
		tmp = cos((x * 2.0)) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e-35], N[(1.0 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000002e-35

   1. Initial program 61.8%

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

   2. Taylor expanded in x around 0 52.5%

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

      1. associate-/r* 52.5%

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

      2. *-commutative 52.5%

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

      3. unpow2 52.5%

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

      4. unpow2 52.5%

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

      5. swap-sqr 69.1%

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

      6. unpow2 69.1%

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

      7. associate-/r* 69.6%

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

      8. rem-square-sqrt 69.6%

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

      9. unpow2 69.6%

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

      10. swap-sqr 75.1%

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

      11. unpow2 75.1%

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

      12. unpow2 75.1%

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

      13. rem-sqrt-square 85.4%

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

      14. *-commutative 85.4%

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

   4. Simplified 85.4%

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

      1. *-commutative 85.4%

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

      2. unpow-prod-down 69.6%

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

      3. add-sqr-sqrt 44.1%

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

      4. fabs-sqr 44.1%

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

      5. add-sqr-sqrt 69.6%

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

      6. pow1 69.6%

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

      7. metadata-eval 69.6%

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

      8. sqrt-pow1 69.6%

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

      9. pow2 69.6%

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

      10. pow2 69.6%

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

      11. swap-sqr 75.1%

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

      12. sqrt-pow1 55.3%

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

      13. metadata-eval 55.3%

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

      14. pow1 55.3%

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

      15. associate-*l* 55.3%

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

      16. sqrt-pow1 84.6%

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

      17. metadata-eval 84.6%

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

      18. pow1 84.6%

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

      19. associate-*l* 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in s around 0 84.6%

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

   8. Taylor expanded in s around 0 85.4%

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

   if 2.00000000000000002e-35 < x

   1. Initial program 63.9%

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

   2. Taylor expanded in c around 0 56.6%

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

      1. *-commutative 56.6%

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

      2. unpow2 56.6%

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

      3. unpow2 56.6%

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

      4. swap-sqr 76.3%

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

      5. unpow2 76.3%

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

      6. unpow2 76.3%

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

      7. rem-square-sqrt 76.3%

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

      8. swap-sqr 83.7%

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

      9. unpow2 83.7%

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

      10. unpow2 83.7%

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

      11. rem-sqrt-square 97.2%

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

      12. *-commutative 97.2%

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

   4. Simplified 97.2%

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

      1. *-commutative 73.2%

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

      2. unpow-prod-down 63.8%

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

      3. add-sqr-sqrt 23.7%

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

      4. fabs-sqr 23.7%

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

      5. add-sqr-sqrt 63.8%

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

      6. pow1 63.8%

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

      7. metadata-eval 63.8%

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

      8. sqrt-pow1 63.8%

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

      9. pow2 63.8%

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

      10. pow2 63.8%

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

      11. swap-sqr 68.8%

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

      12. sqrt-pow1 54.8%

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

      13. metadata-eval 54.8%

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

      14. pow1 54.8%

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

      15. associate-*l* 54.2%

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

      16. sqrt-pow1 73.0%

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

      17. metadata-eval 73.0%

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

      18. pow1 73.0%

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

      19. associate-*l* 73.4%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]

Alternative 4: 79.4% accurate, 24.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: c should be positive before calling this function
NOTE: s should be positive before calling this function
NOTE: c and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

x = abs(x);
c = abs(c);
s = abs(s);
assert(c < s);
double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Fortran

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

Java

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

Python

x = abs(x)
c = abs(c)
s = abs(s)
[c, s] = sort([c, s])
def code(x, c, s):
	t_0 = c * (x * s)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in x around 0 52.5%

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

   1. associate-/r* 52.5%

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

   2. *-commutative 52.5%

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

   3. unpow2 52.5%

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

   4. unpow2 52.5%

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

   5. swap-sqr 67.7%

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

   6. unpow2 67.7%

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

   7. associate-/r* 68.0%

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

   8. rem-square-sqrt 68.0%

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

   9. unpow2 68.0%

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

   10. swap-sqr 73.4%

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

   11. unpow2 73.4%

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

   12. unpow2 73.4%

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

   13. rem-sqrt-square 82.0%

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

   14. *-commutative 82.0%

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

4. Simplified 82.0%

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

   1. *-commutative 82.0%

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

   2. unpow-prod-down 68.0%

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

   3. add-sqr-sqrt 38.4%

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

   4. fabs-sqr 38.4%

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

   5. add-sqr-sqrt 68.0%

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

   6. pow1 68.0%

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

   7. metadata-eval 68.0%

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

   8. sqrt-pow1 68.0%

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

   9. pow2 68.0%

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

   10. pow2 68.0%

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

   11. swap-sqr 73.4%

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

   12. sqrt-pow1 55.1%

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

   13. metadata-eval 55.1%

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

   14. pow1 55.1%

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

   15. associate-*l* 55.0%

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

   16. sqrt-pow1 81.4%

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

   17. metadata-eval 81.4%

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

   18. pow1 81.4%

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

   19. associate-*l* 82.4%

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

6. Applied egg-rr 82.4%

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

7. Taylor expanded in s around 0 81.4%

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

8. Taylor expanded in s around 0 82.0%

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

9. Final simplification 82.0%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))