mixedcos

Percentage Accurate: 66.8% → 97.1%

Time: 13.3s

Alternatives: 6

Speedup: 24.1×

• 30.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% 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: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \frac{\frac{1}{c\_m} \cdot \left(\frac{1}{s} \cdot \frac{\cos \left(x\_m \cdot 2\right)}{x\_m}\right)}{c\_m \cdot \left|s \cdot x\_m\right|} \end{array} \]

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	return ((1.0 / c_m) * ((1.0 / s) * (cos((x_m * 2.0)) / x_m))) / (c_m * fabs((s * x_m)));
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = ((1.0d0 / c_m) * ((1.0d0 / s) * (cos((x_m * 2.0d0)) / x_m))) / (c_m * abs((s * x_m)))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	return ((1.0 / c_m) * ((1.0 / s) * (Math.cos((x_m * 2.0)) / x_m))) / (c_m * Math.abs((s * x_m)));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	return ((1.0 / c_m) * ((1.0 / s) * (math.cos((x_m * 2.0)) / x_m))) / (c_m * math.fabs((s * x_m)))

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	return Float64(Float64(Float64(1.0 / c_m) * Float64(Float64(1.0 / s) * Float64(cos(Float64(x_m * 2.0)) / x_m))) / Float64(c_m * abs(Float64(s * x_m))))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp = code(x_m, c_m, s)
	tmp = ((1.0 / c_m) * ((1.0 / s) * (cos((x_m * 2.0)) / x_m))) / (c_m * abs((s * x_m)));
end

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. *-un-lft-identity 68.6%

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

   2. add-sqr-sqrt 68.6%

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

   3. times-frac 68.6%

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

   4. sqrt-prod 68.6%

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

   5. sqrt-pow1 49.8%

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

   6. metadata-eval 49.8%

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

   7. pow1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-*r* 45.3%

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

   10. unpow2 45.3%

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

   11. pow-prod-down 49.8%

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

   12. sqrt-prod 49.7%

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

4. Applied egg-rr 86.7%

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

   1. associate-*l/ 86.7%

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

   2. *-lft-identity 86.7%

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

   3. unpow2 86.7%

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

   4. rem-sqrt-square 86.7%

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

   5. unpow2 86.7%

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

   6. rem-sqrt-square 97.3%

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

6. Simplified 97.3%

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

   1. *-un-lft-identity 97.3%

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

   2. add-sqr-sqrt 56.4%

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

   3. fabs-sqr 56.4%

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

   4. add-sqr-sqrt 66.5%

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

   5. times-frac 66.3%

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

   6. *-commutative 66.3%

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

8. Applied egg-rr 66.3%

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

   1. *-un-lft-identity 66.3%

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

   2. times-frac 66.3%

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

10. Applied egg-rr 66.3%

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

Alternative 2: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \frac{\frac{1}{c\_m} \cdot \frac{\cos \left(x\_m \cdot 2\right)}{s \cdot x\_m}}{c\_m \cdot \left|s \cdot x\_m\right|} \end{array} \]

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	return ((1.0 / c_m) * (cos((x_m * 2.0)) / (s * x_m))) / (c_m * fabs((s * x_m)));
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = ((1.0d0 / c_m) * (cos((x_m * 2.0d0)) / (s * x_m))) / (c_m * abs((s * x_m)))
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	return ((1.0 / c_m) * (Math.cos((x_m * 2.0)) / (s * x_m))) / (c_m * Math.abs((s * x_m)));
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	return ((1.0 / c_m) * (math.cos((x_m * 2.0)) / (s * x_m))) / (c_m * math.fabs((s * x_m)))

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	return Float64(Float64(Float64(1.0 / c_m) * Float64(cos(Float64(x_m * 2.0)) / Float64(s * x_m))) / Float64(c_m * abs(Float64(s * x_m))))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp = code(x_m, c_m, s)
	tmp = ((1.0 / c_m) * (cos((x_m * 2.0)) / (s * x_m))) / (c_m * abs((s * x_m)));
end

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. *-un-lft-identity 68.6%

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

   2. add-sqr-sqrt 68.6%

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

   3. times-frac 68.6%

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

   4. sqrt-prod 68.6%

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

   5. sqrt-pow1 49.8%

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

   6. metadata-eval 49.8%

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

   7. pow1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-*r* 45.3%

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

   10. unpow2 45.3%

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

   11. pow-prod-down 49.8%

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

   12. sqrt-prod 49.7%

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

4. Applied egg-rr 86.7%

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

   1. associate-*l/ 86.7%

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

   2. *-lft-identity 86.7%

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

   3. unpow2 86.7%

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

   4. rem-sqrt-square 86.7%

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

   5. unpow2 86.7%

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

   6. rem-sqrt-square 97.3%

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

6. Simplified 97.3%

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

   1. *-un-lft-identity 97.3%

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

   2. add-sqr-sqrt 56.4%

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

   3. fabs-sqr 56.4%

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

   4. add-sqr-sqrt 66.5%

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

   5. times-frac 66.3%

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

   6. *-commutative 66.3%

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

8. Applied egg-rr 66.3%

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

Alternative 3: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s \cdot x\_m\right)\\ \frac{\cos \left(x\_m \cdot 2\right)}{t\_0} \cdot \frac{1}{t\_0} \end{array} \end{array} \]

FPCore

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

C

x_m = fabs(x);
c_m = fabs(c);
assert(x_m < c_m && c_m < s);
double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (s * x_m);
	return (cos((x_m * 2.0)) / t_0) * (1.0 / t_0);
}

Fortran

x_m = abs(x)
c_m = abs(c)
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c_m * (s * x_m)
    code = (cos((x_m * 2.0d0)) / t_0) * (1.0d0 / t_0)
end function

Java

x_m = Math.abs(x);
c_m = Math.abs(c);
assert x_m < c_m && c_m < s;
public static double code(double x_m, double c_m, double s) {
	double t_0 = c_m * (s * x_m);
	return (Math.cos((x_m * 2.0)) / t_0) * (1.0 / t_0);
}

Python

x_m = math.fabs(x)
c_m = math.fabs(c)
[x_m, c_m, s] = sort([x_m, c_m, s])
def code(x_m, c_m, s):
	t_0 = c_m * (s * x_m)
	return (math.cos((x_m * 2.0)) / t_0) * (1.0 / t_0)

Julia

x_m = abs(x)
c_m = abs(c)
x_m, c_m, s = sort([x_m, c_m, s])
function code(x_m, c_m, s)
	t_0 = Float64(c_m * Float64(s * x_m))
	return Float64(Float64(cos(Float64(x_m * 2.0)) / t_0) * Float64(1.0 / t_0))
end

MATLAB

x_m = abs(x);
c_m = abs(c);
x_m, c_m, s = num2cell(sort([x_m, c_m, s])){:}
function tmp = code(x_m, c_m, s)
	t_0 = c_m * (s * x_m);
	tmp = (cos((x_m * 2.0)) / t_0) * (1.0 / t_0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
NOTE: x_m, c_m, and s should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s_] := Block[{t$95$0 = N[(c$95$m * N[(s * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 68.6%

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

   1. *-un-lft-identity 68.6%

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

   2. add-sqr-sqrt 68.6%

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

   3. times-frac 68.6%

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

   4. sqrt-prod 68.6%

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

   5. sqrt-pow1 49.8%

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

   6. metadata-eval 49.8%

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

   7. pow1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-*r* 45.3%

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

   10. unpow2 45.3%

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

   11. pow-prod-down 49.8%

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

   12. sqrt-prod 49.7%

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

4. Applied egg-rr 86.7%

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

   1. associate-*l/ 86.7%

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

   2. *-lft-identity 86.7%

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

   3. unpow2 86.7%

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

   4. rem-sqrt-square 86.7%

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

   5. unpow2 86.7%

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

   6. rem-sqrt-square 97.3%

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

6. Simplified 97.3%

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

   1. div-inv 97.3%

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

   2. *-commutative 97.3%

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

   3. add-sqr-sqrt 56.4%

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

   4. fabs-sqr 56.4%

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

   5. add-sqr-sqrt 66.5%

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

   6. add-sqr-sqrt 49.0%

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

   7. fabs-sqr 49.0%

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

   8. add-sqr-sqrt 97.3%

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

8. Applied egg-rr 97.3%

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

Alternative 4: 96.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. *-un-lft-identity 68.6%

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

   2. add-sqr-sqrt 68.6%

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

   3. times-frac 68.6%

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

   4. sqrt-prod 68.6%

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

   5. sqrt-pow1 49.8%

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

   6. metadata-eval 49.8%

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

   7. pow1 49.8%

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

   8. *-commutative 49.8%

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

   9. associate-*r* 45.3%

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

   10. unpow2 45.3%

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

   11. pow-prod-down 49.8%

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

   12. sqrt-prod 49.7%

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

4. Applied egg-rr 86.7%

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

   1. associate-*l/ 86.7%

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

   2. *-lft-identity 86.7%

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

   3. unpow2 86.7%

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

   4. rem-sqrt-square 86.7%

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

   5. unpow2 86.7%

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

   6. rem-sqrt-square 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in x around inf 80.3%

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

   1. *-commutative 80.3%

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

   2. unpow2 80.3%

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

   3. unpow2 80.3%

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

   4. sqr-abs 80.3%

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

   5. swap-sqr 97.3%

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

   6. unpow2 97.3%

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

   7. associate-*r* 97.5%

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

   8. *-commutative 97.5%

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

9. Simplified 97.5%

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

    1. unpow2 97.5%

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

11. Applied egg-rr 97.5%

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

Alternative 5: 79.2% accurate, 20.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \begin{array}{l} t_0 := \frac{1}{c\_m \cdot \left(s \cdot x\_m\right)}\\ t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in x around 0 54.8%

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

   1. associate-/r* 54.0%

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

   2. unpow2 54.0%

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

   3. unpow2 54.0%

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

   4. swap-sqr 66.6%

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

   5. unpow2 66.6%

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

   6. associate-/r* 67.4%

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

   7. unpow2 67.4%

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

   8. rem-square-sqrt 67.4%

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

   9. swap-sqr 73.4%

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

   10. unpow2 73.4%

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

   11. unpow2 73.4%

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

   12. rem-sqrt-square 78.7%

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

5. Simplified 78.7%

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

   1. inv-pow 78.7%

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

   2. unpow2 78.7%

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

   3. unpow-prod-down 78.7%

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

   4. inv-pow 78.7%

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

   5. add-sqr-sqrt 47.5%

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

   6. fabs-sqr 47.5%

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

   7. add-sqr-sqrt 60.9%

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

   8. inv-pow 60.9%

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

   9. add-sqr-sqrt 40.0%

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

   10. fabs-sqr 40.0%

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

   11. add-sqr-sqrt 78.7%

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

7. Applied egg-rr 78.7%

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

Alternative 6: 79.1% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ c_m = \left|c\right| \\ [x_m, c_m, s] = \mathsf{sort}([x_m, c_m, s])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(s \cdot x\_m\right)\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in x around 0 54.8%

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

   1. associate-/r* 54.0%

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

   2. unpow2 54.0%

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

   3. unpow2 54.0%

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

   4. swap-sqr 66.6%

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

   5. unpow2 66.6%

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

   6. associate-/r* 67.4%

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

   7. unpow2 67.4%

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

   8. rem-square-sqrt 67.4%

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

   9. swap-sqr 73.4%

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

   10. unpow2 73.4%

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

   11. unpow2 73.4%

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

   12. rem-sqrt-square 78.7%

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

5. Simplified 78.7%

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

   1. unpow-prod-down 67.4%

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

   2. add-sqr-sqrt 67.3%

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

   3. unpow-prod-down 67.4%

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

   4. sqrt-pow1 51.4%

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

   5. metadata-eval 51.4%

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

   6. pow1 51.4%

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

   7. add-sqr-sqrt 30.2%

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

   8. fabs-sqr 30.2%

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

   9. add-sqr-sqrt 53.7%

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

   10. unpow-prod-down 60.3%

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

   11. sqrt-pow1 60.9%

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

   12. metadata-eval 60.9%

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

   13. pow1 60.9%

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

   14. add-sqr-sqrt 40.0%

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

   15. fabs-sqr 40.0%

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

   16. add-sqr-sqrt 78.7%

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

7. Applied egg-rr 78.7%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))