mixedcos

Percentage Accurate: 66.5% → 97.4%

Time: 38.3s

Alternatives: 7

Speedup: 24.1×

• 31.6% 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: 66.5% 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: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-un-lft-identity 70.6%

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

   2. add-sqr-sqrt 70.6%

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

   3. times-frac 70.6%

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

   4. sqrt-prod 70.6%

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

   5. sqrt-pow1 50.0%

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

   6. metadata-eval 50.0%

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

   7. pow1 50.0%

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

   8. *-commutative 50.0%

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

   9. associate-*r* 46.2%

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

   10. unpow2 46.2%

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

   11. pow-prod-down 50.0%

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

   12. sqrt-prod 50.0%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 88.8%

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

   2. *-lft-identity 88.8%

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

   3. unpow2 88.8%

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

   4. rem-sqrt-square 88.8%

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

   5. unpow2 88.8%

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

   6. rem-sqrt-square 97.1%

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

6. Simplified 97.1%

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

   1. div-inv 97.1%

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

   2. *-commutative 97.1%

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

   3. add-sqr-sqrt 53.5%

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

   4. fabs-sqr 53.5%

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

   5. add-sqr-sqrt 64.7%

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

   6. add-sqr-sqrt 47.5%

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

   7. fabs-sqr 47.5%

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

   8. add-sqr-sqrt 97.1%

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

8. Applied egg-rr 97.1%

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

   1. clear-num 97.1%

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

   2. associate-*l/ 97.1%

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

   3. *-un-lft-identity 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-/r* 97.1%

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

   6. associate-/r* 97.2%

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

10. Applied egg-rr 97.2%

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

Alternative 2: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-un-lft-identity 70.6%

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

   2. add-sqr-sqrt 70.6%

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

   3. times-frac 70.6%

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

   4. sqrt-prod 70.6%

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

   5. sqrt-pow1 50.0%

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

   6. metadata-eval 50.0%

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

   7. pow1 50.0%

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

   8. *-commutative 50.0%

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

   9. associate-*r* 46.2%

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

   10. unpow2 46.2%

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

   11. pow-prod-down 50.0%

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

   12. sqrt-prod 50.0%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 88.8%

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

   2. *-lft-identity 88.8%

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

   3. unpow2 88.8%

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

   4. rem-sqrt-square 88.8%

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

   5. unpow2 88.8%

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

   6. rem-sqrt-square 97.1%

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

6. Simplified 97.1%

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

   1. div-inv 97.1%

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

   2. *-commutative 97.1%

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

   3. add-sqr-sqrt 53.5%

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

   4. fabs-sqr 53.5%

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

   5. add-sqr-sqrt 64.7%

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

   6. add-sqr-sqrt 47.5%

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

   7. fabs-sqr 47.5%

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

   8. add-sqr-sqrt 97.1%

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

8. Applied egg-rr 97.1%

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

Alternative 3: 97.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-un-lft-identity 70.6%

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

   2. add-sqr-sqrt 70.6%

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

   3. times-frac 70.6%

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

   4. sqrt-prod 70.6%

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

   5. sqrt-pow1 50.0%

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

   6. metadata-eval 50.0%

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

   7. pow1 50.0%

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

   8. *-commutative 50.0%

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

   9. associate-*r* 46.2%

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

   10. unpow2 46.2%

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

   11. pow-prod-down 50.0%

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

   12. sqrt-prod 50.0%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 88.8%

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

   2. *-lft-identity 88.8%

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

   3. unpow2 88.8%

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

   4. rem-sqrt-square 88.8%

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

   5. unpow2 88.8%

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

   6. rem-sqrt-square 97.1%

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

6. Simplified 97.1%

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

   1. div-inv 97.1%

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

   2. *-commutative 97.1%

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

   3. add-sqr-sqrt 53.5%

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

   4. fabs-sqr 53.5%

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

   5. add-sqr-sqrt 64.7%

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

   6. add-sqr-sqrt 47.5%

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

   7. fabs-sqr 47.5%

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

   8. add-sqr-sqrt 97.1%

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

8. Applied egg-rr 97.1%

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

   1. un-div-inv 97.1%

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

10. Applied egg-rr 97.1%

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

Alternative 4: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

4. Applied egg-rr 80.1%

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

   1. associate-*r/ 80.5%

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

   2. unpow2 80.5%

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

   3. *-rgt-identity 80.5%

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

   4. add-sqr-sqrt 69.0%

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

   5. pow2 69.0%

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

   6. sqrt-prod 59.6%

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

   7. sqrt-pow1 59.6%

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

   8. metadata-eval 59.6%

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

   9. inv-pow 59.6%

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

   10. div-inv 59.6%

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

   11. pow2 59.6%

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

   12. frac-times 72.2%

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

   13. associate-/l/ 72.2%

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

6. Applied egg-rr 97.2%

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

7. Final simplification 97.2%

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

Alternative 5: 80.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

   1. *-un-lft-identity 70.6%

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

   2. add-sqr-sqrt 70.6%

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

   3. times-frac 70.6%

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

   4. sqrt-prod 70.6%

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

   5. sqrt-pow1 50.0%

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

   6. metadata-eval 50.0%

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

   7. pow1 50.0%

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

   8. *-commutative 50.0%

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

   9. associate-*r* 46.2%

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

   10. unpow2 46.2%

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

   11. pow-prod-down 50.0%

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

   12. sqrt-prod 50.0%

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

4. Applied egg-rr 88.8%

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

   1. associate-*l/ 88.8%

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

   2. *-lft-identity 88.8%

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

   3. unpow2 88.8%

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

   4. rem-sqrt-square 88.8%

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

   5. unpow2 88.8%

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

   6. rem-sqrt-square 97.1%

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

6. Simplified 97.1%

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

   1. div-inv 97.1%

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

   2. *-commutative 97.1%

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

   3. add-sqr-sqrt 53.5%

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

   4. fabs-sqr 53.5%

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

   5. add-sqr-sqrt 64.7%

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

   6. add-sqr-sqrt 47.5%

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

   7. fabs-sqr 47.5%

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

   8. add-sqr-sqrt 97.1%

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

8. Applied egg-rr 97.1%

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

   1. clear-num 97.1%

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

   2. associate-*l/ 97.1%

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

   3. *-un-lft-identity 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-/r* 97.1%

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

   6. associate-/r* 97.2%

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

10. Applied egg-rr 97.2%

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

11. Taylor expanded in x around 0 79.3%

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

Alternative 6: 80.3% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in x around 0 59.5%

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

   1. associate-/r* 59.5%

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

   2. unpow2 59.5%

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

   3. unpow2 59.5%

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

   4. swap-sqr 69.8%

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

   5. unpow2 69.8%

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

   6. associate-/r* 70.0%

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

   7. unpow2 70.0%

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

   8. rem-square-sqrt 70.0%

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

   9. swap-sqr 76.0%

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

   10. unpow2 76.0%

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

   11. unpow2 76.0%

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

   12. rem-sqrt-square 79.2%

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

5. Simplified 79.2%

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

   1. add-sqr-sqrt 50.7%

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

   2. sqrt-prod 79.2%

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

   3. unpow2 79.2%

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

   4. pow2 79.2%

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

   5. unpow2 79.2%

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

   6. sqrt-prod 50.7%

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

   7. add-sqr-sqrt 56.8%

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

   8. add-sqr-sqrt 30.9%

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

   9. fabs-sqr 30.9%

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

   10. add-sqr-sqrt 60.1%

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

   11. unpow2 60.1%

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

   12. sqrt-prod 37.4%

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

   13. add-sqr-sqrt 58.7%

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

   14. add-sqr-sqrt 39.4%

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

   15. fabs-sqr 39.4%

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

   16. add-sqr-sqrt 79.2%

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

7. Applied egg-rr 79.2%

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

Alternative 7: 77.2% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.6%

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

3. Taylor expanded in x around 0 59.5%

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

   1. associate-/r* 59.5%

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

   2. unpow2 59.5%

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

   3. unpow2 59.5%

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

   4. swap-sqr 69.8%

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

   5. unpow2 69.8%

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

   6. associate-/r* 70.0%

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

   7. unpow2 70.0%

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

   8. rem-square-sqrt 70.0%

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

   9. swap-sqr 76.0%

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

   10. unpow2 76.0%

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

   11. unpow2 76.0%

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

   12. rem-sqrt-square 79.2%

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

5. Simplified 79.2%

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

   1. add-sqr-sqrt 50.7%

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

   2. sqrt-prod 79.2%

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

   3. unpow2 79.2%

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

   4. pow2 79.2%

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

   5. add-sqr-sqrt 79.2%

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

   6. /-rgt-identity 79.2%

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

   7. clear-num 79.2%

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

   8. pow-flip 79.2%

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

   9. add-sqr-sqrt 45.3%

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

   10. fabs-sqr 45.3%

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

   11. add-sqr-sqrt 79.2%

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

   12. metadata-eval 79.2%

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

7. Applied egg-rr 79.2%

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

   1. pow-flip 79.2%

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

   2. metadata-eval 79.2%

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

   3. pow2 79.2%

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

   4. associate-*r* 78.8%

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

   5. associate-*l* 78.0%

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

9. Applied egg-rr 78.0%

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

10. Final simplification 78.0%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))