mixedcos

Percentage Accurate: 66.5% → 97.3%

Time: 14.4s

Alternatives: 10

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: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. associate-/r* 57.0%

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

   2. *-commutative 57.0%

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

   3. associate-*r* 52.8%

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

   4. unpow2 52.8%

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

   5. associate-/r* 53.2%

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

   6. add-sqr-sqrt 28.0%

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

   7. sqrt-unprod 44.9%

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

   8. swap-sqr 44.9%

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

   9. metadata-eval 44.9%

      \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   10. metadata-eval 44.9%

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

   11. swap-sqr 44.9%

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

   12. *-commutative 44.9%

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

   13. *-commutative 44.9%

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

   14. sqrt-unprod 22.6%

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

   15. add-sqr-sqrt 53.2%

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

   16. clear-num 53.2%

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

   17. inv-pow 53.2%

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

4. Applied egg-rr 96.5%

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

   1. unpow-1 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-/r/ 96.4%

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

   4. pow-flip 96.5%

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

   5. *-commutative 96.5%

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

   6. metadata-eval 96.5%

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

   7. *-commutative 96.5%

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

6. Applied egg-rr 96.5%

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

   1. metadata-eval 96.5%

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

   2. pow-div 96.4%

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

   3. inv-pow 96.4%

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

   4. pow1 96.4%

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

   5. associate-*l/ 96.4%

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

   6. associate-/r* 96.5%

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

8. Applied egg-rr 96.5%

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

9. Final simplification 96.5%

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

Alternative 2: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-29}:\\ \;\;\;\;{\left(\frac{\frac{1}{c_m}}{x \cdot s}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c_m}}{s \cdot \left(\left(x \cdot s\right) \cdot \left(c_m \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= x 5e-29)
   (pow (/ (/ 1.0 c_m) (* x s)) 2.0)
   (/ (/ (cos (* x 2.0)) c_m) (* s (* (* x s) (* c_m x))))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if (x <= 5e-29) {
		tmp = pow(((1.0 / c_m) / (x * s)), 2.0);
	} else {
		tmp = (cos((x * 2.0)) / c_m) / (s * ((x * s) * (c_m * x)));
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 5d-29) then
        tmp = ((1.0d0 / c_m) / (x * s)) ** 2.0d0
    else
        tmp = (cos((x * 2.0d0)) / c_m) / (s * ((x * s) * (c_m * x)))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if (x <= 5e-29) {
		tmp = Math.pow(((1.0 / c_m) / (x * s)), 2.0);
	} else {
		tmp = (Math.cos((x * 2.0)) / c_m) / (s * ((x * s) * (c_m * x)));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if x <= 5e-29:
		tmp = math.pow(((1.0 / c_m) / (x * s)), 2.0)
	else:
		tmp = (math.cos((x * 2.0)) / c_m) / (s * ((x * s) * (c_m * x)))
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (x <= 5e-29)
		tmp = Float64(Float64(1.0 / c_m) / Float64(x * s)) ^ 2.0;
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / c_m) / Float64(s * Float64(Float64(x * s) * Float64(c_m * x))));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if (x <= 5e-29)
		tmp = ((1.0 / c_m) / (x * s)) ^ 2.0;
	else
		tmp = (cos((x * 2.0)) / c_m) / (s * ((x * s) * (c_m * x)));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[x, 5e-29], N[Power[N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / c$95$m), $MachinePrecision] / N[(s * N[(N[(x * s), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999986e-29

   1. Initial program 57.0%

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

   3. Taylor expanded in x around 0 49.6%

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

      1. associate-/r* 49.5%

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

      2. *-commutative 49.5%

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

      3. unpow2 49.5%

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

      4. unpow2 49.5%

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

      5. swap-sqr 65.6%

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

      6. unpow2 65.6%

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

      7. associate-/r* 65.6%

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

      8. unpow2 65.6%

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

      9. unpow2 65.6%

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

      10. swap-sqr 85.0%

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

      11. unpow2 85.0%

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

      12. *-commutative 85.0%

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

   5. Simplified 85.0%

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

      1. add-sqr-sqrt 84.9%

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

      2. sqrt-div 84.9%

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

      3. metadata-eval 84.9%

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

      4. *-commutative 84.9%

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

      5. unpow2 84.9%

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

      6. sqrt-prod 57.5%

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

      7. add-sqr-sqrt 55.5%

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

      8. sqrt-div 55.5%

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

      9. metadata-eval 55.5%

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

      10. *-commutative 55.5%

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

      11. unpow2 55.5%

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

      12. sqrt-prod 42.3%

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

      13. add-sqr-sqrt 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. pow2 84.9%

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

      2. associate-/r* 84.9%

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

   9. Applied egg-rr 84.9%

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

   if 4.99999999999999986e-29 < x

   1. Initial program 58.7%

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

      1. associate-/r* 57.2%

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

      2. *-commutative 57.2%

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

      3. associate-*r* 53.9%

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

      4. unpow2 53.9%

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

      5. associate-/r* 55.4%

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

      6. add-sqr-sqrt 52.6%

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

      7. sqrt-unprod 39.7%

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

      8. swap-sqr 39.7%

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

      9. metadata-eval 39.7%

         \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      10. metadata-eval 39.7%

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

      11. swap-sqr 39.7%

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

      12. *-commutative 39.7%

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

      13. *-commutative 39.7%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 55.4%

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

      16. clear-num 55.3%

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

      17. inv-pow 55.3%

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

   4. Applied egg-rr 97.0%

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

      1. unpow-1 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-/r/ 96.9%

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

      4. pow-flip 97.0%

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

      5. *-commutative 97.0%

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

      6. metadata-eval 97.0%

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

      7. *-commutative 97.0%

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

   6. Applied egg-rr 97.0%

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

      1. associate-*r* 99.7%

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

      2. pow-prod-down 69.7%

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

      3. *-commutative 69.7%

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

      4. pow-prod-down 99.7%

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

      5. metadata-eval 99.7%

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

      6. pow-flip 99.6%

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

      7. pow-prod-down 70.8%

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

      8. associate-*l/ 70.8%

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

      9. *-un-lft-identity 70.8%

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

      10. *-commutative 70.8%

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

      11. pow-prod-down 99.6%

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

      12. associate-*r* 97.0%

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

      13. pow2 97.0%

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

      14. associate-*l* 94.4%

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

      15. *-commutative 94.4%

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

      16. associate-*r* 93.1%

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

      17. associate-/r* 93.0%

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

      18. *-commutative 93.0%

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

   8. Applied egg-rr 93.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-29}:\\ \;\;\;\;{\left(\frac{\frac{1}{c}}{x \cdot s}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c}}{s \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{\frac{1}{c_m}}{x \cdot s}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c_m \cdot s}}{\left(x \cdot s\right) \cdot \left(c_m \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= x 1.75e-22)
   (pow (/ (/ 1.0 c_m) (* x s)) 2.0)
   (/ (/ (cos (* x 2.0)) (* c_m s)) (* (* x s) (* c_m x)))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if (x <= 1.75e-22) {
		tmp = pow(((1.0 / c_m) / (x * s)), 2.0);
	} else {
		tmp = (cos((x * 2.0)) / (c_m * s)) / ((x * s) * (c_m * x));
	}
	return tmp;
}

Fortran

c_m = abs(c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 1.75d-22) then
        tmp = ((1.0d0 / c_m) / (x * s)) ** 2.0d0
    else
        tmp = (cos((x * 2.0d0)) / (c_m * s)) / ((x * s) * (c_m * x))
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if (x <= 1.75e-22) {
		tmp = Math.pow(((1.0 / c_m) / (x * s)), 2.0);
	} else {
		tmp = (Math.cos((x * 2.0)) / (c_m * s)) / ((x * s) * (c_m * x));
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if x <= 1.75e-22:
		tmp = math.pow(((1.0 / c_m) / (x * s)), 2.0)
	else:
		tmp = (math.cos((x * 2.0)) / (c_m * s)) / ((x * s) * (c_m * x))
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (x <= 1.75e-22)
		tmp = Float64(Float64(1.0 / c_m) / Float64(x * s)) ^ 2.0;
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / Float64(c_m * s)) / Float64(Float64(x * s) * Float64(c_m * x)));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if (x <= 1.75e-22)
		tmp = ((1.0 / c_m) / (x * s)) ^ 2.0;
	else
		tmp = (cos((x * 2.0)) / (c_m * s)) / ((x * s) * (c_m * x));
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[x, 1.75e-22], N[Power[N[(N[(1.0 / c$95$m), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(c$95$m * s), $MachinePrecision]), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75000000000000003e-22

   1. Initial program 57.1%

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

   3. Taylor expanded in x around 0 49.8%

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

      1. associate-/r* 49.8%

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

      2. *-commutative 49.8%

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

      3. unpow2 49.8%

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

      4. unpow2 49.8%

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

      5. swap-sqr 65.6%

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

      6. unpow2 65.6%

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

      7. associate-/r* 65.6%

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

      8. unpow2 65.6%

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

      9. unpow2 65.6%

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

      10. swap-sqr 85.2%

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

      11. unpow2 85.2%

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

      12. *-commutative 85.2%

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

   5. Simplified 85.2%

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

      1. add-sqr-sqrt 85.1%

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

      2. sqrt-div 85.1%

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

      3. metadata-eval 85.1%

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

      4. *-commutative 85.1%

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

      5. unpow2 85.1%

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

      6. sqrt-prod 57.1%

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

      7. add-sqr-sqrt 55.1%

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

      8. sqrt-div 55.1%

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

      9. metadata-eval 55.1%

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

      10. *-commutative 55.1%

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

      11. unpow2 55.1%

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

      12. sqrt-prod 42.1%

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

      13. add-sqr-sqrt 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. pow2 85.1%

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

      2. associate-/r* 85.2%

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

   9. Applied egg-rr 85.2%

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

   if 1.75000000000000003e-22 < x

   1. Initial program 58.4%

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

      1. associate-/r* 56.7%

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

      2. *-commutative 56.7%

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

      3. associate-*r* 53.3%

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

      4. unpow2 53.3%

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

      5. associate-/r* 54.9%

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

      6. add-sqr-sqrt 52.0%

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

      7. sqrt-unprod 38.6%

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

      8. swap-sqr 38.6%

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

      9. metadata-eval 38.6%

         \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      10. metadata-eval 38.6%

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

      11. swap-sqr 38.6%

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

      12. *-commutative 38.6%

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

      13. *-commutative 38.6%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 54.9%

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

      16. clear-num 54.9%

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

      17. inv-pow 54.9%

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

   4. Applied egg-rr 96.9%

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

      1. unpow-1 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-/r/ 96.9%

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

      4. pow-flip 96.9%

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

      5. *-commutative 96.9%

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

      6. metadata-eval 96.9%

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

      7. *-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. associate-*r* 99.7%

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

      2. pow-prod-down 69.9%

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

      3. *-commutative 69.9%

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

      4. pow-prod-down 99.7%

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

      5. metadata-eval 99.7%

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

      6. pow-flip 99.6%

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

      7. pow-prod-down 71.0%

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

      8. associate-/r/ 71.0%

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

      9. clear-num 71.0%

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

      10. *-commutative 71.0%

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

      11. pow-prod-down 99.7%

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

      12. associate-*r* 96.9%

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

      13. pow2 96.9%

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

      14. associate-*l* 94.2%

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

      15. *-commutative 94.2%

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

      16. associate-*r* 92.8%

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

      17. associate-*l* 90.1%

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

   8. Applied egg-rr 90.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-22}:\\ \;\;\;\;{\left(\frac{\frac{1}{c}}{x \cdot s}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot s}}{\left(x \cdot s\right) \cdot \left(c \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. associate-/r* 57.0%

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

   2. *-commutative 57.0%

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

   3. associate-*r* 52.8%

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

   4. unpow2 52.8%

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

   5. associate-/r* 53.2%

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

   6. add-sqr-sqrt 28.0%

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

   7. sqrt-unprod 44.9%

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

   8. swap-sqr 44.9%

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

   9. metadata-eval 44.9%

      \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   10. metadata-eval 44.9%

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

   11. swap-sqr 44.9%

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

   12. *-commutative 44.9%

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

   13. *-commutative 44.9%

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

   14. sqrt-unprod 22.6%

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

   15. add-sqr-sqrt 53.2%

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

   16. clear-num 53.2%

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

   17. inv-pow 53.2%

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

4. Applied egg-rr 96.5%

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

   1. unpow2 96.5%

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

   2. *-commutative 96.5%

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

   3. associate-*r* 93.6%

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

   4. associate-*l* 90.1%

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

6. Applied egg-rr 90.1%

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

   1. unpow-1 90.1%

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

   2. *-commutative 90.1%

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

   3. clear-num 90.1%

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

   4. *-un-lft-identity 90.1%

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

   5. associate-*l* 90.7%

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

   6. associate-*r* 92.4%

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

   7. *-commutative 92.4%

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

   8. associate-*l* 96.5%

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

   9. times-frac 96.4%

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

   10. associate-/r* 96.4%

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

8. Applied egg-rr 96.4%

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

9. Final simplification 96.4%

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

Alternative 5: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

   1. associate-/r* 57.0%

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

   2. *-commutative 57.0%

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

   3. associate-*r* 52.8%

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

   4. unpow2 52.8%

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

   5. associate-/r* 53.2%

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

   6. add-sqr-sqrt 28.0%

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

   7. sqrt-unprod 44.9%

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

   8. swap-sqr 44.9%

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

   9. metadata-eval 44.9%

      \[\leadsto \frac{\cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

   10. metadata-eval 44.9%

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

   11. swap-sqr 44.9%

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

   12. *-commutative 44.9%

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

   13. *-commutative 44.9%

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

   14. sqrt-unprod 22.6%

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

   15. add-sqr-sqrt 53.2%

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

   16. clear-num 53.2%

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

   17. inv-pow 53.2%

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

4. Applied egg-rr 96.5%

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

   1. unpow-1 96.5%

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

   2. clear-num 96.5%

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

   3. unpow2 96.5%

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

   4. associate-/r* 96.4%

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

   5. *-commutative 96.4%

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

6. Applied egg-rr 96.4%

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

7. Final simplification 96.4%

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

Alternative 6: 79.5% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in x around 0 48.5%

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

   1. associate-/r* 48.0%

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

   2. *-commutative 48.0%

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

   3. unpow2 48.0%

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

   4. unpow2 48.0%

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

   5. swap-sqr 61.1%

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

   6. unpow2 61.1%

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

   7. associate-/r* 61.5%

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

   8. unpow2 61.5%

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

   9. unpow2 61.5%

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

   10. swap-sqr 78.6%

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

   11. unpow2 78.6%

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

   12. *-commutative 78.6%

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

5. Simplified 78.6%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-div 78.6%

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

   3. metadata-eval 78.6%

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

   4. *-commutative 78.6%

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

   5. unpow2 78.6%

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

   6. sqrt-prod 50.0%

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

   7. add-sqr-sqrt 57.3%

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

   8. sqrt-div 57.3%

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

   9. metadata-eval 57.3%

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

   10. *-commutative 57.3%

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

   11. unpow2 57.3%

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

   12. sqrt-prod 39.8%

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

   13. add-sqr-sqrt 78.6%

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

7. Applied egg-rr 78.6%

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

8. Final simplification 78.6%

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

Alternative 7: 79.5% accurate, 20.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in x around 0 48.5%

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

   1. associate-/r* 48.0%

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

   2. *-commutative 48.0%

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

   3. unpow2 48.0%

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

   4. unpow2 48.0%

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

   5. swap-sqr 61.1%

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

   6. unpow2 61.1%

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

   7. associate-/r* 61.5%

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

   8. unpow2 61.5%

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

   9. unpow2 61.5%

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

   10. swap-sqr 78.6%

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

   11. unpow2 78.6%

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

   12. *-commutative 78.6%

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

5. Simplified 78.6%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-div 78.6%

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

   3. metadata-eval 78.6%

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

   4. *-commutative 78.6%

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

   5. unpow2 78.6%

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

   6. sqrt-prod 50.0%

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

   7. add-sqr-sqrt 57.3%

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

   8. sqrt-div 57.3%

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

   9. metadata-eval 57.3%

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

   10. *-commutative 57.3%

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

   11. unpow2 57.3%

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

   12. sqrt-prod 39.8%

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

   13. add-sqr-sqrt 78.6%

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

7. Applied egg-rr 78.6%

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

8. Taylor expanded in c around 0 78.6%

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

   1. associate-/r* 78.6%

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

10. Simplified 78.6%

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

11. Final simplification 78.6%

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

Alternative 8: 78.4% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in x around 0 48.5%

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

   1. associate-/r* 48.0%

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

   2. *-commutative 48.0%

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

   3. unpow2 48.0%

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

   4. unpow2 48.0%

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

   5. swap-sqr 61.1%

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

   6. unpow2 61.1%

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

   7. associate-/r* 61.5%

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

   8. unpow2 61.5%

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

   9. unpow2 61.5%

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

   10. swap-sqr 78.6%

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

   11. unpow2 78.6%

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

   12. *-commutative 78.6%

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

5. Simplified 78.6%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-div 78.6%

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

   3. metadata-eval 78.6%

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

   4. *-commutative 78.6%

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

   5. unpow2 78.6%

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

   6. sqrt-prod 50.0%

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

   7. add-sqr-sqrt 57.3%

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

   8. sqrt-div 57.3%

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

   9. metadata-eval 57.3%

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

   10. *-commutative 57.3%

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

   11. unpow2 57.3%

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

   12. sqrt-prod 39.8%

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

   13. add-sqr-sqrt 78.6%

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

7. Applied egg-rr 78.6%

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

   1. pow2 78.6%

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

   2. associate-/r* 78.6%

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

9. Applied egg-rr 78.6%

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

    1. unpow2 78.6%

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

    2. associate-/r* 78.6%

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

    3. frac-2neg 78.6%

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

    4. metadata-eval 78.6%

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

    5. frac-times 76.1%

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

    6. neg-mul-1 76.1%

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

    7. distribute-neg-frac 76.1%

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

    8. metadata-eval 76.1%

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

    9. distribute-rgt-neg-in 76.1%

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

    10. *-commutative 76.1%

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

    11. distribute-rgt-neg-in 76.1%

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

11. Applied egg-rr 76.1%

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

12. Final simplification 76.1%

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

Alternative 9: 77.0% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in x around 0 48.5%

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

   1. associate-/r* 48.0%

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

   2. *-commutative 48.0%

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

   3. unpow2 48.0%

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

   4. unpow2 48.0%

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

   5. swap-sqr 61.1%

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

   6. unpow2 61.1%

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

   7. associate-/r* 61.5%

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

   8. unpow2 61.5%

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

   9. unpow2 61.5%

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

   10. swap-sqr 78.6%

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

   11. unpow2 78.6%

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

   12. *-commutative 78.6%

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

5. Simplified 78.6%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-div 78.6%

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

   3. metadata-eval 78.6%

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

   4. *-commutative 78.6%

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

   5. unpow2 78.6%

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

   6. sqrt-prod 50.0%

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

   7. add-sqr-sqrt 57.3%

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

   8. sqrt-div 57.3%

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

   9. metadata-eval 57.3%

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

   10. *-commutative 57.3%

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

   11. unpow2 57.3%

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

   12. sqrt-prod 39.8%

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

   13. add-sqr-sqrt 78.6%

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

7. Applied egg-rr 78.6%

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

   1. pow2 78.6%

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

   2. associate-/r* 78.6%

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

9. Applied egg-rr 78.6%

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

    1. unpow2 78.6%

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

    2. associate-/r* 78.6%

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

    3. frac-times 76.1%

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

    4. associate-/r/ 76.1%

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

    5. clear-num 76.1%

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

    6. associate-*r* 75.0%

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

    7. *-commutative 75.0%

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

11. Applied egg-rr 75.0%

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

12. Final simplification 75.0%

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

Alternative 10: 78.8% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in x around 0 48.5%

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

   1. associate-/r* 48.0%

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

   2. *-commutative 48.0%

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

   3. unpow2 48.0%

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

   4. unpow2 48.0%

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

   5. swap-sqr 61.1%

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

   6. unpow2 61.1%

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

   7. associate-/r* 61.5%

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

   8. unpow2 61.5%

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

   9. unpow2 61.5%

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

   10. swap-sqr 78.6%

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

   11. unpow2 78.6%

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

   12. *-commutative 78.6%

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

5. Simplified 78.6%

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

   1. add-sqr-sqrt 78.5%

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

   2. sqrt-div 78.6%

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

   3. metadata-eval 78.6%

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

   4. *-commutative 78.6%

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

   5. unpow2 78.6%

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

   6. sqrt-prod 50.0%

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

   7. add-sqr-sqrt 57.3%

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

   8. sqrt-div 57.3%

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

   9. metadata-eval 57.3%

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

   10. *-commutative 57.3%

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

   11. unpow2 57.3%

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

   12. sqrt-prod 39.8%

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

   13. add-sqr-sqrt 78.6%

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

7. Applied egg-rr 78.6%

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

   1. pow2 78.6%

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

   2. associate-/r* 78.6%

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

9. Applied egg-rr 78.6%

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

    1. unpow2 78.6%

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

    2. associate-/r* 78.6%

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

    3. un-div-inv 78.6%

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

    4. *-commutative 78.6%

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

    5. associate-/r* 77.0%

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

    6. associate-/r* 76.8%

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

    7. associate-/r* 76.8%

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

    8. *-commutative 76.8%

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

    9. associate-*r* 77.5%

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

    10. associate-*r* 79.0%

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

    11. *-commutative 79.0%

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

11. Applied egg-rr 79.0%

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

12. Final simplification 79.0%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))