mixedcos

Percentage Accurate: 66.3% → 97.5%

Time: 12.5s

Alternatives: 9

Speedup: 24.1×

• 30.8% 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: 66.3% 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: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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((x * 2.0d0)) * ((x * (s * c)) ** (-2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. *-un-lft-identity 61.7%

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

   2. add-sqr-sqrt 61.7%

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

   3. times-frac 61.7%

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

   4. sqrt-prod 61.7%

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

   5. sqrt-pow1 46.0%

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

   6. metadata-eval 46.0%

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

   7. pow1 46.0%

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

   8. *-commutative 46.0%

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

   9. associate-*r* 43.6%

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

   10. unpow2 43.6%

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

   11. pow-prod-down 46.0%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-/r* 97.3%

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

   3. frac-times 92.6%

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

   4. *-commutative 92.6%

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

6. Applied egg-rr 92.6%

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

   1. *-rgt-identity 92.6%

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

   2. *-rgt-identity 92.6%

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

   3. *-rgt-identity 92.6%

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

   4. div-inv 92.6%

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

   5. associate-/l* 92.6%

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

   6. associate-/r* 92.3%

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

   7. associate-*l* 96.9%

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

   8. pow2 96.9%

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

   9. pow-flip 97.2%

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

   10. metadata-eval 97.2%

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

   11. *-commutative 97.2%

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

   12. associate-*l* 98.3%

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

8. Applied egg-rr 98.3%

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

Alternative 2: 96.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_0}{x}}{c}}{s \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (* x 2.0))) (t_1 (* c (* x s))))
   (if (<= x 1.5e+91)
     (/ (/ t_0 t_1) t_1)
     (/ (/ (/ t_0 x) c) (* s (* x (* s c)))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 1.5e+91) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = ((t_0 / x) / c) / (s * (x * (s * c)));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x * 2.0d0))
    t_1 = c * (x * s)
    if (x <= 1.5d+91) then
        tmp = (t_0 / t_1) / t_1
    else
        tmp = ((t_0 / x) / c) / (s * (x * (s * c)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0));
	double t_1 = c * (x * s);
	double tmp;
	if (x <= 1.5e+91) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = ((t_0 / x) / c) / (s * (x * (s * c)));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x * 2.0))
	t_1 = c * (x * s)
	tmp = 0
	if x <= 1.5e+91:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = ((t_0 / x) / c) / (s * (x * (s * c)))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x * 2.0))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (x <= 1.5e+91)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(Float64(Float64(t_0 / x) / c) / Float64(s * Float64(x * Float64(s * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0));
	t_1 = c * (x * s);
	tmp = 0.0;
	if (x <= 1.5e+91)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = ((t_0 / x) / c) / (s * (x * (s * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.5e+91], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(t$95$0 / x), $MachinePrecision] / c), $MachinePrecision] / N[(s * N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.50000000000000003e91

   1. Initial program 62.0%

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

      1. *-un-lft-identity 62.0%

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

      2. add-sqr-sqrt 62.0%

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

      3. times-frac 62.0%

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

      4. sqrt-prod 62.0%

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

      5. sqrt-pow1 45.1%

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

      6. metadata-eval 45.1%

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

      7. pow1 45.1%

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

      8. *-commutative 45.1%

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

      9. associate-*r* 42.8%

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

      10. unpow2 42.8%

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

      11. pow-prod-down 45.1%

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

      12. sqrt-pow1 44.6%

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

      13. metadata-eval 44.6%

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

      14. pow1 44.6%

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

      15. *-commutative 44.6%

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

   4. Applied egg-rr 97.1%

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

      1. associate-*l/ 97.1%

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

      2. *-un-lft-identity 97.1%

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

      3. *-commutative 97.1%

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

   6. Applied egg-rr 97.1%

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

   if 1.50000000000000003e91 < x

   1. Initial program 60.2%

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

      1. *-un-lft-identity 60.2%

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

      2. add-sqr-sqrt 60.3%

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

      3. times-frac 60.3%

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

      4. sqrt-prod 60.2%

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

      5. sqrt-pow1 49.6%

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

      6. metadata-eval 49.6%

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

      7. pow1 49.6%

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

      8. *-commutative 49.6%

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

      9. associate-*r* 47.3%

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

      10. unpow2 47.3%

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

      11. pow-prod-down 49.7%

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

      12. sqrt-pow1 57.9%

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

      13. metadata-eval 57.9%

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

      14. pow1 57.9%

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

      15. *-commutative 57.9%

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

   4. Applied egg-rr 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/r* 97.6%

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

      3. frac-times 93.8%

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

      4. *-commutative 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. *-rgt-identity 93.8%

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

      2. *-commutative 93.8%

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

      3. *-un-lft-identity 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. div-inv 93.8%

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

      6. associate-*l* 87.9%

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

      7. times-frac 87.5%

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

      8. *-commutative 87.5%

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

      9. associate-*l* 87.6%

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

   8. Applied egg-rr 87.6%

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

      1. associate-*r/ 82.9%

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

      2. un-div-inv 83.0%

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

   10. Applied egg-rr 83.0%

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

Alternative 3: 95.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c \leq 8 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{t\_1}{x}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))) (t_1 (cos (* x 2.0))))
   (if (<= c 8e-236)
     (/ (/ t_1 x) (* (* s c) (* x (* s c))))
     (/ (/ t_1 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = cos((x * 2.0));
	double tmp;
	if (c <= 8e-236) {
		tmp = (t_1 / x) / ((s * c) * (x * (s * c)));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = cos((x * 2.0d0))
    if (c <= 8d-236) then
        tmp = (t_1 / x) / ((s * c) * (x * (s * c)))
    else
        tmp = (t_1 / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double t_1 = Math.cos((x * 2.0));
	double tmp;
	if (c <= 8e-236) {
		tmp = (t_1 / x) / ((s * c) * (x * (s * c)));
	} else {
		tmp = (t_1 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	t_1 = math.cos((x * 2.0))
	tmp = 0
	if c <= 8e-236:
		tmp = (t_1 / x) / ((s * c) * (x * (s * c)))
	else:
		tmp = (t_1 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	t_1 = cos(Float64(x * 2.0))
	tmp = 0.0
	if (c <= 8e-236)
		tmp = Float64(Float64(t_1 / x) / Float64(Float64(s * c) * Float64(x * Float64(s * c))));
	else
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	t_1 = cos((x * 2.0));
	tmp = 0.0;
	if (c <= 8e-236)
		tmp = (t_1 / x) / ((s * c) * (x * (s * c)));
	else
		tmp = (t_1 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, 8e-236], N[(N[(t$95$1 / x), $MachinePrecision] / N[(N[(s * c), $MachinePrecision] * N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 8.0000000000000004e-236

   1. Initial program 61.1%

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

      1. *-un-lft-identity 61.1%

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

      2. add-sqr-sqrt 61.1%

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

      3. times-frac 61.1%

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

      4. sqrt-prod 61.1%

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

      5. sqrt-pow1 28.9%

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

      6. metadata-eval 28.9%

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

      7. pow1 28.9%

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

      8. *-commutative 28.9%

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

      9. associate-*r* 28.0%

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

      10. unpow2 28.0%

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

      11. pow-prod-down 28.9%

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

      12. sqrt-pow1 45.3%

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

      13. metadata-eval 45.3%

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

      14. pow1 45.3%

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

      15. *-commutative 45.3%

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

   4. Applied egg-rr 95.4%

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

      1. *-commutative 95.4%

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

      2. associate-/r* 95.5%

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

      3. frac-times 91.6%

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

      4. *-commutative 91.6%

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

   6. Applied egg-rr 91.6%

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

      1. *-rgt-identity 91.6%

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

      2. *-commutative 91.6%

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

      3. *-un-lft-identity 91.6%

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

      4. *-rgt-identity 91.6%

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

      5. div-inv 91.5%

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

      6. associate-*l* 89.4%

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

      7. times-frac 90.3%

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

      8. *-commutative 90.3%

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

      9. associate-*l* 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. clear-num 92.2%

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

      2. un-div-inv 92.3%

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

      3. *-commutative 92.3%

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

      4. *-un-lft-identity 92.3%

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

      5. times-frac 94.5%

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

      6. /-rgt-identity 94.5%

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

      7. div-inv 94.4%

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

      8. clear-num 94.5%

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

      9. /-rgt-identity 94.5%

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

   10. Applied egg-rr 94.5%

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

   if 8.0000000000000004e-236 < c

   1. Initial program 62.2%

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

      1. *-un-lft-identity 62.2%

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

      2. add-sqr-sqrt 62.2%

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

      3. times-frac 62.2%

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

      4. sqrt-prod 62.2%

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

      5. sqrt-pow1 62.2%

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

      6. metadata-eval 62.2%

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

      7. pow1 62.2%

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

      8. *-commutative 62.2%

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

      9. associate-*r* 58.6%

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

      10. unpow2 58.6%

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

      11. pow-prod-down 62.3%

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

      12. sqrt-pow1 48.6%

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

      13. metadata-eval 48.6%

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

      14. pow1 48.6%

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

      15. *-commutative 48.6%

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

   4. Applied egg-rr 98.9%

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

      1. associate-*l/ 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. *-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 5.8e-10)
   (/ (/ 1.0 c) (* (* x s) (* c (* x s))))
   (/ (/ (cos (* x 2.0)) x) (* (* s c) (* x (* s c))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 5.8e-10) {
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)));
	} else {
		tmp = (cos((x * 2.0)) / x) / ((s * c) * (x * (s * c)));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 5.8d-10) then
        tmp = (1.0d0 / c) / ((x * s) * (c * (x * s)))
    else
        tmp = (cos((x * 2.0d0)) / x) / ((s * c) * (x * (s * c)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5.8e-10) {
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)));
	} else {
		tmp = (Math.cos((x * 2.0)) / x) / ((s * c) * (x * (s * c)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 5.8e-10:
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)))
	else:
		tmp = (math.cos((x * 2.0)) / x) / ((s * c) * (x * (s * c)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 5.8e-10)
		tmp = Float64(Float64(1.0 / c) / Float64(Float64(x * s) * Float64(c * Float64(x * s))));
	else
		tmp = Float64(Float64(cos(Float64(x * 2.0)) / x) / Float64(Float64(s * c) * Float64(x * Float64(s * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5.8e-10)
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)));
	else
		tmp = (cos((x * 2.0)) / x) / ((s * c) * (x * (s * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 5.8e-10], N[(N[(1.0 / c), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[(N[(s * c), $MachinePrecision] * N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.79999999999999962e-10

   1. Initial program 61.5%

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

      1. *-un-lft-identity 61.5%

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

      2. add-sqr-sqrt 61.5%

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

      3. times-frac 61.5%

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

      4. sqrt-prod 61.5%

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

      5. sqrt-pow1 44.3%

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

      6. metadata-eval 44.3%

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

      7. pow1 44.3%

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

      8. *-commutative 44.3%

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

      9. associate-*r* 41.7%

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

      10. unpow2 41.7%

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

      11. pow-prod-down 44.3%

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

      12. sqrt-pow1 44.8%

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

      13. metadata-eval 44.8%

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

      14. pow1 44.8%

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

      15. *-commutative 44.8%

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

   4. Applied egg-rr 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-/r* 97.0%

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

      3. frac-times 92.2%

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

      4. *-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in x around 0 78.3%

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

   if 5.79999999999999962e-10 < x

   1. Initial program 62.2%

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

      1. *-un-lft-identity 62.2%

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

      2. add-sqr-sqrt 62.3%

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

      3. times-frac 62.3%

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

      4. sqrt-prod 62.2%

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

      5. sqrt-pow1 51.1%

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

      6. metadata-eval 51.1%

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

      7. pow1 51.1%

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

      8. *-commutative 51.1%

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

      9. associate-*r* 49.4%

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

      10. unpow2 49.4%

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

      11. pow-prod-down 51.2%

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

      12. sqrt-pow1 53.9%

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

      13. metadata-eval 53.9%

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

      14. pow1 53.9%

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

      15. *-commutative 53.9%

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

   4. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-/r* 98.1%

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

      3. frac-times 93.8%

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

      4. *-commutative 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. *-rgt-identity 93.8%

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

      2. *-commutative 93.8%

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

      3. *-un-lft-identity 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. div-inv 93.9%

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

      6. associate-*l* 89.4%

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

      7. times-frac 89.2%

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

      8. *-commutative 89.2%

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

      9. associate-*l* 89.3%

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

   8. Applied egg-rr 89.3%

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

      1. clear-num 89.2%

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

      2. un-div-inv 89.3%

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

      3. *-commutative 89.3%

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

      4. *-un-lft-identity 89.3%

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

      5. times-frac 90.8%

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

      6. /-rgt-identity 90.8%

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

      7. div-inv 90.7%

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

      8. clear-num 90.6%

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

      9. /-rgt-identity 90.6%

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

   10. Applied egg-rr 90.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 3.5e-10)
   (/ (/ 1.0 c) (* (* x s) (* c (* x s))))
   (/ (cos (* x 2.0)) (* x (* (* s c) (* x (* s c)))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.5e-10) {
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)));
	} else {
		tmp = cos((x * 2.0)) / (x * ((s * c) * (x * (s * c))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 3.5d-10) then
        tmp = (1.0d0 / c) / ((x * s) * (c * (x * s)))
    else
        tmp = cos((x * 2.0d0)) / (x * ((s * c) * (x * (s * c))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 3.5e-10) {
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)));
	} else {
		tmp = Math.cos((x * 2.0)) / (x * ((s * c) * (x * (s * c))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 3.5e-10:
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)))
	else:
		tmp = math.cos((x * 2.0)) / (x * ((s * c) * (x * (s * c))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 3.5e-10)
		tmp = Float64(Float64(1.0 / c) / Float64(Float64(x * s) * Float64(c * Float64(x * s))));
	else
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(x * Float64(Float64(s * c) * Float64(x * Float64(s * c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 3.5e-10)
		tmp = (1.0 / c) / ((x * s) * (c * (x * s)));
	else
		tmp = cos((x * 2.0)) / (x * ((s * c) * (x * (s * c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 3.5e-10], N[(N[(1.0 / c), $MachinePrecision] / N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(N[(s * c), $MachinePrecision] * N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999998e-10

   1. Initial program 61.5%

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

      1. *-un-lft-identity 61.5%

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

      2. add-sqr-sqrt 61.5%

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

      3. times-frac 61.5%

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

      4. sqrt-prod 61.5%

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

      5. sqrt-pow1 44.3%

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

      6. metadata-eval 44.3%

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

      7. pow1 44.3%

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

      8. *-commutative 44.3%

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

      9. associate-*r* 41.7%

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

      10. unpow2 41.7%

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

      11. pow-prod-down 44.3%

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

      12. sqrt-pow1 44.8%

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

      13. metadata-eval 44.8%

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

      14. pow1 44.8%

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

      15. *-commutative 44.8%

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

   4. Applied egg-rr 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-/r* 97.0%

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

      3. frac-times 92.2%

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

      4. *-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in x around 0 78.3%

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

   if 3.4999999999999998e-10 < x

   1. Initial program 62.2%

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

      1. *-un-lft-identity 62.2%

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

      2. add-sqr-sqrt 62.3%

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

      3. times-frac 62.3%

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

      4. sqrt-prod 62.2%

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

      5. sqrt-pow1 51.1%

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

      6. metadata-eval 51.1%

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

      7. pow1 51.1%

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

      8. *-commutative 51.1%

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

      9. associate-*r* 49.4%

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

      10. unpow2 49.4%

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

      11. pow-prod-down 51.2%

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

      12. sqrt-pow1 53.9%

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

      13. metadata-eval 53.9%

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

      14. pow1 53.9%

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

      15. *-commutative 53.9%

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

   4. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-/r* 98.1%

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

      3. frac-times 93.8%

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

      4. *-commutative 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. *-rgt-identity 93.8%

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

      2. *-commutative 93.8%

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

      3. *-un-lft-identity 93.8%

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

      4. *-rgt-identity 93.8%

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

      5. div-inv 93.9%

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

      6. associate-*l* 89.4%

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

      7. times-frac 89.2%

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

      8. *-commutative 89.2%

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

      9. associate-*l* 89.3%

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

   8. Applied egg-rr 89.3%

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

      1. *-commutative 89.3%

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

      2. clear-num 89.2%

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

      3. frac-times 88.8%

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

      4. *-un-lft-identity 88.8%

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

      5. *-commutative 88.8%

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

      6. *-un-lft-identity 88.8%

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

      7. times-frac 90.3%

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

      8. /-rgt-identity 90.3%

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

      9. div-inv 90.2%

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

      10. clear-num 90.1%

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

      11. /-rgt-identity 90.1%

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

   10. Applied egg-rr 90.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c)))) (* (/ (cos (* x 2.0)) t_0) (/ 1.0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	return (cos((x * 2.0)) / t_0) * (1.0 / t_0);
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = x * (s * c)
    code = (cos((x * 2.0d0)) / t_0) * (1.0d0 / t_0)
end function

Java

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

Python

def code(x, c, s):
	t_0 = x * (s * c)
	return (math.cos((x * 2.0)) / t_0) * (1.0 / t_0)

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	return Float64(Float64(cos(Float64(x * 2.0)) / t_0) * Float64(1.0 / t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (s * c);
	tmp = (cos((x * 2.0)) / t_0) * (1.0 / t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

   1. *-un-lft-identity 61.7%

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

   2. add-sqr-sqrt 61.7%

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

   3. times-frac 61.7%

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

   4. sqrt-prod 61.7%

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

   5. sqrt-pow1 46.0%

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

   6. metadata-eval 46.0%

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

   7. pow1 46.0%

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

   8. *-commutative 46.0%

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

   9. associate-*r* 43.6%

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

   10. unpow2 43.6%

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

   11. pow-prod-down 46.0%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 97.2%

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

   1. *-commutative 97.2%

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

   2. associate-/r* 97.3%

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

   3. frac-times 92.6%

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

   4. *-commutative 92.6%

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

6. Applied egg-rr 92.6%

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

   1. *-rgt-identity 92.6%

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

   2. frac-times 97.3%

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

   3. *-rgt-identity 97.3%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 96.0%

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

   6. *-commutative 96.0%

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

   7. associate-*l* 98.3%

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

8. Applied egg-rr 98.3%

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

Alternative 7: 78.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

   12. *-commutative 76.5%

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

5. Simplified 76.5%

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

   1. pow-flip 76.7%

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

   2. *-commutative 76.7%

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

   3. metadata-eval 76.7%

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

7. Applied egg-rr 76.7%

   \[\leadsto \color{blue}{{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}} \]
8. Add Preprocessing

Alternative 8: 78.4% accurate, 20.9× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* c (* x s))))) (* t_0 t_0)))

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = 1.0d0 / (c * (x * s))
    code = t_0 * t_0
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = 1.0 / (c * (x * s));
	return t_0 * t_0;
}

Python

def code(x, c, s):
	t_0 = 1.0 / (c * (x * s))
	return t_0 * t_0

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = 1.0 / (c * (x * s));
	tmp = t_0 * t_0;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

   12. *-commutative 76.5%

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

5. Simplified 76.5%

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

   1. *-commutative 76.5%

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

   2. pow-flip 76.7%

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

   3. metadata-eval 76.7%

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

   4. metadata-eval 76.7%

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

   5. pow-prod-up 76.7%

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

   6. inv-pow 76.7%

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

   7. inv-pow 76.7%

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

7. Applied egg-rr 76.7%

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

Alternative 9: 78.3% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in x around 0 51.3%

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

   1. associate-/r* 50.6%

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

   2. *-commutative 50.6%

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

   3. unpow2 50.6%

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

   4. unpow2 50.6%

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

   5. swap-sqr 62.7%

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

   6. unpow2 62.7%

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

   7. associate-/r* 63.4%

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

   8. unpow2 63.4%

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

   9. unpow2 63.4%

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

   10. swap-sqr 76.5%

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

   11. unpow2 76.5%

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

   12. *-commutative 76.5%

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

5. Simplified 76.5%

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

   1. *-commutative 76.5%

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

   2. pow2 76.5%

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

7. Applied egg-rr 76.5%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))