mixedcos

Percentage Accurate: 67.4% → 97.4%

Time: 16.6s

Alternatives: 9

Speedup: 24.1×

• 31.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	return (cos((x * 2.0)) / 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 = x * (c * s)
    code = (cos((x * 2.0d0)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. *-un-lft-identity 68.0%

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

   2. associate-*r* 70.6%

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

   3. times-frac 70.6%

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

   4. *-commutative 70.6%

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

   5. associate-*r* 68.6%

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

   6. pow-prod-down 84.6%

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

4. Applied egg-rr 84.6%

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

   1. unpow2 84.6%

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

   2. *-commutative 84.6%

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

   3. associate-*r* 83.1%

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

6. Applied egg-rr 83.1%

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

   1. *-commutative 83.1%

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

   2. frac-times 83.1%

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

   3. *-un-lft-identity 83.1%

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

   4. associate-*l* 85.3%

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

   5. associate-*l* 90.2%

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

   6. associate-*l* 91.6%

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

   7. *-commutative 91.6%

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

   8. associate-*r* 88.8%

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

   9. *-commutative 88.8%

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

   10. associate-*r* 91.2%

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

   11. *-commutative 91.2%

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

   12. associate-*r* 94.6%

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

   13. associate-*r* 96.3%

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

   14. *-commutative 96.3%

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

   15. associate-/l/ 96.7%

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

8. Applied egg-rr 97.6%

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

9. Final simplification 97.6%

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

Alternative 2: 94.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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)) / c) / ((x * s) * (c * (x * s)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. *-un-lft-identity 68.0%

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

   2. add-sqr-sqrt 68.0%

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

   3. times-frac 68.0%

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

   4. sqrt-prod 68.0%

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

   5. sqrt-pow1 51.0%

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

   6. metadata-eval 51.0%

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

   7. pow1 51.0%

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

   8. *-commutative 51.0%

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

   9. associate-*r* 46.4%

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

   10. unpow2 46.4%

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

   11. pow-prod-down 51.0%

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

   12. sqrt-pow1 49.1%

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

   13. metadata-eval 49.1%

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

   14. pow1 49.1%

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

   15. *-commutative 49.1%

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

4. Applied egg-rr 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-/r* 96.8%

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

   3. frac-times 94.2%

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

   4. div-inv 94.2%

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

   5. *-commutative 94.2%

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

6. Applied egg-rr 94.2%

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

7. Final simplification 94.2%

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

Alternative 3: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (cos((x * 2.0)) / 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 = (cos((x * 2.0d0)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. *-un-lft-identity 68.0%

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

   2. add-sqr-sqrt 68.0%

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

   3. times-frac 68.0%

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

   4. sqrt-prod 68.0%

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

   5. sqrt-pow1 51.0%

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

   6. metadata-eval 51.0%

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

   7. pow1 51.0%

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

   8. *-commutative 51.0%

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

   9. associate-*r* 46.4%

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

   10. unpow2 46.4%

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

   11. pow-prod-down 51.0%

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

   12. sqrt-pow1 49.1%

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

   13. metadata-eval 49.1%

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

   14. pow1 49.1%

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

   15. *-commutative 49.1%

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

4. Applied egg-rr 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*l/ 96.7%

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

   3. div-inv 96.7%

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

   4. *-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

7. Final simplification 96.7%

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

Alternative 4: 76.5% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;s \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* x s))))
   (if (<= s 2e+165)
     (/ 1.0 (* (* x c) (* s t_0)))
     (/ 1.0 (* (* c s) (* x t_0))))))

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	double tmp;
	if (s <= 2e+165) {
		tmp = 1.0 / ((x * c) * (s * t_0));
	} else {
		tmp = 1.0 / ((c * s) * (x * 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) :: tmp
    t_0 = c * (x * s)
    if (s <= 2d+165) then
        tmp = 1.0d0 / ((x * c) * (s * t_0))
    else
        tmp = 1.0d0 / ((c * s) * (x * 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 tmp;
	if (s <= 2e+165) {
		tmp = 1.0 / ((x * c) * (s * t_0));
	} else {
		tmp = 1.0 / ((c * s) * (x * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = c * (x * s)
	tmp = 0
	if s <= 2e+165:
		tmp = 1.0 / ((x * c) * (s * t_0))
	else:
		tmp = 1.0 / ((c * s) * (x * t_0))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (s <= 2e+165)
		tmp = Float64(1.0 / Float64(Float64(x * c) * Float64(s * t_0)));
	else
		tmp = Float64(1.0 / Float64(Float64(c * s) * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = c * (x * s);
	tmp = 0.0;
	if (s <= 2e+165)
		tmp = 1.0 / ((x * c) * (s * t_0));
	else
		tmp = 1.0 / ((c * s) * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 2e+165], N[(1.0 / N[(N[(x * c), $MachinePrecision] * N[(s * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.9999999999999998e165

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0 51.1%

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

      1. associate-/r* 51.1%

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

      2. *-commutative 51.1%

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

      3. unpow2 51.1%

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

      4. unpow2 51.1%

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

      5. swap-sqr 62.8%

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

      6. unpow2 62.8%

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

      7. associate-/r* 62.8%

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

      8. unpow2 62.8%

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

      9. unpow2 62.8%

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

      10. swap-sqr 73.4%

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

      11. unpow2 73.4%

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

      12. *-commutative 73.4%

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

   5. Simplified 73.4%

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

      1. unpow-prod-down 62.8%

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

      2. *-commutative 62.8%

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

      3. unpow-prod-down 73.4%

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

      4. unpow2 73.4%

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

      5. associate-*r* 73.3%

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

      6. associate-*l* 72.4%

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

   7. Applied egg-rr 72.4%

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

   if 1.9999999999999998e165 < s

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0 59.3%

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

      1. associate-/r* 59.3%

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

      2. *-commutative 59.3%

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

      3. unpow2 59.3%

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

      4. unpow2 59.3%

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

      5. swap-sqr 75.4%

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

      6. unpow2 75.4%

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

      7. associate-/r* 74.3%

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

      8. unpow2 74.3%

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

      9. unpow2 74.3%

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

      10. swap-sqr 93.7%

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

      11. unpow2 93.7%

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

      12. *-commutative 93.7%

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

   5. Simplified 93.7%

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

      1. unpow2 93.7%

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

      2. associate-*r* 93.8%

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

      3. *-commutative 93.8%

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

      4. associate-*l* 90.9%

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

   7. Applied egg-rr 90.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2 \cdot 10^{+165}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{\frac{1}{x}}{s}}{c} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(1.0 / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] / s), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

   1. *-un-lft-identity 68.0%

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

   2. add-sqr-sqrt 68.0%

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

   3. times-frac 68.0%

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

   4. sqrt-prod 68.0%

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

   5. sqrt-pow1 51.0%

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

   6. metadata-eval 51.0%

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

   7. pow1 51.0%

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

   8. *-commutative 51.0%

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

   9. associate-*r* 46.4%

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

   10. unpow2 46.4%

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

   11. pow-prod-down 51.0%

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

   12. sqrt-pow1 49.1%

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

   13. metadata-eval 49.1%

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

   14. pow1 49.1%

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

   15. *-commutative 49.1%

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

4. Applied egg-rr 96.7%

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

5. Taylor expanded in x around 0 76.2%

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

   1. associate-/r* 76.3%

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

   2. *-commutative 76.3%

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

   3. *-rgt-identity 76.3%

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

   4. associate-*r/ 76.2%

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

   5. associate-*l/ 76.2%

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

   6. *-lft-identity 76.2%

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

   7. associate-/r* 76.2%

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

7. Simplified 76.2%

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

8. Final simplification 76.2%

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

Alternative 6: 78.8% accurate, 20.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{x \cdot s}}{c} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(x * s), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

3. Taylor expanded in x around 0 52.1%

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

   1. associate-/r* 52.1%

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

   2. *-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. swap-sqr 64.3%

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

   6. unpow2 64.3%

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

   7. associate-/r* 64.2%

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

   8. unpow2 64.2%

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

   9. unpow2 64.2%

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

   10. swap-sqr 75.8%

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

   11. unpow2 75.8%

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

   12. *-commutative 75.8%

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

5. Simplified 75.8%

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

   1. unpow-prod-down 64.2%

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

   2. *-commutative 64.2%

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

   3. unpow-prod-down 75.8%

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

   4. unpow2 75.8%

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

   5. associate-*r* 75.4%

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

   6. associate-*l* 74.3%

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

7. Applied egg-rr 74.3%

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

   1. metadata-eval 74.3%

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

   2. associate-*r* 75.4%

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

   3. *-commutative 75.4%

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

   4. associate-*r* 74.7%

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

   5. associate-*r* 75.9%

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

   6. *-commutative 75.9%

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

   7. associate-*r* 76.4%

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

   8. frac-times 76.5%

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

   9. *-commutative 76.5%

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

   10. associate-*r* 74.9%

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

   11. *-commutative 74.9%

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

   12. associate-/l/ 75.0%

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

   13. div-inv 75.0%

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

   14. div-inv 75.0%

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

   15. *-commutative 75.0%

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

   16. *-commutative 75.0%

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

   17. associate-*r* 76.2%

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

   18. times-frac 76.3%

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

9. Applied egg-rr 76.3%

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

10. Final simplification 76.3%

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

Alternative 7: 75.7% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

3. Taylor expanded in x around 0 52.1%

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

   1. associate-/r* 52.1%

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

   2. *-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. swap-sqr 64.3%

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

   6. unpow2 64.3%

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

   7. associate-/r* 64.2%

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

   8. unpow2 64.2%

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

   9. unpow2 64.2%

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

   10. swap-sqr 75.8%

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

   11. unpow2 75.8%

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

   12. *-commutative 75.8%

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

5. Simplified 75.8%

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

   1. unpow2 75.8%

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

   2. associate-*r* 74.7%

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

   3. *-commutative 74.7%

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

   4. associate-*l* 72.1%

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

7. Applied egg-rr 72.1%

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

8. Final simplification 72.1%

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

Alternative 8: 78.9% 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 68.0%

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

3. Taylor expanded in x around 0 52.1%

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

   1. associate-/r* 52.1%

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

   2. *-commutative 52.1%

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

   3. unpow2 52.1%

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

   4. unpow2 52.1%

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

   5. swap-sqr 64.3%

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

   6. unpow2 64.3%

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

   7. associate-/r* 64.2%

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

   8. unpow2 64.2%

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

   9. unpow2 64.2%

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

   10. swap-sqr 75.8%

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

   11. unpow2 75.8%

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

   12. *-commutative 75.8%

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

5. Simplified 75.8%

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

   1. unpow-prod-down 64.2%

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

   2. *-commutative 64.2%

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

   3. unpow-prod-down 75.8%

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

   4. unpow2 75.8%

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

7. Applied egg-rr 75.8%

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

8. Final simplification 75.8%

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

Alternative 9: 79.0% accurate, 24.1× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 68.0%

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

   1. *-un-lft-identity 68.0%

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

   2. add-sqr-sqrt 68.0%

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

   3. times-frac 68.0%

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

   4. sqrt-prod 68.0%

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

   5. sqrt-pow1 51.0%

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

   6. metadata-eval 51.0%

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

   7. pow1 51.0%

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

   8. *-commutative 51.0%

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

   9. associate-*r* 46.4%

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

   10. unpow2 46.4%

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

   11. pow-prod-down 51.0%

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

   12. sqrt-pow1 49.1%

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

   13. metadata-eval 49.1%

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

   14. pow1 49.1%

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

   15. *-commutative 49.1%

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

4. Applied egg-rr 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-*l/ 96.7%

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

   3. div-inv 96.7%

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

   4. *-commutative 96.7%

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

6. Applied egg-rr 96.7%

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

7. Taylor expanded in x around 0 76.2%

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

8. Final simplification 76.2%

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

Reproduce

herbie shell --seed 2024059 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))