mixedcos

Percentage Accurate: 66.8% → 97.4%

Time: 8.3s

Alternatives: 7

Speedup: 9.0×

• 31.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow2 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 98.2

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

   7. lift-*.f64 N/A

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

   8. count-2 N/A

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

   9. lift-+.f64 98.2

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 96.8

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-*l* N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 N/A

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

   24. *-commutative N/A

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

   25. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

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

Alternative 2: 84.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 1.3e-40)
   (/ 1.0 (* c (* (* x s) (* c (* x s)))))
   (/ (cos (+ x x)) (* (* x (* s (* x c))) (* s c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.3e-40) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = cos((x + x)) / ((x * (s * (x * c))) * (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 <= 1.3d-40) then
        tmp = 1.0d0 / (c * ((x * s) * (c * (x * s))))
    else
        tmp = cos((x + x)) / ((x * (s * (x * c))) * (s * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 1.3e-40) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = Math.cos((x + x)) / ((x * (s * (x * c))) * (s * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 1.3e-40:
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))))
	else:
		tmp = math.cos((x + x)) / ((x * (s * (x * c))) * (s * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 1.3e-40)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * Float64(c * Float64(x * s)))));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(x * Float64(s * Float64(x * c))) * Float64(s * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 1.3e-40)
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	else
		tmp = cos((x + x)) / ((x * (s * (x * c))) * (s * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 1.3e-40], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(x * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3000000000000001e-40

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 75.8

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

   5. Applied rewrites 75.8%

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

   7. Applied rewrites 83.9%

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

   9. Applied rewrites 84.0%

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

   if 1.3000000000000001e-40 < x

   1. Initial program 70.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-prod-down N/A

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

      10. pow2 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-*.f64 N/A

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

      2. count-2 N/A

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

      3. lift-+.f64 98.1

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 98.2

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 4e-17)
   (/ 1.0 (* c (* (* x s) (* c (* x s)))))
   (/ (cos (+ x x)) (* x (* x (* s (* c (* s c))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 4e-17) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = cos((x + x)) / (x * (x * (s * (c * (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 <= 4d-17) then
        tmp = 1.0d0 / (c * ((x * s) * (c * (x * s))))
    else
        tmp = cos((x + x)) / (x * (x * (s * (c * (s * c)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 4e-17) {
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	} else {
		tmp = Math.cos((x + x)) / (x * (x * (s * (c * (s * c)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 4e-17:
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))))
	else:
		tmp = math.cos((x + x)) / (x * (x * (s * (c * (s * c)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 4e-17)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s) * Float64(c * Float64(x * s)))));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(x * Float64(x * Float64(s * Float64(c * Float64(s * c))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 4e-17)
		tmp = 1.0 / (c * ((x * s) * (c * (x * s))));
	else
		tmp = cos((x + x)) / (x * (x * (s * (c * (s * c)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 4e-17], N[(1.0 / N[(c * N[(N[(x * s), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(s * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000029e-17

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   7. Applied rewrites 83.7%

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

   9. Applied rewrites 83.8%

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

   if 4.00000000000000029e-17 < x

   1. Initial program 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-prod-down N/A

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

      10. pow2 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 98.1

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

   4. Applied rewrites 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 98.2

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 98.2

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.2

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-*.f64 96.5

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

   6. Applied rewrites 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 96.5

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

      9. lift-*.f64 N/A

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

      10. count-2 N/A

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

      11. lift-+.f64 96.5

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. unpow-prod-down N/A

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

      20. pow2 N/A

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

      21. pow2 N/A

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

   8. Applied rewrites 90.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow-prod-down N/A

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

   10. pow2 N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.1

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 96.7

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lift-+.f64 98.1

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

8. Applied rewrites 98.1%

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

9. Final simplification 98.1%

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

Alternative 5: 78.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	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 = x * (s * c)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 80.4%

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

9. Applied rewrites 83.9%

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

Alternative 6: 76.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return 1.0 / (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 = 1.0d0 / (c * ((x * s) * (c * (x * s))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 80.4%

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

9. Applied rewrites 80.6%

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

10. Final simplification 80.6%

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

Alternative 7: 75.8% accurate, 9.0× speedup

Math

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

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(c * Float64(Float64(s * Float64(x * 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[(c * N[(N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 74.7

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

5. Applied rewrites 74.7%

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

7. Applied rewrites 80.4%

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

8. Final simplification 80.4%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))