mixedcos

Percentage Accurate: 66.7% → 97.2%

Time: 10.1s

Alternatives: 17

Speedup: 9.0×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% 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: 97.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 66.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. *-commutative N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-down N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s \cdot c\right) \cdot x\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -5e-121)
   (/ (fma -2.0 (* x x) 1.0) (* (* s x) (* (* (* s c) c) x)))
   (/ 1.0 (pow (* (* s c) x) 2.0))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -5e-121) {
		tmp = fma(-2.0, (x * x), 1.0) / ((s * x) * (((s * c) * c) * x));
	} else {
		tmp = 1.0 / pow(((s * c) * x), 2.0);
	}
	return tmp;
}

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -5e-121)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(s * x) * Float64(Float64(Float64(s * c) * c) * x)));
	else
		tmp = Float64(1.0 / (Float64(Float64(s * c) * x) ^ 2.0));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-121], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * N[(N[(N[(s * c), $MachinePrecision] * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999989e-121

   1. Initial program 52.5%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 24.7

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

   9. Applied rewrites 24.7%

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

   if -4.99999999999999989e-121 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s \cdot c\right) \cdot x\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot x\right) \cdot s} \cdot \frac{\frac{1}{s}}{c \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -5e-121)
   (/ (fma -2.0 (* x x) 1.0) (* (* s x) (* (* (* s c) c) x)))
   (* (/ 1.0 (* (* c x) s)) (/ (/ 1.0 s) (* c x)))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -5e-121) {
		tmp = fma(-2.0, (x * x), 1.0) / ((s * x) * (((s * c) * c) * x));
	} else {
		tmp = (1.0 / ((c * x) * s)) * ((1.0 / s) / (c * x));
	}
	return tmp;
}

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -5e-121)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(s * x) * Float64(Float64(Float64(s * c) * c) * x)));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(c * x) * s)) * Float64(Float64(1.0 / s) / Float64(c * x)));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-121], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * N[(N[(N[(s * c), $MachinePrecision] * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / s), $MachinePrecision] / N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999989e-121

   1. Initial program 52.5%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 24.7

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

   9. Applied rewrites 24.7%

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

   if -4.99999999999999989e-121 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 98.4

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 96.5

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

   11. Applied rewrites 85.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot x\right) \cdot s} \cdot \frac{\frac{1}{s}}{c \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c x) s)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -5e-121)
     (/ (fma -2.0 (* x x) 1.0) (* (* s x) (* (* (* s c) c) x)))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -5e-121) {
		tmp = fma(-2.0, (x * x), 1.0) / ((s * x) * (((s * c) * c) * x));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * x) * s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -5e-121)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(s * x) * Float64(Float64(Float64(s * c) * c) * x)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-121], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * N[(N[(N[(s * c), $MachinePrecision] * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999989e-121

   1. Initial program 52.5%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 24.7

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

   9. Applied rewrites 24.7%

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

   if -4.99999999999999989e-121 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 85.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(s \cdot x\right) \cdot \left(\left(\left(s \cdot c\right) \cdot c\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c x) s)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -5e-121)
     (/ (fma -2.0 (* x x) 1.0) (* (* (* (* c c) x) s) (* s x)))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -5e-121) {
		tmp = fma(-2.0, (x * x), 1.0) / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * x) * s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -5e-121)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(Float64(c * c) * x) * s) * Float64(s * x)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-121], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999989e-121

   1. Initial program 52.5%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   if -4.99999999999999989e-121 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 85.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{-2 \cdot \left(x \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c x) s)))
   (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) -5e-121)
     (/ (* -2.0 (* x x)) (* (* (* (* c c) x) s) (* s x)))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= -5e-121) {
		tmp = (-2.0 * (x * x)) / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * x) * s
    if ((cos((2.0d0 * x)) / ((((s ** 2.0d0) * x) * x) * (c ** 2.0d0))) <= (-5d-121)) then
        tmp = ((-2.0d0) * (x * x)) / ((((c * c) * x) * s) * (s * x))
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * x) * s;
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c, 2.0))) <= -5e-121) {
		tmp = (-2.0 * (x * x)) / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * x) * s
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s, 2.0) * x) * x) * math.pow(c, 2.0))) <= -5e-121:
		tmp = (-2.0 * (x * x)) / ((((c * c) * x) * s) * (s * x))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * x) * s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -5e-121)
		tmp = Float64(Float64(-2.0 * Float64(x * x)) / Float64(Float64(Float64(Float64(c * c) * x) * s) * Float64(s * x)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * x) * s;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= -5e-121)
		tmp = (-2.0 * (x * x)) / ((((c * c) * x) * s) * (s * x));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[Power[s, 2.0], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-121], N[(N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -4.99999999999999989e-121

   1. Initial program 52.5%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 24.6%

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

   if -4.99999999999999989e-121 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. associate-*r* N/A

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

      2. associate-/l/ N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 73.9

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

   5. Applied rewrites 73.9%

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

   7. Applied rewrites 78.5%

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

   9. Applied rewrites 85.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot x\right) \cdot x\right) \cdot {c}^{2}} \leq -5 \cdot 10^{-121}:\\ \;\;\;\;\frac{-2 \cdot \left(x \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (* (* (pow s 2.0) x) x) (pow c 2.0))) INFINITY)
   (/ 1.0 (* (* (* (* (* c c) x) x) s) s))
   (/ 1.0 (* (* (* (* (* x x) c) c) s) s))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (((pow(s, 2.0) * x) * x) * pow(c, 2.0))) <= ((double) INFINITY)) {
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s);
	} else {
		tmp = 1.0 / (((((x * x) * c) * c) * s) * s);
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (((Math.pow(s, 2.0) * x) * x) * Math.pow(c, 2.0))) <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s);
	} else {
		tmp = 1.0 / (((((x * x) * c) * c) * s) * s);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (((math.pow(s, 2.0) * x) * x) * math.pow(c, 2.0))) <= math.inf:
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s)
	else:
		tmp = 1.0 / (((((x * x) * c) * c) * s) * s)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64(Float64(Float64((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= Inf)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c * c) * x) * x) * s) * s));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(x * x) * c) * c) * s) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((((s ^ 2.0) * x) * x) * (c ^ 2.0))) <= Inf)
		tmp = 1.0 / (((((c * c) * x) * x) * s) * s);
	else
		tmp = 1.0 / (((((x * x) * c) * c) * s) * s);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 81.0%

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

      1. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 85.7

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

   4. Applied rewrites 85.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 73.9

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

   10. Applied rewrites 73.9%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

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

      1. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 21.5

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

   10. Applied rewrites 21.5%

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

   12. Applied rewrites 56.0%

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

4. Final simplification 70.6%

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

Alternative 8: 83.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 1.75 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{\left({\left(s \cdot x\right)}^{2} \cdot c\right) \cdot c}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{t\_0}{\left(\left(x \cdot x\right) \cdot \left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)\right) \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= x 1.75e-131)
     (/ 1.0 (* (* (pow (* s x) 2.0) c) c))
     (if (<= x 3.3e+153)
       (/ t_0 (* (* (* x x) (* s c)) (* s c)))
       (/ t_0 (* (* (* s x) (* (* c x) s)) c))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double tmp;
	if (x <= 1.75e-131) {
		tmp = 1.0 / ((pow((s * x), 2.0) * c) * c);
	} else if (x <= 3.3e+153) {
		tmp = t_0 / (((x * x) * (s * c)) * (s * c));
	} else {
		tmp = t_0 / (((s * x) * ((c * x) * s)) * c);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (x <= 1.75d-131) then
        tmp = 1.0d0 / ((((s * x) ** 2.0d0) * c) * c)
    else if (x <= 3.3d+153) then
        tmp = t_0 / (((x * x) * (s * c)) * (s * c))
    else
        tmp = t_0 / (((s * x) * ((c * x) * s)) * c)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (x <= 1.75e-131) {
		tmp = 1.0 / ((Math.pow((s * x), 2.0) * c) * c);
	} else if (x <= 3.3e+153) {
		tmp = t_0 / (((x * x) * (s * c)) * (s * c));
	} else {
		tmp = t_0 / (((s * x) * ((c * x) * s)) * c);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x + x))
	tmp = 0
	if x <= 1.75e-131:
		tmp = 1.0 / ((math.pow((s * x), 2.0) * c) * c)
	elif x <= 3.3e+153:
		tmp = t_0 / (((x * x) * (s * c)) * (s * c))
	else:
		tmp = t_0 / (((s * x) * ((c * x) * s)) * c)
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 1.75e-131)
		tmp = Float64(1.0 / Float64(Float64((Float64(s * x) ^ 2.0) * c) * c));
	elseif (x <= 3.3e+153)
		tmp = Float64(t_0 / Float64(Float64(Float64(x * x) * Float64(s * c)) * Float64(s * c)));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(s * x) * Float64(Float64(c * x) * s)) * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (x <= 1.75e-131)
		tmp = 1.0 / ((((s * x) ^ 2.0) * c) * c);
	elseif (x <= 3.3e+153)
		tmp = t_0 / (((x * x) * (s * c)) * (s * c));
	else
		tmp = t_0 / (((s * x) * ((c * x) * s)) * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.75e-131], N[(1.0 / N[(N[(N[Power[N[(s * x), $MachinePrecision], 2.0], $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e+153], N[(t$95$0 / N[(N[(N[(x * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(s * x), $MachinePrecision] * N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.7500000000000001e-131

   1. Initial program 62.5%

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

      1. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 76.2%

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

   if 1.7500000000000001e-131 < x < 3.29999999999999994e153

   1. Initial program 72.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.5

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.9

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

   6. Applied rewrites 96.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 96.9

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

   8. Applied rewrites 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. unswap-sqr N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 3.29999999999999994e153 < x

   1. Initial program 69.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 95.0

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 95.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 94.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.8

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

   10. Applied rewrites 94.8%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{\left({\left(s \cdot x\right)}^{2} \cdot c\right) \cdot c}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+153}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(x \cdot x\right) \cdot \left(s \cdot c\right)\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 28:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{{\left(\left(s \cdot c\right) \cdot x\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)\right) \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 28.0)
   (/ (fma -2.0 (* x x) 1.0) (pow (* (* s c) x) 2.0))
   (/ (cos (+ x x)) (* (* (* s x) (* (* c x) s)) c))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 28.0) {
		tmp = fma(-2.0, (x * x), 1.0) / pow(((s * c) * x), 2.0);
	} else {
		tmp = cos((x + x)) / (((s * x) * ((c * x) * s)) * c);
	}
	return tmp;
}

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 28.0)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / (Float64(Float64(s * c) * x) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(s * x) * Float64(Float64(c * x) * s)) * c));
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 28.0], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Power[N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s * x), $MachinePrecision] * N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 28

   1. Initial program 63.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.8

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.3

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

   7. Applied rewrites 71.3%

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

   if 28 < x

   1. Initial program 72.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow-prod-down N/A

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

      16. pow-prod-down N/A

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

      17. lower-pow.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 95.9

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2 N/A

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

      4. lower-+.f64 98.3

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

   8. Applied rewrites 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.3

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.3

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

   10. Applied rewrites 92.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 28:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{{\left(\left(s \cdot c\right) \cdot x\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)\right) \cdot c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 4 \cdot 10^{-322}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (pow c 2.0) 4e-322)
   (/ 1.0 (* (* (* (* s c) (* s c)) x) x))
   (/ 1.0 (* (* (* c c) (* s x)) (* s x)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (pow(c, 2.0) <= 4e-322) {
		tmp = 1.0 / ((((s * c) * (s * c)) * x) * x);
	} else {
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x));
	}
	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 ((c ** 2.0d0) <= 4d-322) then
        tmp = 1.0d0 / ((((s * c) * (s * c)) * x) * x)
    else
        tmp = 1.0d0 / (((c * c) * (s * x)) * (s * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (Math.pow(c, 2.0) <= 4e-322) {
		tmp = 1.0 / ((((s * c) * (s * c)) * x) * x);
	} else {
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if math.pow(c, 2.0) <= 4e-322:
		tmp = 1.0 / ((((s * c) * (s * c)) * x) * x)
	else:
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if ((c ^ 2.0) <= 4e-322)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * c) * Float64(s * c)) * x) * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(c * c) * Float64(s * x)) * Float64(s * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((c ^ 2.0) <= 4e-322)
		tmp = 1.0 / ((((s * c) * (s * c)) * x) * x);
	else
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[Power[c, 2.0], $MachinePrecision], 4e-322], N[(1.0 / N[(N[(N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(c * c), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 4.00193e-322

   1. Initial program 55.0%

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

      1. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 77.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.7

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

   9. Applied rewrites 77.7%

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

   if 4.00193e-322 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 69.5%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 83.5

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

   4. Applied rewrites 83.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 68.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.9

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

   9. Applied rewrites 68.9%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 4 \cdot 10^{-322}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 66.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. *-commutative N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-down N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 98.1

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 96.4

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 97.8

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

6. Applied rewrites 97.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2 N/A

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

   4. lower-+.f64 97.8

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

8. Applied rewrites 97.8%

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

Alternative 12: 71.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 1.05e-160)
   (/ 1.0 (* (* (* s x) (* (* s c) c)) x))
   (/ 1.0 (* (* (* c c) (* s x)) (* s x)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 1.05e-160) {
		tmp = 1.0 / (((s * x) * ((s * c) * c)) * x);
	} else {
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x));
	}
	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 (c <= 1.05d-160) then
        tmp = 1.0d0 / (((s * x) * ((s * c) * c)) * x)
    else
        tmp = 1.0d0 / (((c * c) * (s * x)) * (s * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 1.05e-160) {
		tmp = 1.0 / (((s * x) * ((s * c) * c)) * x);
	} else {
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 1.05e-160:
		tmp = 1.0 / (((s * x) * ((s * c) * c)) * x)
	else:
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 1.05e-160)
		tmp = Float64(1.0 / Float64(Float64(Float64(s * x) * Float64(Float64(s * c) * c)) * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(c * c) * Float64(s * x)) * Float64(s * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 1.05e-160)
		tmp = 1.0 / (((s * x) * ((s * c) * c)) * x);
	else
		tmp = 1.0 / (((c * c) * (s * x)) * (s * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 1.05e-160], N[(1.0 / N[(N[(N[(s * x), $MachinePrecision] * N[(N[(s * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(c * c), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.05e-160

   1. Initial program 63.2%

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

      1. 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. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*r* N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 74.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 74.0

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

   9. Applied rewrites 74.0%

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

   if 1.05e-160 < c

   1. Initial program 71.0%

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

      1. 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. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 84.1

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 69.4

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot s\right) \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot s\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(x \cdot s\right)} \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)} \]

      10. lower-*.f64 69.4

          \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot c\right)\right) \cdot \left(x \cdot s\right)} \]

   9. Applied rewrites 69.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot x\right) \cdot \left(c \cdot c\right)\right)} \cdot \left(x \cdot s\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot c\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right) \cdot \left(s \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

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(Float64(c * 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[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. 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. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

   2. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{1}{{x}^{2}}}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{{x}^{2}}}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{2}}}{c}}{c \cdot {s}^{2}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{{x}^{2}}}{c}}{c \cdot {s}^{2}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{{x}^{2}}}{c}}}{c \cdot {s}^{2}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{x \cdot x}}}{c}}{c \cdot {s}^{2}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{x}}}{c}}{c \cdot {s}^{2}} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{x}}{x}}}{c}}{c \cdot {s}^{2}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{x}}}{x}}{c}}{c \cdot {s}^{2}} \]

   12. unpow2 N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{c \cdot \color{blue}{\left(s \cdot s\right)}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(c \cdot s\right) \cdot s}} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right) \cdot s}} \]

   16. lower-*.f64 68.2

       \[\leadsto \frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\color{blue}{\left(s \cdot c\right)} \cdot s} \]

5. Applied rewrites 68.2%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{x}}{x}}{c}}{\left(s \cdot c\right) \cdot s}} \]
6. Step-by-step derivation

7. Applied rewrites 72.4%

   \[\leadsto \frac{\frac{-1}{x} \cdot \frac{-1}{\left(s \cdot c\right) \cdot s}}{\color{blue}{c \cdot x}} \]
8. Step-by-step derivation

9. Applied rewrites 78.9%

   \[\leadsto \frac{\frac{1}{s \cdot \left(c \cdot x\right)}}{\color{blue}{s \cdot \left(c \cdot x\right)}} \]

10. Final simplification 78.9%

    \[\leadsto \frac{\frac{1}{\left(c \cdot x\right) \cdot s}}{\left(c \cdot x\right) \cdot s} \]
11. Add Preprocessing

Alternative 14: 78.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

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(Float64(c * 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[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. lower-*.f64 66.2

      \[\leadsto \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right) \cdot {c}^{2}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \cdot {c}^{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \cdot {c}^{2}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot {c}^{2}} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)} \cdot {c}^{2}} \]

   10. associate-*l* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({s}^{2} \cdot {c}^{2}\right)}} \]

   11. pow2 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{x}^{2}} \cdot \left({s}^{2} \cdot {c}^{2}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{\left({c}^{2} \cdot {s}^{2}\right)}} \]

   13. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right)} \]

   14. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \left({c}^{2} \cdot \color{blue}{{s}^{2}}\right)} \]

   15. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{x}^{2} \cdot \color{blue}{{\left(c \cdot s\right)}^{2}}} \]

   16. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}}^{2}} \]

   19. lower-*.f64 98.1

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)}^{2}} \]

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{\cos \left(x \cdot 2\right)}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. lower-*.f64 98.1

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot s\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right)} \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   9. lower-*.f64 96.4

      \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(c \cdot s\right)\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}} \]

   15. lower-*.f64 97.8

       \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)} \]

6. Applied rewrites 97.8%

   \[\leadsto \frac{\cos \left(x \cdot 2\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)} \]
8. Step-by-step derivation

9. Applied rewrites 78.9%

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)} \]
10. Add Preprocessing

Alternative 15: 75.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* (* s c) (* c x)) (* s x))))

C

double code(double x, double c, double s) {
	return 1.0 / (((s * c) * (c * x)) * (s * 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 = 1.0d0 / (((s * c) * (c * x)) * (s * x))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (((s * c) * (c * x)) * (s * x));
}

Python

def code(x, c, s):
	return 1.0 / (((s * c) * (c * x)) * (s * x))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(Float64(s * c) * Float64(c * x)) * Float64(s * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (((s * c) * (c * x)) * (s * x));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(N[(s * c), $MachinePrecision] * N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. 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. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left({s}^{2} \cdot x\right)}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(\color{blue}{{s}^{2}} \cdot x\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot x\right)} \]

   9. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot \left(s \cdot x\right)\right)}} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot s\right)} \cdot \left(s \cdot x\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot {c}^{2}\right)} \cdot s\right) \cdot \left(s \cdot x\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot {c}^{2}\right)} \cdot s\right) \cdot \left(s \cdot x\right)} \]

   15. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{{c}^{2}}\right) \cdot s\right) \cdot \left(s \cdot x\right)} \]

   16. unpow2 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot s\right) \cdot \left(s \cdot x\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot s\right) \cdot \left(s \cdot x\right)} \]

   18. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}} \]

   19. lower-*.f64 79.4

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}} \]

4. Applied rewrites 79.4%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right) \cdot \left(x \cdot s\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right) \cdot \left(x \cdot s\right)} \]
6. Step-by-step derivation

7. Applied rewrites 66.2%

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right) \cdot \left(x \cdot s\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right)} \cdot \left(x \cdot s\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(x \cdot \left(c \cdot c\right)\right)} \cdot s\right) \cdot \left(x \cdot s\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(x \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot s\right) \cdot \left(x \cdot s\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\left(x \cdot c\right) \cdot c\right)} \cdot s\right) \cdot \left(x \cdot s\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\left(\left(\color{blue}{\left(c \cdot x\right)} \cdot c\right) \cdot s\right) \cdot \left(x \cdot s\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(\color{blue}{\left(c \cdot x\right)} \cdot c\right) \cdot s\right) \cdot \left(x \cdot s\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot s\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\left(\left(c \cdot x\right) \cdot \color{blue}{\left(s \cdot c\right)}\right) \cdot \left(x \cdot s\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(c \cdot x\right) \cdot \color{blue}{\left(s \cdot c\right)}\right) \cdot \left(x \cdot s\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right)} \cdot \left(x \cdot s\right)} \]

   11. lower-*.f64 75.7

       \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right)} \cdot \left(x \cdot s\right)} \]

9. Applied rewrites 75.7%

   \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right)} \cdot \left(x \cdot s\right)} \]

10. Final simplification 75.7%

    \[\leadsto \frac{1}{\left(\left(s \cdot c\right) \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot x\right)} \]
11. Add Preprocessing

Alternative 16: 70.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(\left(\left(c \cdot x\right) \cdot c\right) \cdot x\right) \cdot s\right) \cdot s} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* (* (* (* c x) c) x) s) s)))

C

double code(double x, double c, double s) {
	return 1.0 / (((((c * x) * c) * x) * s) * 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) * c) * x) * s) * s)
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (((((c * x) * c) * x) * s) * s);
}

Python

def code(x, c, s):
	return 1.0 / (((((c * x) * c) * x) * s) * s)

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c * x) * c) * x) * s) * s))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (((((c * x) * c) * x) * s) * s);
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(N[(N[(N[(c * x), $MachinePrecision] * c), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. 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. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot x} \]

   6. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot x\right)}\right) \cdot x} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)} \cdot x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)} \cdot x} \]

   9. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({s}^{2} \cdot {c}^{2}\right)} \cdot x\right) \cdot x} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{s}^{2}} \cdot {c}^{2}\right) \cdot x\right) \cdot x} \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot \color{blue}{{c}^{2}}\right) \cdot x\right) \cdot x} \]

   12. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{\left(s \cdot c\right)}^{2}} \cdot x\right) \cdot x} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{\left(s \cdot c\right)}^{2}} \cdot x\right) \cdot x} \]

   14. lower-*.f64 84.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({\color{blue}{\left(s \cdot c\right)}}^{2} \cdot x\right) \cdot x} \]

4. Applied rewrites 84.5%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(s \cdot c\right)}^{2} \cdot x\right) \cdot x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\left({\left(s \cdot c\right)}^{2} \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 72.5%

   \[\leadsto \frac{\color{blue}{1}}{\left({\left(s \cdot c\right)}^{2} \cdot x\right) \cdot x} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{{c}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {x}^{2}\right) \cdot {s}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {x}^{2}\right) \cdot s\right) \cdot s}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left({c}^{2} \cdot {x}^{2}\right)\right)} \cdot s} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left({c}^{2} \cdot {x}^{2}\right)\right) \cdot s}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {x}^{2}\right) \cdot s\right)} \cdot s} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {x}^{2}\right) \cdot s\right)} \cdot s} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\left(\left({c}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot s\right) \cdot s} \]

   10. associate-*r* N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot x\right)} \cdot s\right) \cdot s} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot x\right)} \cdot s\right) \cdot s} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot x\right) \cdot s\right) \cdot s} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\left(\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot x\right) \cdot s\right) \cdot s} \]

   14. lower-*.f64 64.3

       \[\leadsto \frac{1}{\left(\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot x\right) \cdot s\right) \cdot s} \]

10. Applied rewrites 64.3%

    \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot s\right) \cdot s}} \]
11. Step-by-step derivation

12. Applied rewrites 73.3%

    \[\leadsto \frac{1}{\left(\left(\left(\left(c \cdot x\right) \cdot c\right) \cdot x\right) \cdot s\right) \cdot s} \]
13. Add Preprocessing

Alternative 17: 65.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(\left(\left(x \cdot x\right) \cdot c\right) \cdot c\right) \cdot s\right) \cdot s} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* (* (* (* x x) c) c) s) s)))

C

double code(double x, double c, double s) {
	return 1.0 / (((((x * x) * c) * c) * s) * 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 / (((((x * x) * c) * c) * s) * s)
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (((((x * x) * c) * c) * s) * s);
}

Python

def code(x, c, s):
	return 1.0 / (((((x * x) * c) * c) * s) * s)

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(Float64(Float64(Float64(x * x) * c) * c) * s) * s))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (((((x * x) * c) * c) * s) * s);
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(N[(N[(N[(x * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. 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. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}\right) \cdot x} \]

   6. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot x\right)}\right) \cdot x} \]

   7. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)} \cdot x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)} \cdot x} \]

   9. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({s}^{2} \cdot {c}^{2}\right)} \cdot x\right) \cdot x} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{s}^{2}} \cdot {c}^{2}\right) \cdot x\right) \cdot x} \]

   11. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({s}^{2} \cdot \color{blue}{{c}^{2}}\right) \cdot x\right) \cdot x} \]

   12. pow-prod-down N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{\left(s \cdot c\right)}^{2}} \cdot x\right) \cdot x} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{\left(s \cdot c\right)}^{2}} \cdot x\right) \cdot x} \]

   14. lower-*.f64 84.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({\color{blue}{\left(s \cdot c\right)}}^{2} \cdot x\right) \cdot x} \]

4. Applied rewrites 84.5%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(s \cdot c\right)}^{2} \cdot x\right) \cdot x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\left({\left(s \cdot c\right)}^{2} \cdot x\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 72.5%

   \[\leadsto \frac{\color{blue}{1}}{\left({\left(s \cdot c\right)}^{2} \cdot x\right) \cdot x} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{{c}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {s}^{2}\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {x}^{2}\right) \cdot {s}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\left({c}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {x}^{2}\right) \cdot s\right) \cdot s}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left({c}^{2} \cdot {x}^{2}\right)\right)} \cdot s} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left({c}^{2} \cdot {x}^{2}\right)\right) \cdot s}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {x}^{2}\right) \cdot s\right)} \cdot s} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {x}^{2}\right) \cdot s\right)} \cdot s} \]

   9. unpow2 N/A

      \[\leadsto \frac{1}{\left(\left({c}^{2} \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot s\right) \cdot s} \]

   10. associate-*r* N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot x\right)} \cdot s\right) \cdot s} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{1}{\left(\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot x\right)} \cdot s\right) \cdot s} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot x\right) \cdot s\right) \cdot s} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\left(\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot x\right) \cdot s\right) \cdot s} \]

   14. lower-*.f64 64.3

       \[\leadsto \frac{1}{\left(\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot x\right) \cdot s\right) \cdot s} \]

10. Applied rewrites 64.3%

    \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot s\right) \cdot s}} \]
11. Step-by-step derivation

12. Applied rewrites 69.0%

    \[\leadsto \frac{1}{\left(\left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right) \cdot s\right) \cdot s} \]

13. Final simplification 69.0%

    \[\leadsto \frac{1}{\left(\left(\left(\left(x \cdot x\right) \cdot c\right) \cdot c\right) \cdot s\right) \cdot s} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))