mixedcos

Percentage Accurate: 65.6% → 96.9%

Time: 10.3s

Alternatives: 9

Speedup: 2.3×

• 31.1% 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: 65.6% 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: 96.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-pow.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-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow2 N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 98.5

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

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))))
   (if (<= t_0 -2e-178)
     (/ -2.0 (* c (* c (* s s))))
     (if (<= t_0 INFINITY)
       (/ 1.0 (* c (* (* x c) (* s (* x s)))))
       (/ 1.0 (* c (* s (* c (* s (* x x))))))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))));
	double tmp;
	if (t_0 <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 / (c * ((x * c) * (s * (x * s))));
	} else {
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))));
	double tmp;
	if (t_0 <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (c * ((x * c) * (s * (x * s))));
	} else {
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))
	tmp = 0
	if t_0 <= -2e-178:
		tmp = -2.0 / (c * (c * (s * s)))
	elif t_0 <= math.inf:
		tmp = 1.0 / (c * ((x * c) * (s * (x * s))))
	else:
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= -2e-178)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * c) * Float64(s * Float64(x * s)))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(s * Float64(c * Float64(s * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= -2e-178)
		tmp = -2.0 / (c * (c * (s * s)));
	elseif (t_0 <= Inf)
		tmp = 1.0 / (c * ((x * c) * (s * (x * s))));
	else
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-178], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(1.0 / N[(c * N[(N[(x * c), $MachinePrecision] * N[(s * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(s * N[(c * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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))) < -1.9999999999999999e-178

   1. Initial program 74.4%

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

      1. lift-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 99.5

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 99.5

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.5

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

   9. Simplified 49.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 49.4

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

   12. Simplified 49.4%

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

   if -1.9999999999999999e-178 < (/.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 86.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 85.7

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

   5. Simplified 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

   7. Applied egg-rr 86.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 91.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.6

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

   9. Applied egg-rr 91.6%

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

   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-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 98.0

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

   4. Applied egg-rr 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 98.0

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 98.0

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 91.5

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

   6. Applied egg-rr 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 94.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 94.4

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

   8. Applied egg-rr 94.4%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 53.2

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

   11. Simplified 53.2%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{elif}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s))))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -2e-178)
     (/ -2.0 (* c (* c (* s s))))
     (/ (/ 1.0 t_0) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} 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 = x * (c * s)
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-2d-178)) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    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 = x * (c * s);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -2e-178:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -2e-178)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (c * s);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -2e-178)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-178], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $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))) < -1.9999999999999999e-178

   1. Initial program 74.4%

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

      1. lift-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 99.5

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 99.5

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.5

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

   9. Simplified 49.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 49.4

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

   12. Simplified 49.4%

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

   if -1.9999999999999999e-178 < (/.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 70.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 80.7

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

   5. Simplified 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(c \cdot s\right)}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s))))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -2e-178)
     (/ -2.0 (* c (* c (* s s))))
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} 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 = x * (c * s)
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-2d-178)) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    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 = x * (c * s);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -2e-178:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -2e-178)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (c * s);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -2e-178)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-178], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$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))) < -1.9999999999999999e-178

   1. Initial program 74.4%

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

      1. lift-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 99.5

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 99.5

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.5

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

   9. Simplified 49.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 49.4

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

   12. Simplified 49.4%

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

   if -1.9999999999999999e-178 < (/.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 70.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 80.7

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

   5. Simplified 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 89.8

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

   7. Applied egg-rr 89.8%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(c \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -2e-178)
   (/ -2.0 (* c (* c (* s s))))
   (/ 1.0 (* x (* s (* c (* x (* c s))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (x * (s * (c * (x * (c * s)))));
	}
	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 ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-2d-178)) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    else
        tmp = 1.0d0 / (x * (s * (c * (x * (c * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -2e-178) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (x * (s * (c * (x * (c * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -2e-178:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = 1.0 / (x * (s * (c * (x * (c * s)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -2e-178)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(s * Float64(c * Float64(x * Float64(c * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -2e-178)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = 1.0 / (x * (s * (c * (x * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-178], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(s * N[(c * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -1.9999999999999999e-178

   1. Initial program 74.4%

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

      1. lift-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 99.5

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 99.5

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 49.5

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

   9. Simplified 49.5%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 49.4

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

   12. Simplified 49.4%

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

   if -1.9999999999999999e-178 < (/.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 70.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 80.7

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

   5. Simplified 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   7. Applied egg-rr 85.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-178}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(c \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-41}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -2e-41)
   (/ -2.0 (* c (* c (* s s))))
   (/ 1.0 (* c (* s (* c (* s (* x x))))))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -2e-41) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (c * (s * (c * (s * (x * 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 ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-2d-41)) then
        tmp = (-2.0d0) / (c * (c * (s * s)))
    else
        tmp = 1.0d0 / (c * (s * (c * (s * (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -2e-41) {
		tmp = -2.0 / (c * (c * (s * s)));
	} else {
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -2e-41:
		tmp = -2.0 / (c * (c * (s * s)))
	else:
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -2e-41)
		tmp = Float64(-2.0 / Float64(c * Float64(c * Float64(s * s))));
	else
		tmp = Float64(1.0 / Float64(c * Float64(s * Float64(c * Float64(s * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -2e-41)
		tmp = -2.0 / (c * (c * (s * s)));
	else
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-41], N[(-2.0 / N[(c * N[(c * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(s * N[(c * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < -2.00000000000000001e-41

   1. Initial program 71.4%

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

      1. lift-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.5

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

   4. Applied egg-rr 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 99.5

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 99.5

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 55.0

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

   9. Simplified 55.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 54.8

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

   12. Simplified 54.8%

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

   if -2.00000000000000001e-41 < (/.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 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-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 98.4

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

   4. Applied egg-rr 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 98.4

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 98.4

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.2

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

   6. Applied egg-rr 96.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 95.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 95.5

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

   8. Applied egg-rr 95.5%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f64 77.4

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

   11. Simplified 77.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-41}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 5e-165)
   (/ 1.0 (* (* (* x s) (* x s)) (* c c)))
   (/ (cos (+ x x)) (* (* c s) (* x (* x (* c s)))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-165) {
		tmp = 1.0 / (((x * s) * (x * s)) * (c * c));
	} else {
		tmp = cos((x + x)) / ((c * s) * (x * (x * (c * s))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 5d-165) then
        tmp = 1.0d0 / (((x * s) * (x * s)) * (c * c))
    else
        tmp = cos((x + x)) / ((c * s) * (x * (x * (c * s))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5e-165) {
		tmp = 1.0 / (((x * s) * (x * s)) * (c * c));
	} else {
		tmp = Math.cos((x + x)) / ((c * s) * (x * (x * (c * s))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 5e-165:
		tmp = 1.0 / (((x * s) * (x * s)) * (c * c))
	else:
		tmp = math.cos((x + x)) / ((c * s) * (x * (x * (c * s))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 5e-165)
		tmp = Float64(1.0 / Float64(Float64(Float64(x * s) * Float64(x * s)) * Float64(c * c)));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(c * s) * Float64(x * Float64(x * Float64(c * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5e-165)
		tmp = 1.0 / (((x * s) * (x * s)) * (c * c));
	else
		tmp = cos((x + x)) / ((c * s) * (x * (x * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 5e-165], N[(1.0 / N[(N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(c * s), $MachinePrecision] * N[(x * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999981e-165

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 75.5

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

   5. Simplified 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. swap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. swap-sqr N/A

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

      21. lift-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   7. Applied egg-rr 72.0%

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

   if 4.99999999999999981e-165 < x

   1. Initial program 76.2%

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

      1. lift-pow.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-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 98.8

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

   4. Applied egg-rr 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-/.f64 98.8

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

      7. lift-*.f64 N/A

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

      8. count-2 N/A

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

      9. lift-+.f64 98.8

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.9

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

   6. Applied egg-rr 96.9%

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

4. Final simplification 81.0%

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

Alternative 8: 84.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* c s))))
   (if (<= x 4e-5)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (+ x x)) (* x (* x (* s (* c (* c s)))))))))

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if (x <= 4e-5) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x + x)) / (x * (x * (s * (c * (c * s)))));
	}
	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 = x * (c * s)
    if (x <= 4d-5) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x + x)) / (x * (x * (s * (c * (c * s)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	double tmp;
	if (x <= 4e-5) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x + x)) / (x * (x * (s * (c * (c * s)))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = x * (c * s)
	tmp = 0
	if x <= 4e-5:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x + x)) / (x * (x * (s * (c * (c * s)))))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (x <= 4e-5)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(x * Float64(x * Float64(s * Float64(c * Float64(c * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = x * (c * s);
	tmp = 0.0;
	if (x <= 4e-5)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x + x)) / (x * (x * (s * (c * (c * s)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e-5], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x * N[(s * N[(c * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000033e-5

   1. Initial program 70.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 77.0

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

   5. Simplified 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   7. Applied egg-rr 87.6%

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

   if 4.00000000000000033e-5 < x

   1. Initial program 73.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 86.5

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

   5. Simplified 86.5%

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

      1. count-2 N/A

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

      2. lift-+.f64 86.5

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 87.4%

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

Alternative 9: 27.3% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, c, s):
	return -2.0 / (c * (c * (s * s)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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-pow.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-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-prod-down N/A

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

   11. pow2 N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 98.5

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

4. Applied egg-rr 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-/.f64 98.5

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

   7. lift-*.f64 N/A

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

   8. count-2 N/A

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

   9. lift-+.f64 98.5

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 96.5

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

6. Applied egg-rr 96.5%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 62.8

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

9. Simplified 62.8%

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

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 32.3

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

12. Simplified 32.3%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))