mixedcos

Percentage Accurate: 67.2% → 97.0%

Time: 10.9s

Alternatives: 16

Speedup: 2.4×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 2.3× speedup

Math

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

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c)))) (/ (/ (cos (+ x x)) t_0) t_0)))

C

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

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = s * (x * c)
    code = (cos((x + x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	t_0 = s * (x * c);
	tmp = (cos((x + x)) / t_0) / t_0;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. pow-prod-down N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. pow-lowering-pow.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 96.8

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

4. Applied egg-rr 96.8%

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

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. count-2 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 N/A

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

   18. *-commutative N/A

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

   19. *-lowering-*.f64 97.4

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

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

Alternative 2: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot c\right) \cdot \left(s \cdot c\right)\\ t_1 := \frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{t\_0}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s c) (* s c)))
        (t_1 (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))))
   (if (<= t_1 -1e-149)
     (/ -2.0 t_0)
     (if (<= t_1 INFINITY)
       (/ 1.0 (* (* x s) (* c (* s (* x c)))))
       (/ 1.0 (* x (* x t_0)))))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = (s * c) * (s * c);
	double t_1 = cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))));
	double tmp;
	if (t_1 <= -1e-149) {
		tmp = -2.0 / t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.0 / ((x * s) * (c * (s * (x * c))));
	} else {
		tmp = 1.0 / (x * (x * t_0));
	}
	return tmp;
}

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = (s * c) * (s * c);
	double t_1 = Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))));
	double tmp;
	if (t_1 <= -1e-149) {
		tmp = -2.0 / t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((x * s) * (c * (s * (x * c))));
	} else {
		tmp = 1.0 / (x * (x * t_0));
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = (s * c) * (s * c)
	t_1 = math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))
	tmp = 0
	if t_1 <= -1e-149:
		tmp = -2.0 / t_0
	elif t_1 <= math.inf:
		tmp = 1.0 / ((x * s) * (c * (s * (x * c))))
	else:
		tmp = 1.0 / (x * (x * t_0))
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(Float64(s * c) * Float64(s * c))
	t_1 = Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-149)
		tmp = Float64(-2.0 / t_0);
	elseif (t_1 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(x * s) * Float64(c * Float64(s * Float64(x * c)))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = (s * c) * (s * c);
	t_1 = cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= -1e-149)
		tmp = -2.0 / t_0;
	elseif (t_1 <= Inf)
		tmp = 1.0 / ((x * s) * (c * (s * (x * c))));
	else
		tmp = 1.0 / (x * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(x * 2.0), $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$1, -1e-149], N[(-2.0 / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(1.0 / N[(N[(x * s), $MachinePrecision] * N[(c * N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(x * t$95$0), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 82.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 89.6

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

   7. Applied egg-rr 89.6%

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

   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. 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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 54.4

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

   5. Simplified 54.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 65.9

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

   7. Applied egg-rr 65.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{elif}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \left(s \cdot c\right) \cdot \left(s \cdot c\right)\\ t_1 := \frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{t\_0}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s c) (* s c)))
        (t_1 (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))))
   (if (<= t_1 -1e-149)
     (/ -2.0 t_0)
     (if (<= t_1 INFINITY)
       (/ 1.0 (* c (* c (* (* x s) (* x s)))))
       (/ 1.0 (* x (* x t_0)))))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = (s * c) * (s * c);
	double t_1 = cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))));
	double tmp;
	if (t_1 <= -1e-149) {
		tmp = -2.0 / t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else {
		tmp = 1.0 / (x * (x * t_0));
	}
	return tmp;
}

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = (s * c) * (s * c);
	double t_1 = Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))));
	double tmp;
	if (t_1 <= -1e-149) {
		tmp = -2.0 / t_0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else {
		tmp = 1.0 / (x * (x * t_0));
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = (s * c) * (s * c)
	t_1 = math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))
	tmp = 0
	if t_1 <= -1e-149:
		tmp = -2.0 / t_0
	elif t_1 <= math.inf:
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))))
	else:
		tmp = 1.0 / (x * (x * t_0))
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(Float64(s * c) * Float64(s * c))
	t_1 = Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= -1e-149)
		tmp = Float64(-2.0 / t_0);
	elseif (t_1 <= Inf)
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * s) * Float64(x * s)))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(x * t_0)));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = (s * c) * (s * c);
	t_1 = cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))));
	tmp = 0.0;
	if (t_1 <= -1e-149)
		tmp = -2.0 / t_0;
	elseif (t_1 <= Inf)
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	else
		tmp = 1.0 / (x * (x * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(x * 2.0), $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$1, -1e-149], N[(-2.0 / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(x * t$95$0), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 82.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

   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. 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. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 54.4

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

   5. Simplified 54.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 65.9

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

   7. Applied egg-rr 65.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{elif}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))))
   (if (<= t_0 -1e-149)
     (/ -2.0 (* (* s c) (* s c)))
     (if (<= t_0 INFINITY)
       (/ 1.0 (* c (* c (* (* x s) (* x s)))))
       (/ 1.0 (* c (* s (* c (* s (* x x))))))))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))));
	double tmp;
	if (t_0 <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else {
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	}
	return tmp;
}

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))));
	double tmp;
	if (t_0 <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	} else {
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))
	tmp = 0
	if t_0 <= -1e-149:
		tmp = -2.0 / ((s * c) * (s * c))
	elif t_0 <= math.inf:
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))))
	else:
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))))
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= -1e-149)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * 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

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= -1e-149)
		tmp = -2.0 / ((s * c) * (s * c));
	elseif (t_0 <= Inf)
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	else
		tmp = 1.0 / (c * (s * (c * (s * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[Cos[N[(x * 2.0), $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, -1e-149], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * 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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 82.6%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

   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. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 95.6

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

   4. Applied egg-rr 95.6%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      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

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-lowering-*.f64 N/A

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 56.5

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

   7. Simplified 56.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{elif}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))) (t_1 (* x (* s c))))
   (if (<= (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -1e-149)
     (/ (fma x (* x -2.0) 1.0) (* t_0 t_0))
     (/ (/ 1.0 t_1) t_1))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double t_1 = x * (s * c);
	double tmp;
	if ((cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -1e-149) {
		tmp = fma(x, (x * -2.0), 1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_1) / t_1;
	}
	return tmp;
}

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	t_1 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -1e-149)
		tmp = Float64(fma(x, Float64(x * -2.0), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	end
	return tmp
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-149], N[(N[(x * N[(x * -2.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

      1. unpow2 N/A

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

      2. swap-sqr N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. count-2 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. *-lowering-*.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. Applied egg-rr 99.3%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 36.8

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

   9. Simplified 36.8%

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

   if -9.99999999999999979e-150 < (/.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 64.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 77.2

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

   9. Applied egg-rr 77.2%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. associate-/r* N/A

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

       7. /-lowering-/.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 86.2

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

   11. Applied egg-rr 86.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot -2, 1\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -1e-149)
     (/ -2.0 (* (* s c) (* s c)))
     (/ (/ 1.0 t_0) t_0))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if ((cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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 * (s * c)
    if ((cos((x * 2.0d0)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-1d-149)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if ((Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if (math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -1e-149:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -1e-149)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -1e-149)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-149], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 64.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 77.2

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

   9. Applied egg-rr 77.2%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. associate-/r* N/A

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

       7. /-lowering-/.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 86.2

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

   11. Applied egg-rr 86.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -1e-149)
     (/ -2.0 (* (* s c) (* s c)))
     (/ 1.0 (* t_0 t_0)))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if ((cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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 * (s * c)
    if ((cos((x * 2.0d0)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-1d-149)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if ((Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if (math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -1e-149:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -1e-149)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -1e-149)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-149], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 64.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. *-lowering-*.f64 N/A

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 86.1

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot \left(s \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c))))
   (if (<= (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -1e-149)
     (/ -2.0 (* (* s c) (* s c)))
     (/ 1.0 (* t_0 t_0)))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if ((cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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 = s * (x * c)
    if ((cos((x * 2.0d0)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-1d-149)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	double tmp;
	if ((Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	tmp = 0
	if (math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -1e-149:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -1e-149)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = s * (x * c);
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -1e-149)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-149], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 64.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. /-lowering-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-commutative N/A

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

      19. *-lowering-*.f64 86.3

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

   7. Applied egg-rr 86.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* x 2.0)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) -1e-149)
   (/ -2.0 (* (* s c) (* s c)))
   (/ 1.0 (* c (* c (* (* x s) (* x s)))))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double tmp;
	if ((cos((x * 2.0)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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((x * 2.0d0)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= (-1d-149)) then
        tmp = (-2.0d0) / ((s * c) * (s * c))
    else
        tmp = 1.0d0 / (c * (c * ((x * s) * (x * s))))
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((x * 2.0)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= -1e-149) {
		tmp = -2.0 / ((s * c) * (s * c));
	} else {
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	tmp = 0
	if (math.cos((x * 2.0)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= -1e-149:
		tmp = -2.0 / ((s * c) * (s * c))
	else:
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))))
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(x * 2.0)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= -1e-149)
		tmp = Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)));
	else
		tmp = Float64(1.0 / Float64(c * Float64(c * Float64(Float64(x * s) * Float64(x * s)))));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((x * 2.0)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= -1e-149)
		tmp = -2.0 / ((s * c) * (s * c));
	else
		tmp = 1.0 / (c * (c * ((x * s) * (x * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-149], N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(c * N[(N[(x * s), $MachinePrecision] * N[(x * s), $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))) < -9.99999999999999979e-150

   1. Initial program 61.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

   7. Simplified 11.7%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 36.8

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

   10. Simplified 36.8%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 36.9

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

   12. Applied egg-rr 36.9%

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

   if -9.99999999999999979e-150 < (/.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 64.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 75.4

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

   5. Simplified 75.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;{s}^{2} \leq 2 \cdot 10^{+248}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* x (* s c))))
   (if (<= (pow s 2.0) 2e+248)
     (/ (cos (+ x x)) (* c (* (* x c) (* x (* s s)))))
     (/ (/ 1.0 t_0) t_0))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (pow(s, 2.0) <= 2e+248) {
		tmp = cos((x + x)) / (c * ((x * c) * (x * (s * s))));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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 * (s * c)
    if ((s ** 2.0d0) <= 2d+248) then
        tmp = cos((x + x)) / (c * ((x * c) * (x * (s * s))))
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = x * (s * c);
	double tmp;
	if (Math.pow(s, 2.0) <= 2e+248) {
		tmp = Math.cos((x + x)) / (c * ((x * c) * (x * (s * s))));
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = x * (s * c)
	tmp = 0
	if math.pow(s, 2.0) <= 2e+248:
		tmp = math.cos((x + x)) / (c * ((x * c) * (x * (s * s))))
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(x * Float64(s * c))
	tmp = 0.0
	if ((s ^ 2.0) <= 2e+248)
		tmp = Float64(cos(Float64(x + x)) / Float64(c * Float64(Float64(x * c) * Float64(x * Float64(s * s)))));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = x * (s * c);
	tmp = 0.0;
	if ((s ^ 2.0) <= 2e+248)
		tmp = cos((x + x)) / (c * ((x * c) * (x * (s * s))));
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[s, 2.0], $MachinePrecision], 2e+248], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(c * N[(N[(x * c), $MachinePrecision] * N[(x * N[(s * s), $MachinePrecision]), $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 (pow.f64 s #s(literal 2 binary64)) < 2.00000000000000009e248

   1. Initial program 66.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. pow-lowering-pow.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 95.9

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

   4. Applied egg-rr 95.9%

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

      1. unpow2 N/A

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

      2. swap-sqr N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. /-lowering-/.f64 N/A

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

      8. count-2 N/A

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

      9. cos-lowering-cos.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. *-lowering-*.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. Applied egg-rr 97.3%

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

      1. swap-sqr N/A

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

      2. pow2 N/A

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

      3. associate-*l* N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. *-lowering-*.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. *-lowering-*.f64 N/A

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

      21. *-lowering-*.f64 81.3

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

   8. Applied egg-rr 81.3%

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

   if 2.00000000000000009e248 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 58.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 82.8

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

   5. Simplified 82.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 90.1

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

   7. Applied egg-rr 90.1%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.4

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

   9. Applied egg-rr 73.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. associate-/r* N/A

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

       7. /-lowering-/.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 93.8

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

   11. Applied egg-rr 93.8%

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

Alternative 11: 83.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;x \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+114}:\\ \;\;\;\;\frac{t\_0}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{c \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (* x (* s c))))
   (if (<= x 3.9e-7)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x 3.9e+114)
       (/ t_0 (* s (* c (* s (* c (* x x))))))
       (/ t_0 (* c (* (* x c) (* x (* s s)))))))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double t_1 = x * (s * c);
	double tmp;
	if (x <= 3.9e-7) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x <= 3.9e+114) {
		tmp = t_0 / (s * (c * (s * (c * (x * x)))));
	} else {
		tmp = t_0 / (c * ((x * c) * (x * (s * s))));
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x + x))
    t_1 = x * (s * c)
    if (x <= 3.9d-7) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x <= 3.9d+114) then
        tmp = t_0 / (s * (c * (s * (c * (x * x)))))
    else
        tmp = t_0 / (c * ((x * c) * (x * (s * s))))
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double t_1 = x * (s * c);
	double tmp;
	if (x <= 3.9e-7) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x <= 3.9e+114) {
		tmp = t_0 / (s * (c * (s * (c * (x * x)))));
	} else {
		tmp = t_0 / (c * ((x * c) * (x * (s * s))));
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = math.cos((x + x))
	t_1 = x * (s * c)
	tmp = 0
	if x <= 3.9e-7:
		tmp = (1.0 / t_1) / t_1
	elif x <= 3.9e+114:
		tmp = t_0 / (s * (c * (s * (c * (x * x)))))
	else:
		tmp = t_0 / (c * ((x * c) * (x * (s * s))))
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (x <= 3.9e-7)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x <= 3.9e+114)
		tmp = Float64(t_0 / Float64(s * Float64(c * Float64(s * Float64(c * Float64(x * x))))));
	else
		tmp = Float64(t_0 / Float64(c * Float64(Float64(x * c) * Float64(x * Float64(s * s)))));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	t_1 = x * (s * c);
	tmp = 0.0;
	if (x <= 3.9e-7)
		tmp = (1.0 / t_1) / t_1;
	elseif (x <= 3.9e+114)
		tmp = t_0 / (s * (c * (s * (c * (x * x)))));
	else
		tmp = t_0 / (c * ((x * c) * (x * (s * s))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.9e-7], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 3.9e+114], N[(t$95$0 / N[(s * N[(c * N[(s * N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(c * N[(N[(x * c), $MachinePrecision] * N[(x * N[(s * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.90000000000000025e-7

   1. Initial program 64.2%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 71.6

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

   5. Simplified 71.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. unswap-sqr N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 75.6

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

   7. Applied egg-rr 75.6%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 73.9

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

   9. Applied egg-rr 73.9%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. associate-/r* N/A

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

       7. /-lowering-/.f64 N/A

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

       8. /-lowering-/.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. *-lowering-*.f64 N/A

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

       16. *-lowering-*.f64 84.5

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

   11. Applied egg-rr 84.5%

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

   if 3.90000000000000025e-7 < x < 3.9000000000000001e114

   1. Initial program 62.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. pow-prod-down N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      6. pow-prod-down N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      9. *-lowering-*.f64 99.7

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      2. swap-sqr N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(s \cdot c\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      15. swap-sqr N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      16. *-lowering-*.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. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}} \]

   7. Taylor expanded in s around 0

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left({s}^{2} \cdot {x}^{2}\right) \cdot {c}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right) \cdot {c}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)} \cdot {c}^{2}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{s \cdot \left(\left(s \cdot {x}^{2}\right) \cdot {c}^{2}\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left({c}^{2} \cdot \left(s \cdot {x}^{2}\right)\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{s \cdot \left({c}^{2} \cdot \left(s \cdot {x}^{2}\right)\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot \left(s \cdot {x}^{2}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(s \cdot {x}^{2}\right)\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot {x}^{2}\right) \cdot c\right)}\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left(c \cdot \left(\left(s \cdot {x}^{2}\right) \cdot c\right)\right)}} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \color{blue}{\left(s \cdot \left({x}^{2} \cdot c\right)\right)}\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \color{blue}{\left(s \cdot \left({x}^{2} \cdot c\right)\right)}\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(c \cdot {x}^{2}\right)}\right)\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(c \cdot {x}^{2}\right)}\right)\right)} \]

      15. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      16. *-lowering-*.f64 91.4

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   9. Simplified 91.4%

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{s \cdot \left(c \cdot \left(s \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}} \]

   if 3.9000000000000001e114 < x

   1. Initial program 63.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      4. pow-prod-down N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      5. pow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      6. pow-prod-down N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      9. *-lowering-*.f64 93.7

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied egg-rr 93.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      2. swap-sqr N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(s \cdot c\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      13. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      15. swap-sqr N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      16. *-lowering-*.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. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}} \]
   7. Step-by-step derivation

      1. swap-sqr N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)}} \]

      2. pow2 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(s \cdot s\right) \cdot \color{blue}{{\left(c \cdot x\right)}^{2}}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{s \cdot \left(s \cdot {\left(c \cdot x\right)}^{2}\right)}} \]

      4. pow2 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(s \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)}\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(s \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot \left(c \cdot x\right)\right)\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(s \cdot \color{blue}{\left(x \cdot \left(c \cdot \left(c \cdot x\right)\right)\right)}\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(s \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot x\right)}\right)\right)} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot x\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot \left(c \cdot x\right)\right)} \cdot \left(\left(s \cdot x\right) \cdot s\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(\left(c \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(\left(c \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)}} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)}} \]

      16. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)} \]

      18. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot s\right)\right)} \]

      19. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot s\right)\right)}\right)} \]

      20. *-lowering-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot s\right)\right)}\right)} \]

      21. *-lowering-*.f64 78.3

          \[\leadsto \frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)} \]

   8. Applied egg-rr 78.3%

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 96.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \frac{\cos \left(x + x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* s (* x c)))) (/ (cos (+ x x)) (* t_0 t_0))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	return cos((x + x)) / (t_0 * t_0);
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = s * (x * c)
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double t_0 = s * (x * c);
	return Math.cos((x + x)) / (t_0 * t_0);
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	t_0 = s * (x * c)
	return math.cos((x + x)) / (t_0 * t_0)

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(s * Float64(x * c))
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	t_0 = s * (x * c);
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

   4. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

   6. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

   9. *-lowering-*.f64 96.8

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

4. Applied egg-rr 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

   2. swap-sqr N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(s \cdot c\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

   8. count-2 N/A

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \]

   11. associate-*r* N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

   13. associate-*r* N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

   15. swap-sqr N/A

       \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

   16. *-lowering-*.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. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)}} \]

7. Final simplification 97.3%

   \[\leadsto \frac{\cos \left(x + x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \]
8. Add Preprocessing

Alternative 13: 27.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{c \cdot \left(s \cdot \left(s \cdot c\right)\right)} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* c (* s (* s c)))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / (c * (s * (s * c)));
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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 * (s * (s * c)))
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / (c * (s * (s * c)));
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / (c * (s * (s * c)))

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(c * Float64(s * Float64(s * c))))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / (c * (s * (s * c)));
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(c * N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

   4. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

   6. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

   9. *-lowering-*.f64 96.8

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

4. Applied egg-rr 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]

7. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right) \cdot \frac{1}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}}{x \cdot x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-2}{\color{blue}{{s}^{2} \cdot {c}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2}{s \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   10. *-lowering-*.f64 28.9

       \[\leadsto \frac{-2}{s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot c\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-2}{s \cdot \left(\color{blue}{\left(s \cdot c\right)} \cdot c\right)} \]

    3. associate-*r* N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot \left(s \cdot c\right)\right)} \cdot c} \]

    6. *-lowering-*.f64 28.6

       \[\leadsto \frac{-2}{\left(s \cdot \color{blue}{\left(s \cdot c\right)}\right) \cdot c} \]

12. Applied egg-rr 28.6%

    \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}} \]

13. Final simplification 28.6%

    \[\leadsto \frac{-2}{c \cdot \left(s \cdot \left(s \cdot c\right)\right)} \]
14. Add Preprocessing

Alternative 14: 29.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* (* s s) (* c c))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / ((s * s) * (c * c));
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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) / ((s * s) * (c * c))
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / ((s * s) * (c * c));
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / ((s * s) * (c * c))

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(Float64(s * s) * Float64(c * c)))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / ((s * s) * (c * c));
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

   4. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

   6. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

   9. *-lowering-*.f64 96.8

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

4. Applied egg-rr 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]

7. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right) \cdot \frac{1}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}}{x \cdot x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-2}{\color{blue}{{s}^{2} \cdot {c}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2}{s \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   10. *-lowering-*.f64 28.9

       \[\leadsto \frac{-2}{s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot c\right) \cdot \left(c \cdot s\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \color{blue}{\left(s \cdot c\right)}} \]

    3. swap-sqr N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right)} \cdot \left(c \cdot c\right)} \]

    6. *-lowering-*.f64 30.0

       \[\leadsto \frac{-2}{\left(s \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]

12. Applied egg-rr 30.0%

    \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}} \]
13. Add Preprocessing

Alternative 15: 26.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* (* s c) (* s c))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / ((s * c) * (s * c));
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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) / ((s * c) * (s * c))
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / ((s * c) * (s * c));
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / ((s * c) * (s * c))

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(Float64(s * c) * Float64(s * c)))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / ((s * c) * (s * c));
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(N[(s * c), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

   4. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

   6. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

   9. *-lowering-*.f64 96.8

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

4. Applied egg-rr 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]

7. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right) \cdot \frac{1}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}}{x \cdot x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-2}{\color{blue}{{s}^{2} \cdot {c}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2}{s \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   10. *-lowering-*.f64 28.9

       \[\leadsto \frac{-2}{s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot c\right) \cdot \left(c \cdot s\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \color{blue}{\left(s \cdot c\right)}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot c\right)} \cdot \left(s \cdot c\right)} \]

    5. *-lowering-*.f64 27.5

       \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \color{blue}{\left(s \cdot c\right)}} \]

12. Applied egg-rr 27.5%

    \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}} \]
13. Add Preprocessing

Alternative 16: 27.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \frac{-2}{s \cdot \left(c \cdot \left(s \cdot c\right)\right)} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s) :precision binary64 (/ -2.0 (* s (* c (* s c)))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	return -2.0 / (s * (c * (s * c)));
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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) / (s * (c * (s * c)))
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	return -2.0 / (s * (c * (s * c)));
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	return -2.0 / (s * (c * (s * c)))

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	return Float64(-2.0 / Float64(s * Float64(c * Float64(s * c))))
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp = code(x, c, s)
	tmp = -2.0 / (s * (c * (s * c)));
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := N[(-2.0 / N[(s * N[(c * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

   4. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

   5. pow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

   6. pow-prod-down N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

   9. *-lowering-*.f64 96.8

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

4. Applied egg-rr 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} + \frac{1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}}} \]

7. Simplified 38.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right) \cdot \frac{1}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}}{x \cdot x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-2}{\color{blue}{{s}^{2} \cdot {c}^{2}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{-2}{\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}} \]

   4. associate-*l* N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{\color{blue}{s \cdot \left(s \cdot {c}^{2}\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left({c}^{2} \cdot s\right)}} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2}{s \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2}{s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}} \]

   10. *-lowering-*.f64 28.9

       \[\leadsto \frac{-2}{s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)} \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{-2}{s \cdot \left(c \cdot \left(c \cdot s\right)\right)}} \]

11. Final simplification 28.9%

    \[\leadsto \frac{-2}{s \cdot \left(c \cdot \left(s \cdot c\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))