Average Error: 28.3 → 0.9
Time: 7.6s
Precision: binary64
\[ \begin{array}{c}[c, s] = \mathsf{sort}([c, s])\\ \end{array} \]
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}\\ \end{array} \]
(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))
↓
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))) (t_1 (* c (* x s))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))
        INFINITY)
     (/ (/ t_0 t_1) t_1)
     (* t_0 (/ -1.0 (- (pow (* x (* c s)) 2.0)))))))
double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}
↓
double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double t_1 = c * (x * s);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 * (-1.0 / -pow((x * (c * s)), 2.0));
	}
	return tmp;
}

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

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))
↓
def code(x, c, s):
	t_0 = math.cos((x + x))
	t_1 = c * (x * s)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = t_0 * (-1.0 / -math.pow((x * (c * s)), 2.0))
	return tmp

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end
↓
function code(x, c, s)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(t_0 * Float64(-1.0 / Float64(-(Float64(x * Float64(c * s)) ^ 2.0))));
	end
	return tmp
end

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end
↓
function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	t_1 = c * (x * s);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = t_0 * (-1.0 / -((x * (c * s)) ^ 2.0));
	end
	tmp_2 = tmp;
end

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]
↓
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 * N[(-1.0 / (-N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\begin{array}{l}
t_0 := \cos \left(x + x\right)\\
t_1 := c \cdot \left(x \cdot s\right)\\
\mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\
\;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\

\mathbf{else}:\\
\;\;\;\;t_0 \cdot \frac{-1}{-{\left(x \cdot \left(c \cdot s\right)\right)}^{2}}\\


\end{array}

Error

Bits error versus x

Bits error versus c

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

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

    1. Initial program 18.4

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

    2. Simplified2.7

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

    3. Applied egg-rr2.4

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

    4. Applied egg-rr1.4

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

    5. Applied egg-rr0.3

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

    if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) 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. Simplified3.0

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

    3. Applied egg-rr3.0

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2022153 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))