Average Error: 27.4 → 8.5
Time: 9.3s
Precision: binary64
\[[c, s] = \mathsf{sort}([c, s]) \\]
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := c \cdot \left(s \cdot \sqrt{x}\right)\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 2.3092907686599463 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{{\left(\frac{x}{t_1}\right)}^{-1}}{c}}{{s}^{2} \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t_1}{x}}{t_0}}{t_0}\\ \end{array} \]
(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 (* c (* s (sqrt x)))) (t_1 (cos (+ x x))))
   (if (<= x 2.3092907686599463e-249)
     (/ (/ (pow (/ x t_1) -1.0) c) (* (pow s 2.0) (* x c)))
     (/ (/ (/ t_1 x) t_0) t_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 = c * (s * sqrt(x));
	double t_1 = cos((x + x));
	double tmp;
	if (x <= 2.3092907686599463e-249) {
		tmp = (pow((x / t_1), -1.0) / c) / (pow(s, 2.0) * (x * c));
	} else {
		tmp = ((t_1 / x) / t_0) / t_0;
	}
	return tmp;
}

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
↓
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 = c * (s * sqrt(x))
    t_1 = cos((x + x))
    if (x <= 2.3092907686599463d-249) then
        tmp = (((x / t_1) ** (-1.0d0)) / c) / ((s ** 2.0d0) * (x * c))
    else
        tmp = ((t_1 / x) / t_0) / t_0
    end if
    code = tmp
end function

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 = c * (s * Math.sqrt(x));
	double t_1 = Math.cos((x + x));
	double tmp;
	if (x <= 2.3092907686599463e-249) {
		tmp = (Math.pow((x / t_1), -1.0) / c) / (Math.pow(s, 2.0) * (x * c));
	} else {
		tmp = ((t_1 / x) / t_0) / t_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 = c * (s * math.sqrt(x))
	t_1 = math.cos((x + x))
	tmp = 0
	if x <= 2.3092907686599463e-249:
		tmp = (math.pow((x / t_1), -1.0) / c) / (math.pow(s, 2.0) * (x * c))
	else:
		tmp = ((t_1 / x) / t_0) / t_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 = Float64(c * Float64(s * sqrt(x)))
	t_1 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 2.3092907686599463e-249)
		tmp = Float64(Float64((Float64(x / t_1) ^ -1.0) / c) / Float64((s ^ 2.0) * Float64(x * c)));
	else
		tmp = Float64(Float64(Float64(t_1 / x) / t_0) / t_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 = c * (s * sqrt(x));
	t_1 = cos((x + x));
	tmp = 0.0;
	if (x <= 2.3092907686599463e-249)
		tmp = (((x / t_1) ^ -1.0) / c) / ((s ^ 2.0) * (x * c));
	else
		tmp = ((t_1 / x) / t_0) / t_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[(c * N[(s * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.3092907686599463e-249], N[(N[(N[Power[N[(x / t$95$1), $MachinePrecision], -1.0], $MachinePrecision] / c), $MachinePrecision] / N[(N[Power[s, 2.0], $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 / x), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $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 := c \cdot \left(s \cdot \sqrt{x}\right)\\
t_1 := \cos \left(x + x\right)\\
\mathbf{if}\;x \leq 2.3092907686599463 \cdot 10^{-249}:\\
\;\;\;\;\frac{\frac{{\left(\frac{x}{t_1}\right)}^{-1}}{c}}{{s}^{2} \cdot \left(x \cdot c\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{t_1}{x}}{t_0}}{t_0}\\


\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 x < 2.30929076865994628e-249

    1. Initial program 28.2

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

    2. Applied egg-rr26.4

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

    3. Applied egg-rr18.2

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

    4. Taylor expanded in c around 0 15.3

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

    5. Applied egg-rr15.3

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

    if 2.30929076865994628e-249 < x

    1. Initial program 26.6

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

    2. Applied egg-rr25.2

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

    3. Applied egg-rr1.0

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3092907686599463 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{{\left(\frac{x}{\cos \left(x + x\right)}\right)}^{-1}}{c}}{{s}^{2} \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x + x\right)}{x}}{c \cdot \left(s \cdot \sqrt{x}\right)}}{c \cdot \left(s \cdot \sqrt{x}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))