mixedcos

Average Error: 27.8 → 0.8

Time: 6.6s

Precision: binary64

\[ \begin{array}{c}[c, s] = \mathsf{sort}([c, s])\\ \end{array} \]

Math

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

↓

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

FPCore

(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))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))
        INFINITY)
     (/ (pow (* c (* x s)) -2.0) (/ 1.0 t_0))
     (* t_0 (/ -1.0 (- (pow (* x (* c s)) 2.0)))))))

C

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

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));
}

↓

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

Python

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

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

↓

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

MATLAB

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));
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = ((c * (x * s)) ^ -2.0) / (1.0 / t_0);
	else
		tmp = t_0 * (-1.0 / -((x * (c * s)) ^ 2.0));
	end
	tmp_2 = tmp;
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]

↓

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

Error

Bits error versus x

Bits error versus c

Bits error versus s

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 18.3

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

   2. Simplified 2.8

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

   3. Applied egg-rr 0.6

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

   4. Applied egg-rr 2.8

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

   5. Applied egg-rr 0.3

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

   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. Simplified 2.6

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

   3. Applied egg-rr 10.8

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

   4. Applied egg-rr 2.6

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

4. Final simplification 0.8

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

Reproduce

herbie shell --seed 2022159 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))