mixedcos

Average Error: 28.1 → 0.9

Time: 38.7s

Precision: binary64

Cost: 27460

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ t_2 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_2 \cdot t_2}\\ \end{array} \]

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))) (t_1 (* c (* x s))) (t_2 (* (* c s) x)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x)))
        INFINITY)
     (/ (/ t_0 t_1) t_1)
     (/ t_0 (* t_2 t_2)))))

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 t_1 = c * (x * s);
	double t_2 = (c * s) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 / (t_2 * t_2);
	}
	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 t_1 = c * (x * s);
	double t_2 = (c * s) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 / (t_2 * t_2);
	}
	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))
	t_1 = c * (x * s)
	t_2 = (c * s) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= math.inf:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = t_0 / (t_2 * t_2)
	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))
	t_1 = Float64(c * Float64(x * s))
	t_2 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= Inf)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(t_0 / Float64(t_2 * t_2));
	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));
	t_1 = c * (x * s);
	t_2 = (c * s) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= Inf)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = t_0 / (t_2 * t_2);
	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]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[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], Infinity], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

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 18.7

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

   2. Simplified 23.3

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

   3. Applied egg-rr 0.4

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

   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 64.0

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

   3. Applied egg-rr 56.4

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

   4. Applied egg-rr 25.3

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

   5. Applied egg-rr 3.0

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

Alternatives

Alternative 1
Error	14.3
Cost	7888

\[\begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot \left(c \cdot \left(s \cdot s\right)\right)\right) \cdot x\right)}\\ t_1 := \frac{\frac{\frac{1}{s}}{x}}{c}\\ t_2 := \frac{1}{s \cdot \left(x \cdot c\right)}\\ \mathbf{if}\;s \leq -1.45 \cdot 10^{+28}:\\ \;\;\;\;t_1 \cdot t_1\\ \mathbf{elif}\;s \leq -2.7 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 2.6 \cdot 10^{-158}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{elif}\;s \leq 6 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{c} \cdot \frac{-1}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 2
Error	5.8
Cost	7624

\[\begin{array}{l} t_0 := \frac{\cos \left(x + x\right)}{c \cdot \left(\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot s\right) \cdot x\right)}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.6
Cost	7492

\[\begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ t_1 := \cos \left(x + x\right)\\ t_2 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq 1.75 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_1}{t_2 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 4
Error	2.7
Cost	7360

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

Alternative 5
Error	23.8
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)}\\ \mathbf{if}\;c \leq 2.1 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	26.7
Cost	832

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

Alternative 7
Error	16.6
Cost	832

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{1}{t_0 \cdot t_0} \end{array} \]

Alternative 8
Error	16.5
Cost	832

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\frac{1}{t_0}}{t_0} \end{array} \]

Alternative 9
Error	16.4
Cost	832

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

Reproduce

herbie shell --seed 2023010 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))