mixedcos

Average Accuracy: 56.1% → 97.8%

Time: 19.9s

Precision: binary64

Cost: 79172

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ t_1 := {\left(c \cdot \left(x \cdot s\right)\right)}^{2}\\ t_2 := \cos \left(2 \cdot x\right)\\ t_3 := \cos \left(x + x\right)\\ \mathbf{if}\;\frac{t_2}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt[3]{\frac{t_3}{t_1}} \cdot \sqrt[3]{{t_3}^{2}}}{{\left(\sqrt[3]{t_1}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_0 \cdot t_0}\\ \end{array} \]

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 (* x (* c s)))
        (t_1 (pow (* c (* x s)) 2.0))
        (t_2 (cos (* 2.0 x)))
        (t_3 (cos (+ x x))))
   (if (<= (/ t_2 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ (* (cbrt (/ t_3 t_1)) (cbrt (pow t_3 2.0))) (pow (cbrt t_1) 2.0))
     (/ t_2 (* t_0 t_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 = x * (c * s);
	double t_1 = pow((c * (x * s)), 2.0);
	double t_2 = cos((2.0 * x));
	double t_3 = cos((x + x));
	double tmp;
	if ((t_2 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (cbrt((t_3 / t_1)) * cbrt(pow(t_3, 2.0))) / pow(cbrt(t_1), 2.0);
	} else {
		tmp = t_2 / (t_0 * t_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 = x * (c * s);
	double t_1 = Math.pow((c * (x * s)), 2.0);
	double t_2 = Math.cos((2.0 * x));
	double t_3 = Math.cos((x + x));
	double tmp;
	if ((t_2 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (Math.cbrt((t_3 / t_1)) * Math.cbrt(Math.pow(t_3, 2.0))) / Math.pow(Math.cbrt(t_1), 2.0);
	} else {
		tmp = t_2 / (t_0 * t_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 = Float64(x * Float64(c * s))
	t_1 = Float64(c * Float64(x * s)) ^ 2.0
	t_2 = cos(Float64(2.0 * x))
	t_3 = cos(Float64(x + x))
	tmp = 0.0
	if (Float64(t_2 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(cbrt(Float64(t_3 / t_1)) * cbrt((t_3 ^ 2.0))) / (cbrt(t_1) ^ 2.0));
	else
		tmp = Float64(t_2 / Float64(t_0 * t_0));
	end
	return 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[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[Power[N[(t$95$3 / t$95$1), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Power[t$95$3, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(t$95$2 / N[(t$95$0 * t$95$0), $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 71.0%

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

   2. Simplified 74.2%

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

      [Start] 71.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 74.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      unpow2 [=>] 74.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x} \]
      unpow2 [=>] 74.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right) \cdot x} \]

   3. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{{\cos \left(x + x\right)}^{2}}}{{\left(\sqrt[3]{x \cdot {\left(c \cdot \left(s \cdot \sqrt{x}\right)\right)}^{2}}\right)}^{2}} \cdot \sqrt[3]{\frac{\cos \left(x + x\right)}{x \cdot {\left(c \cdot \left(s \cdot \sqrt{x}\right)\right)}^{2}}}} \]
      Proof

      [Start] 74.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x} \]
      add-cube-cbrt [=>] 74.1	\[ \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x} \]
      add-cube-cbrt [=>] 73.9	\[ \frac{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\color{blue}{\left(\sqrt[3]{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x} \cdot \sqrt[3]{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x}\right) \cdot \sqrt[3]{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x}}} \]
      times-frac [=>] 73.9	\[ \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\sqrt[3]{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x} \cdot \sqrt[3]{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x}} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\sqrt[3]{\left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right) \cdot x}}} \]

   4. Simplified 98.4%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\cos \left(x + x\right)}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \cdot \sqrt[3]{{\cos \left(x + x\right)}^{2}}}{{\left(\sqrt[3]{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\right)}^{2}}} \]
      Proof

      [Start] 43.5	\[ \frac{\sqrt[3]{{\cos \left(x + x\right)}^{2}}}{{\left(\sqrt[3]{x \cdot {\left(c \cdot \left(s \cdot \sqrt{x}\right)\right)}^{2}}\right)}^{2}} \cdot \sqrt[3]{\frac{\cos \left(x + x\right)}{x \cdot {\left(c \cdot \left(s \cdot \sqrt{x}\right)\right)}^{2}}} \]
      associate-*l/ [=>] 43.5	\[ \color{blue}{\frac{\sqrt[3]{{\cos \left(x + x\right)}^{2}} \cdot \sqrt[3]{\frac{\cos \left(x + x\right)}{x \cdot {\left(c \cdot \left(s \cdot \sqrt{x}\right)\right)}^{2}}}}{{\left(\sqrt[3]{x \cdot {\left(c \cdot \left(s \cdot \sqrt{x}\right)\right)}^{2}}\right)}^{2}}} \]

   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 0.0%

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

   2. Simplified 95.6%

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

      [Start] 0.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 0.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 0.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 0.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 0.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
      unpow2 [=>] 0.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      unswap-sqr [=>] 65.1	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
      unswap-sqr [=>] 95.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\cos \left(x + x\right)}{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}} \cdot \sqrt[3]{{\cos \left(x + x\right)}^{2}}}{{\left(\sqrt[3]{{\left(c \cdot \left(x \cdot s\right)\right)}^{2}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.9%
Cost	13440

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

Alternative 2
Accuracy	81.4%
Cost	7888

\[\begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ t_1 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;s \leq -5.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\frac{\frac{c}{\frac{1}{x}}}{\frac{\frac{-1}{c \cdot \left(x \cdot s\right)}}{-s}}}\\ \mathbf{elif}\;s \leq -3.55 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 7.8 \cdot 10^{-159}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{elif}\;s \leq 1.06 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 3
Accuracy	90.3%
Cost	7756

\[\begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := \frac{t_0}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ t_2 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	96.3%
Cost	7753

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ t_1 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-100} \lor \neg \left(x \leq 2.55 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\cos \left(x + x\right)}{t_1} \cdot \frac{1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 5
Accuracy	89.9%
Cost	7625

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-7} \lor \neg \left(x \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 6
Accuracy	93.4%
Cost	7625

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-18} \lor \neg \left(x \leq 6.2 \cdot 10^{-234}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{t_0 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{-2}\\ \end{array} \]

Alternative 7
Accuracy	95.7%
Cost	7625

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ t_1 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-68} \lor \neg \left(x \leq 1.05 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 8
Accuracy	93.5%
Cost	7624

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ t_2 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{t_2}{\left(x \cdot \left(c \cdot s\right)\right) \cdot t_0}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-233}:\\ \;\;\;\;{t_1}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_1 \cdot t_0}\\ \end{array} \]

Alternative 9
Accuracy	68.5%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+182} \lor \neg \left(c \leq -4.9 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	69.3%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{elif}\;c \leq -7.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(c \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 11
Accuracy	70.8%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	74.6%
Cost	960

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

Alternative 13
Accuracy	61.9%
Cost	832

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

Alternative 14
Accuracy	74.3%
Cost	832

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

Alternative 15
Accuracy	74.6%
Cost	832

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

Reproduce

herbie shell --seed 2023138 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))