mixedcos

Average Error: 44.19% → 1.23%

Time: 15.7s

Precision: binary64

Cost: 27588

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;\frac{t_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{c \cdot s}}{x \cdot \left(c \cdot s\right)} \cdot \cos \left(x + x\right)\\ \end{array} \]

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 (* c (* x s))) (t_1 (cos (* 2.0 x))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ (/ t_1 t_0) t_0)
     (* (/ (/ (/ 1.0 x) (* c s)) (* x (* c s))) (cos (+ x x))))))

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 = c * (x * s);
	double t_1 = cos((2.0 * x));
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = (((1.0 / x) / (c * s)) / (x * (c * s))) * cos((x + x));
	}
	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 = c * (x * s);
	double t_1 = Math.cos((2.0 * x));
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = (((1.0 / x) / (c * s)) / (x * (c * s))) * Math.cos((x + x));
	}
	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 = c * (x * s)
	t_1 = math.cos((2.0 * x))
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (t_1 / t_0) / t_0
	else:
		tmp = (((1.0 / x) / (c * s)) / (x * (c * s))) * math.cos((x + x))
	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(c * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / x) / Float64(c * s)) / Float64(x * Float64(c * s))) * cos(Float64(x + x)));
	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 = c * (x * s);
	t_1 = cos((2.0 * x));
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (t_1 / t_0) / t_0;
	else
		tmp = (((1.0 / x) / (c * s)) / (x * (c * s))) * cos((x + x));
	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[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 / 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$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision] / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(x + x), $MachinePrecision]], $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 28.85

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

   2. Simplified 25.91

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

      [Start] 28.85	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 25.91	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      unpow2 [=>] 25.91	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x} \]
      unpow2 [=>] 25.91	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right) \cdot x} \]

   3. Applied egg-rr 15.9

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

   4. Applied egg-rr 0.51

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

   5. Taylor expanded in x around inf 0.53

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

   6. Simplified 0.53

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

      [Start] 0.53	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot x\right)}}{\left(x \cdot s\right) \cdot c} \]
      *-commutative [=>] 0.53	\[ \frac{\frac{\cos \color{blue}{\left(x \cdot 2\right)}}{c \cdot \left(s \cdot x\right)}}{\left(x \cdot s\right) \cdot c} \]

   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 100

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

   2. Simplified 4.3

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

      [Start] 100	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 100	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 100	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 99.81	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 99.81	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 99.81	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
      unpow2 [=>] 99.81	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      unswap-sqr [=>] 39.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
      unswap-sqr [=>] 4.3	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Applied egg-rr 3.77

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

   4. Applied egg-rr 3.75

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

4. Final simplification 1.23

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

Alternatives

Alternative 1
Error	10.81%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-7} \lor \neg \left(x \leq 3.3 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 2
Error	6.18%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-42} \lor \neg \left(x \leq 6.5 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 3
Error	5.8%
Cost	7625

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(x \cdot c\right)}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-36} \lor \neg \left(x \leq 8.5 \cdot 10^{-198}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 4
Error	3.51%
Cost	7625

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ t_2 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;c \leq -6 \cdot 10^{+193} \lor \neg \left(c \leq -1.14 \cdot 10^{-163}\right):\\ \;\;\;\;\frac{t_2}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_2}{t_1}}{t_1}\\ \end{array} \]

Alternative 5
Error	3.61%
Cost	7625

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;c \leq -6.4 \cdot 10^{+192} \lor \neg \left(c \leq -1 \cdot 10^{-165}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(c \cdot \left(x \cdot s\right)\right)}}{x \cdot s}\\ \end{array} \]

Alternative 6
Error	25.02%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;s \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{elif}\;s \leq 8.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 7
Error	4.19%
Cost	7488

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

Alternative 8
Error	4.56%
Cost	7360

\[\begin{array}{l} t_0 := x \cdot \left(c \cdot s\right)\\ \frac{\cos \left(2 \cdot x\right)}{t_0 \cdot t_0} \end{array} \]

Alternative 9
Error	35.02%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-131} \lor \neg \left(x \leq 3 \cdot 10^{-158}\right):\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(c \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	25.69%
Cost	964

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 1.6 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \]

Alternative 11
Error	25.56%
Cost	964

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{1}{t_1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \]

Alternative 12
Error	25.55%
Cost	964

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 4 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternative 13
Error	44.24%
Cost	832

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

Alternative 14
Error	31.62%
Cost	832

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

Alternative 15
Error	26.46%
Cost	832

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

Reproduce

herbie shell --seed 2023125 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))