mixedcos

Average Error: 43.78% → 2.04%

Time: 17.8s

Precision: binary64

Cost: 27588

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

Math

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

↓

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

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

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 = (((1.0 / c) / (x * s)) / (c * (x * s))) * t_0;
	} else {
		tmp = t_0 * (1.0 / (x * ((c * s) * (x * (c * s)))));
	}
	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 = (((1.0 / c) / (x * s)) / (c * (x * s))) * t_0;
	} else {
		tmp = t_0 * (1.0 / (x * ((c * s) * (x * (c * s)))));
	}
	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 = (((1.0 / c) / (x * s)) / (c * (x * s))) * t_0
	else:
		tmp = t_0 * (1.0 / (x * ((c * s) * (x * (c * s)))))
	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(Float64(Float64(1.0 / c) / Float64(x * s)) / Float64(c * Float64(x * s))) * t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x * Float64(Float64(c * s) * Float64(x * Float64(c * s))))));
	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 = (((1.0 / c) / (x * s)) / (c * (x * s))) * t_0;
	else
		tmp = t_0 * (1.0 / (x * ((c * s) * (x * (c * s)))));
	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[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x * N[(N[(c * s), $MachinePrecision] * N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 28.86

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

   2. Simplified 21.07

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

      [Start] 28.86	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 25.88	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      *-commutative [=>] 25.88	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
      *-commutative [=>] 25.88	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot x\right)}\right)} \]
      associate-*r* [=>] 29.14	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]
      *-commutative [=>] 29.14	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]
      unpow2 [=>] 29.13	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)\right)} \]
      unpow2 [=>] 29.13	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)} \]
      unswap-sqr [=>] 21.07	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right)} \]

   3. Applied egg-rr 4.81

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

   4. Applied egg-rr 25.67

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)} - 1\right)} \cdot \cos \left(x + x\right) \]

   5. Simplified 2.23

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

      [Start] 25.67	\[ \left(e^{\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)} - 1\right) \cdot \cos \left(x + x\right) \]
      expm1-def [=>] 5.98	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\right)\right)} \cdot \cos \left(x + x\right) \]
      expm1-log1p [=>] 4.37	\[ \color{blue}{{\left(x \cdot \left(c \cdot s\right)\right)}^{-2}} \cdot \cos \left(x + x\right) \]
      associate-*r* [=>] 2.23	\[ {\color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}}^{-2} \cdot \cos \left(x + x\right) \]
      *-commutative [=>] 2.23	\[ {\left(\color{blue}{\left(c \cdot x\right)} \cdot s\right)}^{-2} \cdot \cos \left(x + x\right) \]

   6. Applied egg-rr 0.47

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

   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 100

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

   2. Simplified 23.62

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

      [Start] 100	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 99.62	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      *-commutative [=>] 99.62	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
      *-commutative [=>] 99.62	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left({c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot x\right)}\right)} \]
      associate-*r* [=>] 99.41	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]
      *-commutative [=>] 99.41	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]
      unpow2 [=>] 99.41	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)\right)} \]
      unpow2 [=>] 99.41	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)} \]
      unswap-sqr [=>] 23.62	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right)} \]

   3. Applied egg-rr 5.08

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

   4. Applied egg-rr 7.92

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

4. Final simplification 2.04

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

Alternatives

Alternative 1
Error	14.47%
Cost	20553

\[\begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;{s}^{2} \leq 0 \lor \neg \left({s}^{2} \leq 5 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	15.67%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -0.000195 \lor \neg \left(x \leq 0.00015\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{c \cdot s}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 3
Error	11.05%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-14} \lor \neg \left(x \leq 1.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{c \cdot s}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 4
Error	6.05%
Cost	7625

\[\begin{array}{l} t_0 := \frac{\frac{1}{x \cdot s}}{c}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-174} \lor \neg \left(x \leq 10^{-126}\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 5
Error	4.85%
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 6
Error	34.14%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-151} \lor \neg \left(x \leq 3 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	33.81%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-151} \lor \neg \left(x \leq 3 \cdot 10^{-162}\right):\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)}\\ \end{array} \]

Alternative 8
Error	28.09%
Cost	964

\[\begin{array}{l} \mathbf{if}\;c \leq 7.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot c\right)}\\ \end{array} \]

Alternative 9
Error	27.99%
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	28.63%
Cost	964

\[\begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{+172}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{c \cdot \left(s \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	40.68%
Cost	832

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

Alternative 12
Error	26.16%
Cost	832

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

Alternative 13
Error	26.14%
Cost	832

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

Reproduce

herbie shell --seed 2023103 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))