mixedcos

Average Error: 28.2 → 2.6

Time: 17.5s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

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 (* (cos (+ x x)) (pow (* s (* x c)) -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) {
	return cos((x + x)) * pow((s * (x * c)), -2.0);
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

↓

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((x + x)) * ((s * (x * c)) ** (-2.0d0))
end function

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) {
	return Math.cos((x + x)) * Math.pow((s * (x * c)), -2.0);
}

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):
	return math.cos((x + x)) * math.pow((s * (x * c)), -2.0)

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)
	return Float64(cos(Float64(x + x)) * (Float64(s * Float64(x * c)) ^ -2.0))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

↓

function tmp = code(x, c, s)
	tmp = cos((x + x)) * ((s * (x * c)) ^ -2.0);
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_] := N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] * N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.2

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

2. Simplified 3.0

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

   [Start] 28.2	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   *-commutative [=>] 28.2	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
   associate-*l* [=>] 31.5	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
   associate-*r* [=>] 31.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
   *-commutative [=>] 31.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
   unpow2 [=>] 31.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
   unpow2 [=>] 31.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
   unswap-sqr [=>] 19.3	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
   unswap-sqr [=>] 3.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

3. Taylor expanded in x around inf 31.5

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

4. Simplified 2.6

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

   [Start] 31.5	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]
   count-2 [<=] 31.5	\[ \frac{\cos \color{blue}{\left(x + x\right)}}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]
   associate-*r* [=>] 31.5	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]
   unpow2 [=>] 31.5	\[ \frac{\cos \left(x + x\right)}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]
   associate-/r* [=>] 31.5	\[ \color{blue}{\frac{\frac{\cos \left(x + x\right)}{{c}^{2} \cdot {s}^{2}}}{x \cdot x}} \]
   unpow2 [=>] 31.5	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}}}{x \cdot x} \]
   unpow2 [=>] 31.5	\[ \frac{\frac{\cos \left(x + x\right)}{\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}}}{x \cdot x} \]
   swap-sqr [<=] 19.4	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}}}{x \cdot x} \]
   unpow2 [<=] 19.4	\[ \frac{\frac{\cos \left(x + x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}}}}{x \cdot x} \]
   *-lft-identity [<=] 19.4	\[ \frac{\color{blue}{1 \cdot \frac{\cos \left(x + x\right)}{{\left(c \cdot s\right)}^{2}}}}{x \cdot x} \]
   associate-*l/ [<=] 19.6	\[ \color{blue}{\frac{1}{x \cdot x} \cdot \frac{\cos \left(x + x\right)}{{\left(c \cdot s\right)}^{2}}} \]
   unpow2 [=>] 19.6	\[ \frac{1}{x \cdot x} \cdot \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(c \cdot s\right)}} \]
   associate-/r* [=>] 19.5	\[ \frac{1}{x \cdot x} \cdot \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot s}}{c \cdot s}} \]
   times-frac [<=] 15.5	\[ \color{blue}{\frac{1 \cdot \frac{\cos \left(x + x\right)}{c \cdot s}}{\left(x \cdot x\right) \cdot \left(c \cdot s\right)}} \]
   *-commutative [<=] 15.5	\[ \frac{1 \cdot \frac{\cos \left(x + x\right)}{c \cdot s}}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot x\right)}} \]
   associate-*r* [=>] 6.7	\[ \frac{1 \cdot \frac{\cos \left(x + x\right)}{c \cdot s}}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot x}} \]
   *-commutative [<=] 6.7	\[ \frac{1 \cdot \frac{\cos \left(x + x\right)}{c \cdot s}}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot x} \]

5. Final simplification 2.6

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

Alternatives

Alternative 1
Error	9.8
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \lor \neg \left(x \leq 0.00062\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \end{array} \]

Alternative 2
Error	6.6
Cost	7625

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-7} \lor \neg \left(x \leq 0.00045\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(s \cdot t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_0}^{-2}\\ \end{array} \]

Alternative 3
Error	2.8
Cost	7625

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-114} \lor \neg \left(x \leq 1.5 \cdot 10^{-210}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot \left(x \cdot c\right)}{\frac{\frac{\frac{1}{c}}{x}}{s}}}\\ \end{array} \]

Alternative 4
Error	7.2
Cost	7624

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)\right)}\\ \mathbf{elif}\;x \leq 0.00045:\\ \;\;\;\;{t_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(c \cdot \left(s \cdot t_0\right)\right)}\\ \end{array} \]

Alternative 5
Error	2.8
Cost	7492

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_0}}{t_0}\\ \end{array} \]

Alternative 6
Error	3.0
Cost	7360

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

Alternative 7
Error	2.7
Cost	7360

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

Alternative 8
Error	16.9
Cost	6784

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

Alternative 9
Error	23.6
Cost	1228

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{if}\;s \leq -3.2 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 4 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{elif}\;s \leq 2.05 \cdot 10^{+170}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 10
Error	20.5
Cost	1161

\[\begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+140} \lor \neg \left(c \leq -1.35 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{c}}{\left(s \cdot c\right) \cdot \left(x \cdot \left(-s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	25.0
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-158} \lor \neg \left(x \leq 1.32 \cdot 10^{-143}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 12
Error	24.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-159} \lor \neg \left(x \leq 2.6 \cdot 10^{-143}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(c \cdot c\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	16.5
Cost	1092

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;s \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 14
Error	16.5
Cost	1092

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;s \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{1}{s}}{x \cdot c} \cdot \frac{1}{x \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 15
Error	16.5
Cost	1092

\[\begin{array}{l} t_0 := \frac{\frac{1}{x \cdot s}}{c}\\ \mathbf{if}\;s \leq 20:\\ \;\;\;\;\frac{\frac{1}{s}}{x \cdot c} \cdot \frac{1}{x \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 16
Error	18.3
Cost	960

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

Alternative 17
Error	17.1
Cost	960

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

Alternative 18
Error	17.1
Cost	960

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

Alternative 19
Error	29.0
Cost	832

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

Alternative 20
Error	17.0
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 2023060 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))