mixedcos

Average Error: 28.0 → 2.7

Time: 15.6s

Precision: binary64

Cost: 7488

Math

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

↓

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

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)) (* x (* c s))) (/ (/ 1.0 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) {
	return (cos((x + x)) / (x * (c * s))) * ((1.0 / x) / (c * s));
}

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)) / (x * (c * s))) * ((1.0d0 / x) / (c * s))
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)) / (x * (c * s))) * ((1.0 / x) / (c * s));
}

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)) / (x * (c * s))) * ((1.0 / x) / (c * s))

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(Float64(cos(Float64(x + x)) / Float64(x * Float64(c * s))) * Float64(Float64(1.0 / x) / Float64(c * s)))
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)) / (x * (c * s))) * ((1.0 / x) / (c * s));
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[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.0

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

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] 28.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   *-commutative [=>] 28.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
   associate-*l* [=>] 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* [=>] 31.7	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
   *-commutative [=>] 31.7	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
   unpow2 [=>] 31.7	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
   unpow2 [=>] 31.7	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
   unswap-sqr [=>] 20.1	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
   unswap-sqr [=>] 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. Applied egg-rr 2.8

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

4. Applied egg-rr 2.7

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

5. Final simplification 2.7

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

Alternatives

Alternative 1
Error	10.0
Cost	7625

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(c \cdot \left(-s\right)\right)}\\ \mathbf{if}\;x \leq -1.48 \cdot 10^{-5} \lor \neg \left(x \leq 5.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 2
Error	7.0
Cost	7625

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(c \cdot \left(-s\right)\right)}\\ \mathbf{if}\;x \leq -1.06 \cdot 10^{-16} \lor \neg \left(x \leq 1.1 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 3
Error	5.5
Cost	7625

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;c \leq -2.3 \cdot 10^{-196} \lor \neg \left(c \leq 3.9 \cdot 10^{-218}\right):\\ \;\;\;\;\frac{t_0}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	7.6
Cost	7624

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(c \cdot \left(-s\right)\right)}\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-33}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	5.1
Cost	7624

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := \frac{\frac{1}{x \cdot s}}{c}\\ \mathbf{if}\;s \leq 1.9 \cdot 10^{+139}:\\ \;\;\;\;\frac{t_0}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{elif}\;s \leq 1.4 \cdot 10^{+204}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{t_1}}{t_1}}\\ \end{array} \]

Alternative 6
Error	4.5
Cost	7624

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := x \cdot \left(c \cdot s\right)\\ t_2 := \frac{\frac{1}{x \cdot s}}{c}\\ \mathbf{if}\;s \leq 8.8 \cdot 10^{-181}:\\ \;\;\;\;\frac{t_0}{t_1 \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{elif}\;s \leq 2.1 \cdot 10^{+204}:\\ \;\;\;\;\frac{t_0}{t_1 \cdot \left(s \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{t_2}}{t_2}}\\ \end{array} \]

Alternative 7
Error	2.8
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	3.0
Cost	7360

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

Alternative 9
Error	17.3
Cost	1088

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

Alternative 10
Error	26.6
Cost	832

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

Alternative 11
Error	17.2
Cost	832

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

Alternative 12
Error	17.1
Cost	832

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

Alternative 13
Error	41.2
Cost	576

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

Reproduce

herbie shell --seed 2023057 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))