mixedcos

Average Accuracy: 55.3% → 95.8%

Time: 14.8s

Precision: binary64

Cost: 13440

Math

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

↓

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

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

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 = ((s * (x * c)) ** (-2.0d0)) * cos((x + x))
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.pow((s * (x * c)), -2.0) * Math.cos((x + x));
}

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

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(s * Float64(x * c)) ^ -2.0) * cos(Float64(x + x)))
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 = ((s * (x * c)) ^ -2.0) * cos((x + x));
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[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

2. Simplified 95.4%

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

   [Start] 55.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   *-commutative [=>] 55.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   associate-*r* [=>] 49.7	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}} \]
   associate-*r* [=>] 49.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}}} \]
   unpow2 [=>] 49.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}} \]
   unswap-sqr [=>] 68.2	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot {s}^{2}} \]
   unpow2 [=>] 68.2	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]
   swap-sqr [<=] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   *-commutative [<=] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(\left(c \cdot x\right) \cdot s\right)} \]
   *-commutative [<=] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]
   *-commutative [=>] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \color{blue}{\left(x \cdot c\right)}\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)} \]
   *-commutative [=>] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

3. Applied egg-rr 95.8%

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

   [Start] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \]
   clear-num [=>] 95.4	\[ \color{blue}{\frac{1}{\frac{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}{\cos \left(2 \cdot x\right)}}} \]
   associate-/r/ [=>] 95.4	\[ \color{blue}{\frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \cdot \cos \left(2 \cdot x\right)} \]
   pow2 [=>] 95.4	\[ \frac{1}{\color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}} \cdot \cos \left(2 \cdot x\right) \]
   pow-flip [=>] 95.8	\[ \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{\left(-2\right)}} \cdot \cos \left(2 \cdot x\right) \]
   metadata-eval [=>] 95.8	\[ {\left(s \cdot \left(x \cdot c\right)\right)}^{\color{blue}{-2}} \cdot \cos \left(2 \cdot x\right) \]
   cos-2 [=>] 95.7	\[ {\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \color{blue}{\left(\cos x \cdot \cos x - \sin x \cdot \sin x\right)} \]
   cos-sum [<=] 95.8	\[ {\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \color{blue}{\cos \left(x + x\right)} \]

4. Final simplification 95.8%

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

Alternatives

Alternative 1
Accuracy	77.1%
Cost	7756

\[\begin{array}{l} t_0 := c \cdot \left(s \cdot x\right)\\ t_1 := \frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{\frac{1}{s}}{x \cdot c}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 2
Accuracy	75.9%
Cost	7756

\[\begin{array}{l} t_0 := c \cdot \left(s \cdot x\right)\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(\left(x \cdot \left(c \cdot c\right)\right) \cdot \left(s \cdot s\right)\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{\frac{1}{s}}{x \cdot c}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{t_1}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 3
Accuracy	96.2%
Cost	7488

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

Alternative 4
Accuracy	95.8%
Cost	7488

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

Alternative 5
Accuracy	92.4%
Cost	7360

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

Alternative 6
Accuracy	95.4%
Cost	7360

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

Alternative 7
Accuracy	73.1%
Cost	960

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

Alternative 8
Accuracy	73.1%
Cost	832

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

Alternative 9
Accuracy	72.9%
Cost	832

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

Alternative 10
Accuracy	73.1%
Cost	832

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

Reproduce

herbie shell --seed 2023147 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))