mixedcos

Percentage Accurate: 66.3% → 97.1%

Time: 26.4s

Precision: binary64

Cost: 7488

• Unsound rule application detected in e-graph. Results may not be sound.

• 31.4% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \frac{x \cdot c}{\cos \left(x \cdot 2\right)}} \]

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
 (/ (/ (/ 1.0 s) (* x c)) (* s (/ (* x c) (cos (* x 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 ((1.0 / s) / (x * c)) / (s * ((x * c) / cos((x * 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 = ((1.0d0 / s) / (x * c)) / (s * ((x * c) / cos((x * 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 ((1.0 / s) / (x * c)) / (s * ((x * c) / Math.cos((x * 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 ((1.0 / s) / (x * c)) / (s * ((x * c) / math.cos((x * 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(Float64(Float64(1.0 / s) / Float64(x * c)) / Float64(s * Float64(Float64(x * c) / cos(Float64(x * 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 = ((1.0 / s) / (x * c)) / (s * ((x * c) / cos((x * 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[(N[(1.0 / s), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] / N[(s * N[(N[(x * c), $MachinePrecision] / N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

2. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]
   Step-by-step derivation

   [Start] 66.7%	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   *-commutative [=>] 66.7%	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   associate-*r* [=>] 59.8%	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}} \]
   associate-*r* [=>] 58.6%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}}} \]
   unpow2 [=>] 58.6%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}} \]
   unswap-sqr [=>] 77.1%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot {s}^{2}} \]
   unpow2 [=>] 77.1%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]
   swap-sqr [<=] 97.5%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   *-commutative [<=] 97.5%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(\left(c \cdot x\right) \cdot s\right)} \]
   *-commutative [<=] 97.5%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]
   *-commutative [=>] 97.5%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \color{blue}{\left(x \cdot c\right)}\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)} \]
   *-commutative [=>] 97.5%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

3. Applied egg-rr 98.1%

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

   [Start] 97.5%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \]
   div-inv [=>] 97.5%	\[ \color{blue}{\cos \left(2 \cdot x\right) \cdot \frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}} \]
   *-commutative [=>] 97.5%	\[ \cos \color{blue}{\left(x \cdot 2\right)} \cdot \frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \]
   pow2 [=>] 97.5%	\[ \cos \left(x \cdot 2\right) \cdot \frac{1}{\color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}} \]
   pow-flip [=>] 98.1%	\[ \cos \left(x \cdot 2\right) \cdot \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{\left(-2\right)}} \]
   metadata-eval [=>] 98.1%	\[ \cos \left(x \cdot 2\right) \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{\color{blue}{-2}} \]

4. Applied egg-rr 98.0%

   \[\leadsto \cos \left(x \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\right)} \]
   Step-by-step derivation

   [Start] 98.1%	\[ \cos \left(x \cdot 2\right) \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \]
   metadata-eval [<=] 98.1%	\[ \cos \left(x \cdot 2\right) \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{\color{blue}{\left(-1 + -1\right)}} \]
   pow-prod-up [<=] 98.0%	\[ \cos \left(x \cdot 2\right) \cdot \color{blue}{\left({\left(s \cdot \left(x \cdot c\right)\right)}^{-1} \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-1}\right)} \]
   unpow-1 [=>] 98.0%	\[ \cos \left(x \cdot 2\right) \cdot \left(\color{blue}{\frac{1}{s \cdot \left(x \cdot c\right)}} \cdot {\left(s \cdot \left(x \cdot c\right)\right)}^{-1}\right) \]
   unpow-1 [=>] 98.0%	\[ \cos \left(x \cdot 2\right) \cdot \left(\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \color{blue}{\frac{1}{s \cdot \left(x \cdot c\right)}}\right) \]

5. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{s}}{x \cdot c}}{\frac{x \cdot c}{\cos \left(x \cdot 2\right)} \cdot s}} \]
   Step-by-step derivation

   [Start] 98.0%	\[ \cos \left(x \cdot 2\right) \cdot \left(\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\right) \]
   associate-*r* [=>] 98.0%	\[ \color{blue}{\left(\cos \left(x \cdot 2\right) \cdot \frac{1}{s \cdot \left(x \cdot c\right)}\right) \cdot \frac{1}{s \cdot \left(x \cdot c\right)}} \]
   un-div-inv [=>] 98.0%	\[ \color{blue}{\frac{\cos \left(x \cdot 2\right) \cdot \frac{1}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}} \]
   *-commutative [=>] 98.0%	\[ \frac{\cos \left(x \cdot 2\right) \cdot \frac{1}{s \cdot \left(x \cdot c\right)}}{\color{blue}{\left(x \cdot c\right) \cdot s}} \]
   frac-times [<=] 93.2%	\[ \color{blue}{\frac{\cos \left(x \cdot 2\right)}{x \cdot c} \cdot \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{s}} \]
   clear-num [=>] 93.2%	\[ \color{blue}{\frac{1}{\frac{x \cdot c}{\cos \left(x \cdot 2\right)}}} \cdot \frac{\frac{1}{s \cdot \left(x \cdot c\right)}}{s} \]
   frac-times [=>] 98.1%	\[ \color{blue}{\frac{1 \cdot \frac{1}{s \cdot \left(x \cdot c\right)}}{\frac{x \cdot c}{\cos \left(x \cdot 2\right)} \cdot s}} \]
   *-un-lft-identity [<=] 98.1%	\[ \frac{\color{blue}{\frac{1}{s \cdot \left(x \cdot c\right)}}}{\frac{x \cdot c}{\cos \left(x \cdot 2\right)} \cdot s} \]
   associate-/r* [=>] 98.1%	\[ \frac{\color{blue}{\frac{\frac{1}{s}}{x \cdot c}}}{\frac{x \cdot c}{\cos \left(x \cdot 2\right)} \cdot s} \]

6. Final simplification 98.1%

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

Alternatives

Alternative 1
Accuracy	97.1%
Cost	7488

\[\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \frac{x \cdot c}{\cos \left(x \cdot 2\right)}} \]

Alternative 2
Accuracy	89.3%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00185 \lor \neg \left(x \leq 0.00105\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternative 3
Accuracy	89.4%
Cost	7624

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -0.0018:\\ \;\;\;\;\frac{t_0}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(s \cdot x\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(s \cdot \left(x \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	94.6%
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 5
Accuracy	96.8%
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 6
Accuracy	97.1%
Cost	7360

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

Alternative 7
Accuracy	76.5%
Cost	964

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

Alternative 8
Accuracy	54.6%
Cost	832

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

Alternative 9
Accuracy	54.5%
Cost	832

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

Alternative 10
Accuracy	74.9%
Cost	832

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

Alternative 11
Accuracy	77.4%
Cost	832

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

Alternative 12
Accuracy	78.6%
Cost	832

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))