mixedcos

Percentage Accurate: 67.1% → 97.1%

Time: 12.1s

Precision: binary64

Cost: 7360

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

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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 (* s (* x c)))) (/ (cos (* 2.0 x)) (* t_0 t_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) {
	double t_0 = s * (x * c);
	return cos((2.0 * x)) / (t_0 * t_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
    real(8) :: t_0
    t_0 = s * (x * c)
    code = cos((2.0d0 * x)) / (t_0 * t_0)
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) {
	double t_0 = s * (x * c);
	return Math.cos((2.0 * x)) / (t_0 * t_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):
	t_0 = s * (x * c)
	return math.cos((2.0 * x)) / (t_0 * t_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)
	t_0 = Float64(s * Float64(x * c))
	return Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_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)
	t_0 = s * (x * c);
	tmp = cos((2.0 * x)) / (t_0 * t_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_] := Block[{t$95$0 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 72.7%

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

2. Simplified 98.7%

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

   [Start] 72.7%	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   *-commutative [=>] 72.7%	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   associate-*r* [=>] 68.9%	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}} \]
   associate-*r* [=>] 67.4%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}}} \]
   unpow2 [=>] 67.4%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}} \]
   unswap-sqr [=>] 80.8%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot {s}^{2}} \]
   unpow2 [=>] 80.8%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]
   swap-sqr [<=] 98.7%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
   *-commutative [<=] 98.7%	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(\left(c \cdot x\right) \cdot s\right)} \]
   *-commutative [<=] 98.7%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]
   *-commutative [=>] 98.7%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \color{blue}{\left(x \cdot c\right)}\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)} \]
   *-commutative [=>] 98.7%	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

3. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	97.1%
Cost	7360

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

Alternative 2
Accuracy	91.3%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-8} \lor \neg \left(x \leq 0.000225\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \end{array} \]

Alternative 3
Accuracy	93.9%
Cost	7625

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-11} \lor \neg \left(x \leq 0.00019\right):\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(c \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}\\ \end{array} \]

Alternative 4
Accuracy	79.9%
Cost	7624

\[\begin{array}{l} t_0 := \frac{\frac{\frac{1}{x}}{c}}{s}\\ \mathbf{if}\;c \leq -5.9 \cdot 10^{+85}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{elif}\;c \leq -7.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 5
Accuracy	95.0%
Cost	7360

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

Alternative 6
Accuracy	78.9%
Cost	6912

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

Alternative 7
Accuracy	78.8%
Cost	960

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

Alternative 8
Accuracy	78.7%
Cost	896

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

Alternative 9
Accuracy	55.4%
Cost	832

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

Alternative 10
Accuracy	67.2%
Cost	832

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

Alternative 11
Accuracy	75.8%
Cost	832

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

Alternative 12
Accuracy	75.8%
Cost	832

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

Alternative 13
Accuracy	76.9%
Cost	832

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

Reproduce

herbie shell --seed 2023167 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))