mixedcos

Average Error: 56.2% → 95.7%

Time: 15.7s

Precision: binary64

Cost: 7488.00

Math

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

↓

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

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 (* x (* c s)))) (* (/ (cos (+ x x)) t_0) (/ 1.0 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 = x * (c * s);
	return (cos((x + x)) / t_0) * (1.0 / 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 = x * (c * s)
    code = (cos((x + x)) / t_0) * (1.0d0 / 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 = x * (c * s);
	return (Math.cos((x + x)) / t_0) * (1.0 / 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 = x * (c * s)
	return (math.cos((x + x)) / t_0) * (1.0 / 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(x * Float64(c * s))
	return Float64(Float64(cos(Float64(x + x)) / t_0) * Float64(1.0 / 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 = x * (c * s);
	tmp = (cos((x + x)) / t_0) * (1.0 / 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[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.2

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

2. Simplified 95.4

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

   [Start] 56.2	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   *-commutative [=>] 56.2	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
   associate-*l* [=>] 50.6	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
   associate-*r* [=>] 50.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
   *-commutative [=>] 50.6	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
   unpow2 [=>] 50.6	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
   unpow2 [=>] 50.6	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
   unswap-sqr [=>] 69.0	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
   unswap-sqr [=>] 95.4	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

3. Applied egg-rr 95.7

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

4. Final simplification 95.7

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

Alternatives

Alternative 1
Error	79.3%
Cost	7625.00

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0014 \lor \neg \left(x \leq 0.011\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot s}}{x}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 2
Error	84.5%
Cost	7625.00

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00155 \lor \neg \left(x \leq 0.000205\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(s \cdot \left(s \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot s}}{x}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 3
Error	85.6%
Cost	7624.00

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -0.0014:\\ \;\;\;\;\frac{t_0}{x \cdot \left(s \cdot \left(s \cdot \left(c \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-116}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	95.6%
Cost	7360.00

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

Alternative 5
Error	95.4%
Cost	7360.00

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

Alternative 6
Error	69.5%
Cost	1097.00

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-132} \lor \neg \left(x \leq 500000000\right):\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(c \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 7
Error	50.1%
Cost	832.00

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

Alternative 8
Error	67.6%
Cost	832.00

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

Alternative 9
Error	69.3%
Cost	832.00

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

Alternative 10
Error	74.1%
Cost	832.00

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

Alternative 11
Error	73.9%
Cost	832.00

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

Reproduce

herbie shell --seed 2023093 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))