mixedcos

Average Error: 28.0 → 2.9

Time: 15.4s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

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 (* 2.0 x)) (pow (* s (* x c)) 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 cos((2.0 * x)) / pow((s * (x * c)), 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 = cos((2.0d0 * x)) / ((s * (x * c)) ** 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 Math.cos((2.0 * x)) / Math.pow((s * (x * c)), 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 math.cos((2.0 * x)) / math.pow((s * (x * c)), 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(cos(Float64(2.0 * x)) / (Float64(s * Float64(x * c)) ^ 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 = cos((2.0 * x)) / ((s * (x * c)) ^ 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[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], 2.0], $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 13.7

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}} \]
   Proof
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (*.f64 (*.f64 c s) (*.f64 c s))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 s s)))))): 62 points increase in error, 11 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 s s))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 s 2)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 s 2) (pow.f64 c 2)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (pow.f64 s 2)) (pow.f64 c 2))))): 16 points increase in error, 14 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 2) (*.f64 x (pow.f64 s 2)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (pow.f64 c 2) (*.f64 x (pow.f64 s 2))) x))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))): 17 points increase in error, 9 points decrease in error

3. Applied egg-rr 2.9

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

4. Final simplification 2.9

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

Alternatives

Alternative 1
Error	3.8
Cost	7756

\[\begin{array}{l} t_0 := \frac{\frac{\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot s\right)}}{x}}{c \cdot s}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-140}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 2.756106127162585 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	4.0
Cost	7756

\[\begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x + x\right)}{t_1}}{x}}{c \cdot s}\\ \mathbf{elif}\;x \leq 10^{-232}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{elif}\;x \leq 2.756106127162585 \cdot 10^{+166}:\\ \;\;\;\;\frac{t_0}{\left(c \cdot s\right) \cdot \left(x \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	5.3
Cost	7624

\[\begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-15}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	5.7
Cost	7624

\[\begin{array}{l} t_0 := \frac{\frac{\frac{\cos \left(x + x\right)}{x \cdot \left(c \cdot s\right)}}{s}}{x \cdot c}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-125}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	2.7
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 6
Error	2.6
Cost	7360

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

Alternative 7
Error	16.8
Cost	6784

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

Alternative 8
Error	18.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 1000000000000:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot \left(c \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s}}{x \cdot c}}{x \cdot s}}{c}\\ \end{array} \]

Alternative 9
Error	17.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 7.109133160765928 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{s}}{x \cdot c}}{x \cdot s}}{c}\\ \end{array} \]

Alternative 10
Error	16.8
Cost	960

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

Alternative 11
Error	20.2
Cost	832

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

Alternative 12
Error	16.6
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	17.0
Cost	832

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

Alternative 14
Error	33.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))