mixedcos

Average Error: 27.8 → 2.5

Time: 12.2s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

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

Derivation

1. Initial program 27.8

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

2. Simplified 23.3

   \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(c \cdot \left(x \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}} \]
   Proof
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 c (*.f64 c (*.f64 x (*.f64 x (*.f64 s s)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 c (*.f64 c (*.f64 x (*.f64 x (Rewrite<= unpow2_binary64 (pow.f64 s 2))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 c (*.f64 c (Rewrite<= *-commutative_binary64 (*.f64 (*.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-*l*_binary64 (*.f64 (*.f64 c c) (*.f64 (*.f64 x (pow.f64 s 2)) x)))): 36 points increase in error, 3 points decrease in error
   (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 (*.f64 x (pow.f64 s 2)) x))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 13.4

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

4. Applied egg-rr 2.5

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

5. Final simplification 2.5

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

Alternatives

Alternative 1
Error	11.0
Cost	8548

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot s\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ t_2 := \frac{1}{t_1}\\ t_3 := \cos \left(x \cdot 2\right)\\ t_4 := \frac{t_3}{t_0 \cdot \left(c \cdot \left(c \cdot x\right)\right)}\\ t_5 := \frac{t_3}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ t_6 := \frac{1}{x \cdot \left(c \cdot s\right)}\\ \mathbf{if}\;c \leq -5 \cdot 10^{+226}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq -2.15 \cdot 10^{+153}:\\ \;\;\;\;t_2 \cdot \frac{1}{\frac{c}{\frac{1}{x \cdot s}}}\\ \mathbf{elif}\;c \leq -1.22 \cdot 10^{+36}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{t_3}{c \cdot \left(c \cdot \left(x \cdot t_0\right)\right)}\\ \mathbf{elif}\;c \leq -8.6 \cdot 10^{-203}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-178}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-128}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-116}:\\ \;\;\;\;t_6 \cdot t_6\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{-61}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \end{array} \]

Alternative 2
Error	8.8
Cost	7756

\[\begin{array}{l} t_0 := \cos \left(x \cdot 2\right)\\ t_1 := \frac{t_0}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_0}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.1
Cost	7624

\[\begin{array}{l} t_0 := \frac{\cos \left(x \cdot 2\right)}{c \cdot \left(c \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.5
Cost	7360

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

Alternative 5
Error	18.9
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-263}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{\left(s \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	17.8
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{if}\;x \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(\left(c \cdot s\right) \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	17.5
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(s \cdot \left(c \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	16.6
Cost	960

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

Alternative 9
Error	16.6
Cost	960

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

Alternative 10
Error	22.5
Cost	832

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

Alternative 11
Error	19.3
Cost	832

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

Alternative 12
Error	16.6
Cost	832

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

Reproduce

herbie shell --seed 2022338 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))