mixedcos

Average Error: 27.8 → 3.7

Time: 15.9s

Precision: binary64

Cost: 7625

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-171} \lor \neg \left(x \leq 6 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

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
 (if (or (<= x -1.1e-171) (not (<= x 6e-156)))
   (/ (cos (* x 2.0)) (* (* x (* c s)) (* c (* x s))))
   (/ (/ (/ 1.0 s) (* x c)) (* s (* x c)))))

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 tmp;
	if ((x <= -1.1e-171) || !(x <= 6e-156)) {
		tmp = cos((x * 2.0)) / ((x * (c * s)) * (c * (x * s)));
	} else {
		tmp = ((1.0 / s) / (x * c)) / (s * (x * c));
	}
	return tmp;
}

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) :: tmp
    if ((x <= (-1.1d-171)) .or. (.not. (x <= 6d-156))) then
        tmp = cos((x * 2.0d0)) / ((x * (c * s)) * (c * (x * s)))
    else
        tmp = ((1.0d0 / s) / (x * c)) / (s * (x * c))
    end if
    code = tmp
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 tmp;
	if ((x <= -1.1e-171) || !(x <= 6e-156)) {
		tmp = Math.cos((x * 2.0)) / ((x * (c * s)) * (c * (x * s)));
	} else {
		tmp = ((1.0 / s) / (x * c)) / (s * (x * c));
	}
	return tmp;
}

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):
	tmp = 0
	if (x <= -1.1e-171) or not (x <= 6e-156):
		tmp = math.cos((x * 2.0)) / ((x * (c * s)) * (c * (x * s)))
	else:
		tmp = ((1.0 / s) / (x * c)) / (s * (x * c))
	return tmp

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)
	tmp = 0.0
	if ((x <= -1.1e-171) || !(x <= 6e-156))
		tmp = Float64(cos(Float64(x * 2.0)) / Float64(Float64(x * Float64(c * s)) * Float64(c * Float64(x * s))));
	else
		tmp = Float64(Float64(Float64(1.0 / s) / Float64(x * c)) / Float64(s * Float64(x * c)));
	end
	return tmp
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

↓

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((x <= -1.1e-171) || ~((x <= 6e-156)))
		tmp = cos((x * 2.0)) / ((x * (c * s)) * (c * (x * s)));
	else
		tmp = ((1.0 / s) / (x * c)) / (s * (x * c));
	end
	tmp_2 = tmp;
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_] := If[Or[LessEqual[x, -1.1e-171], N[Not[LessEqual[x, 6e-156]], $MachinePrecision]], N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision] * N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / s), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001e-171 or 6e-156 < x

   1. Initial program 25.3

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

   2. Simplified 1.9

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

      [Start] 25.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 25.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]
      associate-*l* [=>] 26.7	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]
      associate-*r* [=>] 26.8	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]
      *-commutative [=>] 26.8	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot x\right) \cdot \left({c}^{2} \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 26.8	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}\right)} \]
      unpow2 [=>] 26.8	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]
      unswap-sqr [=>] 13.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}} \]
      unswap-sqr [=>] 1.9	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)}} \]

   3. Taylor expanded in s around 0 3.6

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

   if -1.1000000000000001e-171 < x < 6e-156

   1. Initial program 44.7

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

   2. Simplified 40.4

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

      [Start] 44.7	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 40.4	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      unpow2 [=>] 40.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x} \]
      unpow2 [=>] 40.4	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)\right) \cdot x} \]

   3. Taylor expanded in x around 0 63.4

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

   4. Simplified 30.5

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

      [Start] 63.4	\[ \frac{1}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      associate-*r* [=>] 63.4	\[ \frac{1}{\color{blue}{\left({s}^{2} \cdot {c}^{2}\right) \cdot {x}^{2}}} \]
      *-commutative [<=] 63.4	\[ \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right)} \cdot {x}^{2}} \]
      associate-*r* [<=] 63.5	\[ \frac{1}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
      unpow2 [=>] 63.5	\[ \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]
      sqr-pow [=>] 63.5	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left({s}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right)} \cdot {x}^{2}\right)} \]
      unpow2 [=>] 63.5	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left({s}^{\left(\frac{2}{2}\right)} \cdot {s}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]
      unswap-sqr [=>] 26.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left({s}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)}} \]
      metadata-eval [=>] 26.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left({s}^{\color{blue}{1}} \cdot x\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)} \]
      unpow1 [=>] 26.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(\color{blue}{s} \cdot x\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)} \]
      rem-square-sqrt [<=] 43.1	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left({s}^{\left(\frac{2}{2}\right)} \cdot x\right)\right)} \]
      metadata-eval [=>] 43.1	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \cdot \left({s}^{\color{blue}{1}} \cdot x\right)\right)} \]
      unpow1 [=>] 43.1	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \cdot \left(\color{blue}{s} \cdot x\right)\right)} \]
      rem-square-sqrt [<=] 43.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right) \cdot \left(s \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)} \]
      associate-*l* [<=] 43.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)} \cdot \left(s \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right)\right)} \]
      associate-*l* [<=] 43.2	\[ \frac{1}{\left(c \cdot c\right) \cdot \left(\left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(s \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)}\right)} \]

   5. Applied egg-rr 30.4

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\frac{\frac{1}{s}}{x \cdot s}}{c \cdot c}}}} \]

   6. Applied egg-rr 5.0

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

   7. Applied egg-rr 4.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-171} \lor \neg \left(x \leq 6 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.6
Cost	13440

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

Alternative 2
Error	6.6
Cost	7756

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ t_1 := \cos \left(x \cdot 2\right)\\ t_2 := \frac{t_1}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.9
Cost	7756

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ t_1 := \cos \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-76}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+193}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x \cdot \left(c \cdot \left(c \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	6.7
Cost	7625

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-8} \lor \neg \left(x \leq 1.15 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 5
Error	12.6
Cost	7624

\[\begin{array}{l} \mathbf{if}\;s \leq 1.6 \cdot 10^{-160}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2}\\ \mathbf{elif}\;s \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 6
Error	2.7
Cost	7616

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

Alternative 7
Error	3.1
Cost	7492

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 1.18 \cdot 10^{+179}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	2.7
Cost	7488

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

Alternative 9
Error	16.5
Cost	6784

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

Alternative 10
Error	23.8
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-158} \lor \neg \left(x \leq 1.75 \cdot 10^{-157}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	15.8
Cost	1096

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;c \leq -8.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-183}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 12
Error	21.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;c \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	16.5
Cost	960

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

Alternative 14
Error	16.5
Cost	960

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

Alternative 15
Error	28.2
Cost	832

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

Alternative 16
Error	19.4
Cost	832

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

Alternative 17
Error	16.6
Cost	832

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

Reproduce

herbie shell --seed 2023073 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))