mixedcos

Percentage Accurate: 66.6% → 97.5%

Time: 16.0s

Precision: binary64

Cost: 13572

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

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

• 30.8% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;s \leq 4 \cdot 10^{+198}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}}\\ \end{array} \]

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 (cos (+ x x))))
   (if (<= s 4e+198)
     (* (pow (* s (* x c)) -2.0) t_0)
     (/ t_0 (pow (* x (* s 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) {
	double t_0 = cos((x + x));
	double tmp;
	if (s <= 4e+198) {
		tmp = pow((s * (x * c)), -2.0) * t_0;
	} else {
		tmp = t_0 / pow((x * (s * c)), 2.0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (s <= 4d+198) then
        tmp = ((s * (x * c)) ** (-2.0d0)) * t_0
    else
        tmp = t_0 / ((x * (s * c)) ** 2.0d0)
    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 t_0 = Math.cos((x + x));
	double tmp;
	if (s <= 4e+198) {
		tmp = Math.pow((s * (x * c)), -2.0) * t_0;
	} else {
		tmp = t_0 / Math.pow((x * (s * c)), 2.0);
	}
	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):
	t_0 = math.cos((x + x))
	tmp = 0
	if s <= 4e+198:
		tmp = math.pow((s * (x * c)), -2.0) * t_0
	else:
		tmp = t_0 / math.pow((x * (s * c)), 2.0)
	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)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (s <= 4e+198)
		tmp = Float64((Float64(s * Float64(x * c)) ^ -2.0) * t_0);
	else
		tmp = Float64(t_0 / (Float64(x * Float64(s * c)) ^ 2.0));
	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)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (s <= 4e+198)
		tmp = ((s * (x * c)) ^ -2.0) * t_0;
	else
		tmp = t_0 / ((x * (s * c)) ^ 2.0);
	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_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[s, 4e+198], N[(N[Power[N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 / N[Power[N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 4.00000000000000007e198

   1. Initial program 61.5%

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

   2. Simplified 97.5%

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

      [Start] 61.5	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      *-commutative [=>] 61.5	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
      associate-*r* [=>] 52.3	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot {s}^{2}\right)}} \]
      associate-*r* [=>] 55.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}}} \]
      unpow2 [=>] 55.0	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot x\right)\right) \cdot {s}^{2}} \]
      unswap-sqr [=>] 74.0	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot {s}^{2}} \]
      unpow2 [=>] 74.0	\[ \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]
      swap-sqr [<=] 97.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(c \cdot x\right) \cdot s\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)}} \]
      *-commutative [<=] 97.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(s \cdot \left(c \cdot x\right)\right)} \cdot \left(\left(c \cdot x\right) \cdot s\right)} \]
      *-commutative [<=] 97.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(c \cdot x\right)\right) \cdot \color{blue}{\left(s \cdot \left(c \cdot x\right)\right)}} \]
      *-commutative [=>] 97.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \color{blue}{\left(x \cdot c\right)}\right) \cdot \left(s \cdot \left(c \cdot x\right)\right)} \]
      *-commutative [=>] 97.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

   3. Applied egg-rr 97.7%

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

      [Start] 97.5	\[ \frac{\cos \left(2 \cdot x\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \]
      clear-num [=>] 97.4	\[ \color{blue}{\frac{1}{\frac{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)}{\cos \left(2 \cdot x\right)}}} \]
      associate-/r/ [=>] 97.4	\[ \color{blue}{\frac{1}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot \left(x \cdot c\right)\right)} \cdot \cos \left(2 \cdot x\right)} \]
      pow2 [=>] 97.4	\[ \frac{1}{\color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{2}}} \cdot \cos \left(2 \cdot x\right) \]
      pow-flip [=>] 97.7	\[ \color{blue}{{\left(s \cdot \left(x \cdot c\right)\right)}^{\left(-2\right)}} \cdot \cos \left(2 \cdot x\right) \]
      metadata-eval [=>] 97.7	\[ {\left(s \cdot \left(x \cdot c\right)\right)}^{\color{blue}{-2}} \cdot \cos \left(2 \cdot x\right) \]
      cos-2 [=>] 98.0	\[ {\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \color{blue}{\left(\cos x \cdot \cos x - \sin x \cdot \sin x\right)} \]
      cos-sum [<=] 97.7	\[ {\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \color{blue}{\cos \left(x + x\right)} \]

   if 4.00000000000000007e198 < s

   1. Initial program 68.5%

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

   2. Simplified 68.4%

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

      [Start] 68.5	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      associate-*r* [=>] 68.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right) \cdot x}} \]
      *-commutative [=>] 68.5	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
      associate-*r* [=>] 68.4	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot {s}^{2}\right)}} \]
      unpow2 [=>] 68.4	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot {s}^{2}\right)} \]
      unpow2 [=>] 68.4	\[ \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   3. Taylor expanded in x around inf 63.9%

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

   4. Simplified 99.7%

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

      [Start] 63.9	\[ \frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      count-2 [<=] 63.9	\[ \frac{\cos \color{blue}{\left(x + x\right)}}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      associate-*r* [=>] 63.9	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left({s}^{2} \cdot {c}^{2}\right) \cdot {x}^{2}}} \]
      unpow2 [=>] 63.9	\[ \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right) \cdot {x}^{2}} \]
      unpow2 [=>] 63.9	\[ \frac{\cos \left(x + x\right)}{\left(\left(s \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot {x}^{2}} \]
      unswap-sqr [=>] 86.5	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right)} \cdot {x}^{2}} \]
      unpow2 [=>] 86.5	\[ \frac{\cos \left(x + x\right)}{\left(\left(s \cdot c\right) \cdot \left(s \cdot c\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]
      swap-sqr [<=] 99.7	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)}} \]
      rem-square-sqrt [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\left(\left(s \cdot c\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)} \]
      associate-*l* [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(s \cdot c\right) \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)} \cdot \left(\left(s \cdot c\right) \cdot x\right)} \]
      associate-*r* [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(s \cdot \left(c \cdot \sqrt{x}\right)\right)} \cdot \sqrt{x}\right) \cdot \left(\left(s \cdot c\right) \cdot x\right)} \]
      rem-square-sqrt [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\left(\left(s \cdot \left(c \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right) \cdot \left(\left(s \cdot c\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)} \]
      associate-*l* [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\left(\left(s \cdot \left(c \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\left(s \cdot c\right) \cdot \sqrt{x}\right) \cdot \sqrt{x}\right)}} \]
      associate-*r* [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\left(\left(s \cdot \left(c \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(s \cdot \left(c \cdot \sqrt{x}\right)\right)} \cdot \sqrt{x}\right)} \]
      unpow2 [<=] 27.2	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(s \cdot \left(c \cdot \sqrt{x}\right)\right) \cdot \sqrt{x}\right)}^{2}}} \]
      associate-*r* [=>] 27.2	\[ \frac{\cos \left(x + x\right)}{{\left(\color{blue}{\left(\left(s \cdot c\right) \cdot \sqrt{x}\right)} \cdot \sqrt{x}\right)}^{2}} \]
      associate-*l* [=>] 27.1	\[ \frac{\cos \left(x + x\right)}{{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)\right)}}^{2}} \]
      rem-square-sqrt [=>] 99.7	\[ \frac{\cos \left(x + x\right)}{{\left(\left(s \cdot c\right) \cdot \color{blue}{x}\right)}^{2}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4 \cdot 10^{+198}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot \cos \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{{\left(x \cdot \left(s \cdot c\right)\right)}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.5%
Cost	13572

\[\begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ t_1 := \cos \left(x + x\right)\\ \mathbf{if}\;s \leq 2 \cdot 10^{+191}:\\ \;\;\;\;{\left(s \cdot \left(x \cdot c\right)\right)}^{-2} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \end{array} \]

Alternative 2
Accuracy	83.1%
Cost	7888

\[\begin{array}{l} t_0 := \frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot c\right)\right)\right)}\\ t_1 := c \cdot \left(s \cdot x\right)\\ t_2 := \frac{\frac{1}{c}}{s \cdot x}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot c\right) \cdot \left(x \cdot \left|t_1\right|\right)}\\ \end{array} \]

Alternative 3
Accuracy	95.7%
Cost	7625

\[\begin{array}{l} t_0 := \frac{\frac{1}{c}}{s \cdot x}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-32} \lor \neg \left(x \leq 2.6 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(s \cdot \left(x \cdot c\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	7492

\[\begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ t_1 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 10^{+198}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_0}}{t_0}\\ \end{array} \]

Alternative 5
Accuracy	96.8%
Cost	7360

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

Alternative 6
Accuracy	78.2%
Cost	960

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

Alternative 7
Accuracy	67.2%
Cost	832

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

Alternative 8
Accuracy	75.2%
Cost	832

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

Alternative 9
Accuracy	78.1%
Cost	832

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

Reproduce

herbie shell --seed 2023160 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))