mixedcos

Average Error: 28.4 → 2.4

Time: 24.3s

Precision: binary64

Cost: 13832

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

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := {\left(c \cdot \left(s \cdot x\right)\right)}^{2}\\ \mathbf{if}\;c \leq -7.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{t_1} \cdot t_0\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \end{array} \]

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))) (t_1 (pow (* c (* s x)) 2.0)))
   (if (<= c -7.9e-162)
     (* (/ 1.0 t_1) t_0)
     (if (<= c 2.25e-237)
       (* (/ 1.0 (pow (* s (* c x)) 2.0)) t_0)
       (/ t_0 t_1)))))

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 t_1 = pow((c * (s * x)), 2.0);
	double tmp;
	if (c <= -7.9e-162) {
		tmp = (1.0 / t_1) * t_0;
	} else if (c <= 2.25e-237) {
		tmp = (1.0 / pow((s * (c * x)), 2.0)) * t_0;
	} else {
		tmp = t_0 / t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = cos((x + x))
    t_1 = (c * (s * x)) ** 2.0d0
    if (c <= (-7.9d-162)) then
        tmp = (1.0d0 / t_1) * t_0
    else if (c <= 2.25d-237) then
        tmp = (1.0d0 / ((s * (c * x)) ** 2.0d0)) * t_0
    else
        tmp = t_0 / t_1
    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 t_1 = Math.pow((c * (s * x)), 2.0);
	double tmp;
	if (c <= -7.9e-162) {
		tmp = (1.0 / t_1) * t_0;
	} else if (c <= 2.25e-237) {
		tmp = (1.0 / Math.pow((s * (c * x)), 2.0)) * t_0;
	} else {
		tmp = t_0 / t_1;
	}
	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))
	t_1 = math.pow((c * (s * x)), 2.0)
	tmp = 0
	if c <= -7.9e-162:
		tmp = (1.0 / t_1) * t_0
	elif c <= 2.25e-237:
		tmp = (1.0 / math.pow((s * (c * x)), 2.0)) * t_0
	else:
		tmp = t_0 / t_1
	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))
	t_1 = Float64(c * Float64(s * x)) ^ 2.0
	tmp = 0.0
	if (c <= -7.9e-162)
		tmp = Float64(Float64(1.0 / t_1) * t_0);
	elseif (c <= 2.25e-237)
		tmp = Float64(Float64(1.0 / (Float64(s * Float64(c * x)) ^ 2.0)) * t_0);
	else
		tmp = Float64(t_0 / t_1);
	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));
	t_1 = (c * (s * x)) ^ 2.0;
	tmp = 0.0;
	if (c <= -7.9e-162)
		tmp = (1.0 / t_1) * t_0;
	elseif (c <= 2.25e-237)
		tmp = (1.0 / ((s * (c * x)) ^ 2.0)) * t_0;
	else
		tmp = t_0 / t_1;
	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]}, Block[{t$95$1 = N[Power[N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[c, -7.9e-162], N[(N[(1.0 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[c, 2.25e-237], N[(N[(1.0 / N[Power[N[(s * N[(c * x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.9000000000000002e-162

   1. Initial program 24.8

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

   2. Simplified 13.0

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

      [Start] 24.8	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      rational.json-simplify-46 [=>] 24.9	\[ \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]
      rational.json-simplify-46 [=>] 23.2	\[ \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x}} \]
      rational.json-simplify-47 [=>] 23.1	\[ \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}}{x} \]
      rational.json-simplify-43 [=>] 24.5	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}}{x} \]
      exponential.json-simplify-27 [=>] 13.0	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{{\left(s \cdot c\right)}^{2}}}}{x} \]

   3. Taylor expanded in x around inf 28.6

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

   4. Simplified 2.1

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

      [Start] 28.6	\[ \frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      metadata-eval [<=] 28.6	\[ \frac{\cos \left(\color{blue}{\left(1 + 1\right)} \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-7 [<=] 28.6	\[ \frac{\cos \left(\left(1 + 1\right) \cdot \color{blue}{\frac{x}{1}}\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-30 [=>] 28.6	\[ \frac{\cos \color{blue}{\left(x + \frac{x}{1}\right)}}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-7 [=>] 28.6	\[ \frac{\cos \left(x + \color{blue}{x}\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-2 [=>] 28.6	\[ \frac{\cos \left(x + x\right)}{{s}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {c}^{2}\right)}} \]
      rational.json-simplify-43 [<=] 28.5	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
      exponential.json-simplify-27 [=>] 15.3	\[ \frac{\cos \left(x + x\right)}{{c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}} \]
      exponential.json-simplify-27 [=>] 2.1	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]

   5. Applied egg-rr 2.1

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

   if -7.9000000000000002e-162 < c < 2.25000000000000005e-237

   1. Initial program 64.0

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

   2. Simplified 20.2

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

      [Start] 64.0	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      rational.json-simplify-46 [=>] 64.0	\[ \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]
      rational.json-simplify-46 [=>] 64.0	\[ \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x}} \]
      rational.json-simplify-47 [=>] 64.0	\[ \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}}{x} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}}{x} \]
      exponential.json-simplify-27 [=>] 20.2	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{{\left(s \cdot c\right)}^{2}}}}{x} \]

   3. Taylor expanded in x around inf 64.0

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

   4. Simplified 12.7

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

      [Start] 64.0	\[ \frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      metadata-eval [<=] 64.0	\[ \frac{\cos \left(\color{blue}{\left(1 + 1\right)} \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-7 [<=] 64.0	\[ \frac{\cos \left(\left(1 + 1\right) \cdot \color{blue}{\frac{x}{1}}\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-30 [=>] 64.0	\[ \frac{\cos \color{blue}{\left(x + \frac{x}{1}\right)}}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-7 [=>] 64.0	\[ \frac{\cos \left(x + \color{blue}{x}\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\cos \left(x + x\right)}{{s}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {c}^{2}\right)}} \]
      rational.json-simplify-43 [<=] 64.0	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
      exponential.json-simplify-27 [=>] 64.0	\[ \frac{\cos \left(x + x\right)}{{c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}} \]
      exponential.json-simplify-27 [=>] 12.7	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]

   5. Applied egg-rr 12.7

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

   6. Taylor expanded in c around 0 6.0

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

   if 2.25000000000000005e-237 < c

   1. Initial program 23.1

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

   2. Simplified 11.7

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

      [Start] 23.1	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
      rational.json-simplify-46 [=>] 23.6	\[ \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{\left(x \cdot {s}^{2}\right) \cdot x}} \]
      rational.json-simplify-46 [=>] 22.1	\[ \color{blue}{\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{{c}^{2}}}{x \cdot {s}^{2}}}{x}} \]
      rational.json-simplify-47 [=>] 21.6	\[ \frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot {s}^{2}\right)}}}{x} \]
      rational.json-simplify-43 [=>] 23.3	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({s}^{2} \cdot {c}^{2}\right)}}}{x} \]
      exponential.json-simplify-27 [=>] 11.7	\[ \frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \color{blue}{{\left(s \cdot c\right)}^{2}}}}{x} \]

   3. Taylor expanded in x around inf 27.1

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

   4. Simplified 1.5

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

      [Start] 27.1	\[ \frac{\cos \left(2 \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      metadata-eval [<=] 27.1	\[ \frac{\cos \left(\color{blue}{\left(1 + 1\right)} \cdot x\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-7 [<=] 27.1	\[ \frac{\cos \left(\left(1 + 1\right) \cdot \color{blue}{\frac{x}{1}}\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-30 [=>] 27.1	\[ \frac{\cos \color{blue}{\left(x + \frac{x}{1}\right)}}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-7 [=>] 27.1	\[ \frac{\cos \left(x + \color{blue}{x}\right)}{{s}^{2} \cdot \left({c}^{2} \cdot {x}^{2}\right)} \]
      rational.json-simplify-2 [=>] 27.1	\[ \frac{\cos \left(x + x\right)}{{s}^{2} \cdot \color{blue}{\left({x}^{2} \cdot {c}^{2}\right)}} \]
      rational.json-simplify-43 [<=] 27.4	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
      exponential.json-simplify-27 [=>] 16.8	\[ \frac{\cos \left(x + x\right)}{{c}^{2} \cdot \color{blue}{{\left(s \cdot x\right)}^{2}}} \]
      exponential.json-simplify-27 [=>] 1.5	\[ \frac{\cos \left(x + x\right)}{\color{blue}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}} \cdot \cos \left(x + x\right)\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{1}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}} \cdot \cos \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	20036

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;{c}^{2} \leq 0:\\ \;\;\;\;\frac{t_0}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}\\ \end{array} \]

Alternative 2
Error	2.3
Cost	13704

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := {\left(c \cdot \left(s \cdot x\right)\right)}^{2}\\ \mathbf{if}\;c \leq -6 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{t_1} \cdot t_0\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_0}{{\left(s \cdot \left(c \cdot x\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \end{array} \]

Alternative 3
Error	3.1
Cost	13440

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

Alternative 4
Error	16.8
Cost	6912

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

Reproduce

herbie shell --seed 2023074 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))