mixedcos

Average Error: 28.6 → 27.4

Time: 14.8s

Precision: binary64

Cost: 20032

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

Math

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

↓

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

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

Derivation

1. Initial program 28.6

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

2. Simplified 28.8

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

   [Start] 28.6	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 28.6	\[ \frac{\cos \color{blue}{\left(x \cdot 2\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   metadata-eval [<=] 28.6	\[ \frac{\cos \left(x \cdot \color{blue}{\left(1 + 1\right)}\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   metadata-eval [<=] 28.6	\[ \frac{\cos \left(x \cdot \left(\color{blue}{\frac{2}{2}} + 1\right)\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   metadata-eval [<=] 28.6	\[ \frac{\cos \left(x \cdot \left(\frac{2}{2} + \color{blue}{\frac{2}{2}}\right)\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-23 [<=] 28.6	\[ \frac{\cos \color{blue}{\left(\frac{2}{2} \cdot x + x \cdot \frac{2}{2}\right)}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 28.6	\[ \frac{\cos \left(\color{blue}{x \cdot \frac{2}{2}} + x \cdot \frac{2}{2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   metadata-eval [=>] 28.6	\[ \frac{\cos \left(x \cdot \color{blue}{1} + x \cdot \frac{2}{2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 28.6	\[ \frac{\cos \left(\color{blue}{x} + x \cdot \frac{2}{2}\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   metadata-eval [=>] 28.6	\[ \frac{\cos \left(x + x \cdot \color{blue}{1}\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 28.6	\[ \frac{\cos \left(x + \color{blue}{x}\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 28.6	\[ \frac{\cos \left(x + x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.4	\[ \frac{\cos \left(x + x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 28.8	\[ \frac{\cos \left(x + x\right)}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

3. Taylor expanded in x around 0 27.4

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

4. Final simplification 27.4

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

Alternatives

Alternative 1
Error	27.4
Cost	20032

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

Alternative 2
Error	35.3
Cost	19840

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))