mixedcos

Average Error: 27.7 → 26.1

Time: 20.0s

Precision: binary64

Cost: 20160

Math

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

↓

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

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 c 2.0)) (/ (pow s 2.0) (/ 1.0 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(c, 2.0)) / (pow(s, 2.0) / (1.0 / 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) / (c ** 2.0d0)) / ((s ** 2.0d0) / (1.0d0 / 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(c, 2.0)) / (Math.pow(s, 2.0) / (1.0 / 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(c, 2.0)) / (math.pow(s, 2.0) / (1.0 / 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(Float64(Float64(cos(Float64(x + x)) / x) / (c ^ 2.0)) / Float64((s ^ 2.0) / Float64(1.0 / 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) / (c ^ 2.0)) / ((s ^ 2.0) / (1.0 / 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[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[s, 2.0], $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.7

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

2. Simplified 26.3

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

   [Start] 27.7	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   rational.json-simplify-50 [=>] 27.7	\[ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   rational.json-simplify-24 [=>] 26.3	\[ \frac{\cos \left(2 \cdot x\right)}{\color{blue}{x \cdot \left({c}^{2} \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

3. Applied egg-rr 27.7

   \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} + 0} \]

4. Simplified 26.1

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

   [Start] 27.7	\[ \frac{\cos \left(x + x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} + 0 \]
   rational.json-simplify-8 [=>] 27.7	\[ \color{blue}{\frac{\cos \left(x + x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]
   rational.json-simplify-28 [=>] 28.0	\[ \color{blue}{\frac{\frac{\cos \left(x + x\right)}{{c}^{2}}}{x \cdot \left(x \cdot {s}^{2}\right)}} \]
   rational.json-simplify-28 [=>] 26.4	\[ \color{blue}{\frac{\frac{\frac{\cos \left(x + x\right)}{{c}^{2}}}{x}}{x \cdot {s}^{2}}} \]
   rational.json-simplify-15 [=>] 26.1	\[ \frac{\color{blue}{\frac{\frac{\cos \left(x + x\right)}{x}}{{c}^{2}}}}{x \cdot {s}^{2}} \]
   rational.json-simplify-50 [=>] 26.1	\[ \frac{\frac{\frac{\cos \left(x + x\right)}{x}}{{c}^{2}}}{\color{blue}{{s}^{2} \cdot x}} \]

5. Applied egg-rr 26.1

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

6. Final simplification 26.1

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

Alternatives

Alternative 1
Error	28.0
Cost	20032

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

Alternative 2
Error	26.3
Cost	20032

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

Alternative 3
Error	26.3
Cost	20032

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

Alternative 4
Error	26.2
Cost	20032

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

Alternative 5
Error	26.1
Cost	20032

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

Alternative 6
Error	34.6
Cost	13504

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

Alternative 7
Error	31.0
Cost	13504

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

Alternative 8
Error	31.0
Cost	13504

\[\frac{\frac{1}{{c}^{2} \cdot x}}{{s}^{2} \cdot x} \]

Alternative 9
Error	31.0
Cost	13504

\[\frac{\frac{\frac{1}{x}}{{c}^{2}}}{{s}^{2} \cdot x} \]

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))))