arccos

Average Error: 0.02% → 0.02%

Time: 5.4s

Precision: binary64

Cost: 13440

Math

\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]

↓

\[2 \cdot \tan^{-1} \left({\left(\frac{-1 - x}{-1 + x}\right)}^{-0.5}\right) \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))

↓

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (pow (/ (- -1.0 x) (+ -1.0 x)) -0.5))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}

↓

double code(double x) {
	return 2.0 * atan(pow(((-1.0 - x) / (-1.0 + x)), -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(sqrt(((1.0d0 - x) / (1.0d0 + x))))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(((((-1.0d0) - x) / ((-1.0d0) + x)) ** (-0.5d0)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.sqrt(((1.0 - x) / (1.0 + x))));
}

↓

public static double code(double x) {
	return 2.0 * Math.atan(Math.pow(((-1.0 - x) / (-1.0 + x)), -0.5));
}

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))

↓

def code(x):
	return 2.0 * math.atan(math.pow(((-1.0 - x) / (-1.0 + x)), -0.5))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end

↓

function code(x)
	return Float64(2.0 * atan((Float64(Float64(-1.0 - x) / Float64(-1.0 + x)) ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end

↓

function tmp = code(x)
	tmp = 2.0 * atan((((-1.0 - x) / (-1.0 + x)) ^ -0.5));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(2.0 * N[ArcTan[N[Power[N[(N[(-1.0 - x), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.02

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]

2. Applied egg-rr 0.01

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{\frac{-1 - x}{-1 + x}}}\right)} \]

3. Simplified 0.01

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{\frac{-1 - x}{x + -1}}}\right)} \]
   Proof

   [Start] 0.01	\[ 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{-1 - x}{-1 + x}}}\right) \]
   +-commutative [=>] 0.01	\[ 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{-1 - x}{\color{blue}{x + -1}}}}\right) \]

4. Applied egg-rr 0.02

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\frac{-1 - x}{-1 + x}\right)}^{-0.5}\right)} \]

5. Simplified 0.02

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\frac{-1 - x}{x + -1}\right)}^{-0.5}\right)} \]
   Proof

   [Start] 0.02	\[ 2 \cdot \tan^{-1} \left({\left(\frac{-1 - x}{-1 + x}\right)}^{-0.5}\right) \]
   +-commutative [=>] 0.02	\[ 2 \cdot \tan^{-1} \left({\left(\frac{-1 - x}{\color{blue}{x + -1}}\right)}^{-0.5}\right) \]

6. Final simplification 0.02

   \[\leadsto 2 \cdot \tan^{-1} \left({\left(\frac{-1 - x}{-1 + x}\right)}^{-0.5}\right) \]

Alternatives

Alternative 1
Error	0.02%
Cost	13376

\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{x + 1}}\right) \]

Alternative 2
Error	0.47%
Cost	7488

\[2 \cdot \tan^{-1} \left(\frac{1 - x}{1 + \left(1 + \left(-1 + -0.5 \cdot \left(x \cdot x\right)\right)\right)}\right) \]

Alternative 3
Error	0.46%
Cost	7232

\[2 \cdot \tan^{-1} \left(\frac{1 - x}{1 + -0.5 \cdot \left(x \cdot x\right)}\right) \]

Alternative 4
Error	0.65%
Cost	7104

\[2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right) \]

Alternative 5
Error	0.98%
Cost	6720

\[2 \cdot \tan^{-1} \left(1 - x\right) \]

Alternative 6
Error	1.99%
Cost	6592

\[2 \cdot \tan^{-1} 1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))