arccos

Average Accuracy: 100.0% → 100.0%

Time: 4.4s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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((sqrt((1.0d0 - (x * x))) / (1.0d0 + x)))
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.sqrt((1.0 - (x * x))) / (1.0 + x)));
}

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.sqrt((1.0 - (x * x))) / (1.0 + x)))

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(sqrt(Float64(1.0 - Float64(x * x))) / Float64(1.0 + x))))
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((sqrt((1.0 - (x * x))) / (1.0 + x)));
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[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
   frac-2neg [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{-\left(1 - x\right)}{-\left(1 + x\right)}}}\right) \]
   div-inv [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(-\left(1 - x\right)\right) \cdot \frac{1}{-\left(1 + x\right)}}}\right) \]
   *-commutative [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{1}{-\left(1 + x\right)} \cdot \left(-\left(1 - x\right)\right)}}\right) \]
   neg-sub0 [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{0 - \left(1 + x\right)}} \cdot \left(-\left(1 - x\right)\right)}\right) \]
   metadata-eval [<=] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{\log 1} - \left(1 + x\right)} \cdot \left(-\left(1 - x\right)\right)}\right) \]
   associate--r+ [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{\left(\log 1 - 1\right) - x}} \cdot \left(-\left(1 - x\right)\right)}\right) \]
   metadata-eval [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\left(\color{blue}{0} - 1\right) - x} \cdot \left(-\left(1 - x\right)\right)}\right) \]
   metadata-eval [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{-1} - x} \cdot \left(-\left(1 - x\right)\right)}\right) \]
   neg-sub0 [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{-1 - x} \cdot \color{blue}{\left(0 - \left(1 - x\right)\right)}}\right) \]
   metadata-eval [<=] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{-1 - x} \cdot \left(\color{blue}{\log 1} - \left(1 - x\right)\right)}\right) \]
   associate--r- [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{-1 - x} \cdot \color{blue}{\left(\left(\log 1 - 1\right) + x\right)}}\right) \]
   metadata-eval [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{-1 - x} \cdot \left(\left(\color{blue}{0} - 1\right) + x\right)}\right) \]
   metadata-eval [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{-1 - x} \cdot \left(\color{blue}{-1} + x\right)}\right) \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{-1 - x} \cdot \left(-1 + x\right)}\right) \]
   *-commutative [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(-1 + x\right) \cdot \frac{1}{-1 - x}}}\right) \]
   flip-+ [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{-1 \cdot -1 - x \cdot x}{-1 - x}} \cdot \frac{1}{-1 - x}}\right) \]
   frac-times [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{\left(-1 \cdot -1 - x \cdot x\right) \cdot 1}{\left(-1 - x\right) \cdot \left(-1 - x\right)}}}\right) \]
   associate-/r* [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{\frac{\left(-1 \cdot -1 - x \cdot x\right) \cdot 1}{-1 - x}}{-1 - x}}}\right) \]
   metadata-eval [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{\left(\color{blue}{1} - x \cdot x\right) \cdot 1}{-1 - x}}{-1 - x}}\right) \]

4. Taylor expanded in x around 0 100.0%

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

5. Simplified 100.0%

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

   [Start] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{1}{1 + x} \cdot \sqrt{1 - {x}^{2}}\right) \]
   associate-*l/ [=>] 100.0	\[ 2 \cdot \tan^{-1} \color{blue}{\left(\frac{1 \cdot \sqrt{1 - {x}^{2}}}{1 + x}\right)} \]
   unpow2 [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{1 \cdot \sqrt{1 - \color{blue}{x \cdot x}}}{1 + x}\right) \]
   *-lft-identity [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{\color{blue}{\sqrt{1 - x \cdot x}}}{1 + x}\right) \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13376

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

Alternative 2
Accuracy	99.6%
Cost	7232

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

Alternative 3
Accuracy	99.4%
Cost	7104

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

Alternative 4
Accuracy	99.1%
Cost	6720

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

Alternative 5
Accuracy	98.1%
Cost	6592

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

Reproduce

herbie shell --seed 2023133 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))