arccos

Average Accuracy: 100.0% → 100.0%

Time: 4.8s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (/ (- 1.0 x) (sqrt (- 1.0 (* x 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(((1.0 - x) / sqrt((1.0 - (x * 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(((1.0d0 - x) / sqrt((1.0d0 - (x * 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(((1.0 - x) / Math.sqrt((1.0 - (x * 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(((1.0 - x) / math.sqrt((1.0 - (x * 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(Float64(1.0 - x) / sqrt(Float64(1.0 - Float64(x * 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(((1.0 - x) / sqrt((1.0 - (x * 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[(1.0 - x), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $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%

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

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

3. Simplified 100.0%

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

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

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13376

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

Alternative 2
Accuracy	99.5%
Cost	7232

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

Alternative 3
Accuracy	99.3%
Cost	7104

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

Alternative 4
Accuracy	99.0%
Cost	6976

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

Alternative 5
Accuracy	99.0%
Cost	6720

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

Alternative 6
Accuracy	98.0%
Cost	6592

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

Reproduce

herbie shell --seed 2023140 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))