arccos

Average Accuracy: 100.0% → 100.0%

Time: 8.1s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

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

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

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

3. Simplified 100.0%

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

   [Start] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{1 + x}{1 - x}}}\right) \]
   +-commutative [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{\color{blue}{x + 1}}{1 - x}}}\right) \]
   metadata-eval [<=] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{x + \color{blue}{\left(--1\right)}}{1 - x}}}\right) \]
   sub-neg [<=] 100.0	\[ 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{\color{blue}{x - -1}}{1 - x}}}\right) \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13440

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

Alternative 2
Accuracy	100.0%
Cost	13376

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

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	6848

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

Alternative 5
Accuracy	99.1%
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 2023151 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))