arccos

Average Error: 0.0 → 0.3

Time: 3.8s

Precision: binary64

Cost: 7232

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (/ (+ 1.0 (* (* x x) -0.5)) (+ 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 + ((x * x) * -0.5)) / (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 + ((x * x) * (-0.5d0))) / (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 + ((x * x) * -0.5)) / (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 + ((x * x) * -0.5)) / (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(Float64(1.0 + Float64(Float64(x * x) * -0.5)) / 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 + ((x * x) * -0.5)) / (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[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0

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

2. Applied egg-rr 0.0

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

3. Taylor expanded in x around 0 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ 2 \cdot \tan^{-1} \left(\frac{1 + -0.5 \cdot {x}^{2}}{1 + x}\right) \]
   *-commutative [=>] 0.3	\[ 2 \cdot \tan^{-1} \left(\frac{1 + \color{blue}{{x}^{2} \cdot -0.5}}{1 + x}\right) \]
   unpow2 [=>] 0.3	\[ 2 \cdot \tan^{-1} \left(\frac{1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5}{1 + x}\right) \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	7104

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

Alternative 2
Error	0.6
Cost	6720

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

Alternative 3
Error	1.4
Cost	6592

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

Reproduce

herbie shell --seed 2023041 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))