arccos

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Precision: binary64

Cost: 13888

Math

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

↓

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

FPCore

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

↓

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

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\frac{x}{1 + x} \cdot 1\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}}\right) \]
   Step-by-step derivation

   [Start] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
   div-sub [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + x} - \frac{x}{1 + x}}}\right) \]
   flip-+ [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \frac{x}{1 + x}}\right) \]
   associate-/r/ [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(1 - x\right)} - \frac{x}{1 + x}}\right) \]
   *-un-lft-identity [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(1 - x\right) - \color{blue}{1 \cdot \frac{x}{1 + x}}}\right) \]
   prod-diff [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{1 \cdot 1 - x \cdot x}, 1 - x, -\frac{x}{1 + x} \cdot 1\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}}\right) \]
   metadata-eval [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{1}{\color{blue}{1} - x \cdot x}, 1 - x, -\frac{x}{1 + x} \cdot 1\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]

3. Simplified 100.0%

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

   [Start] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\frac{x}{1 + x} \cdot 1\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   *-rgt-identity [=>] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\color{blue}{\frac{x}{1 + x}}\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   fma-neg [<=] 100.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \frac{x}{1 + x}\right)} + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   rem-square-sqrt [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1 + x}\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   associate-*l/ [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \color{blue}{\frac{\sqrt{x}}{1 + x} \cdot \sqrt{x}}\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   /-rgt-identity [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \frac{\sqrt{x}}{1 + x} \cdot \color{blue}{\frac{\sqrt{x}}{1}}\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   unsub-neg [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right)} + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \frac{x}{1 + x} \cdot 1\right)}\right) \]
   *-rgt-identity [=>] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \mathsf{fma}\left(-\frac{x}{1 + x}, 1, \color{blue}{\frac{x}{1 + x}}\right)}\right) \]
   fma-def [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \color{blue}{\left(\left(-\frac{x}{1 + x}\right) \cdot 1 + \frac{x}{1 + x}\right)}}\right) \]
   *-rgt-identity [=>] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \left(\color{blue}{\left(-\frac{x}{1 + x}\right)} + \frac{x}{1 + x}\right)}\right) \]
   neg-mul-1 [=>] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \left(\color{blue}{-1 \cdot \frac{x}{1 + x}} + \frac{x}{1 + x}\right)}\right) \]
   distribute-lft1-in [=>] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{x}{1 + x}}}\right) \]
   rem-square-sqrt [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \left(-1 + 1\right) \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1 + x}}\right) \]
   associate-*l/ [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \left(-1 + 1\right) \cdot \color{blue}{\left(\frac{\sqrt{x}}{1 + x} \cdot \sqrt{x}\right)}}\right) \]
   /-rgt-identity [<=] 50.8	\[ 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) + \left(-\frac{\sqrt{x}}{1 + x} \cdot \frac{\sqrt{x}}{1}\right)\right) + \left(-1 + 1\right) \cdot \left(\frac{\sqrt{x}}{1 + x} \cdot \color{blue}{\frac{\sqrt{x}}{1}}\right)}\right) \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13504

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

Alternative 2
Accuracy	100.0%
Cost	13376

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

Alternative 3
Accuracy	99.6%
Cost	7360

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

Alternative 4
Accuracy	99.5%
Cost	7104

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

Alternative 5
Accuracy	99.1%
Cost	6976

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

Alternative 6
Accuracy	99.1%
Cost	6848

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

Alternative 7
Accuracy	99.1%
Cost	6720

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

Alternative 8
Accuracy	98.1%
Cost	6592

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

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))