arccos

Average Error: 0.0 → 0.0

Time: 8.0s

Precision: binary64

Cost: 14400

Math

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

↓

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

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) (- 1.0 x))
     (* (- 1.0 x) (/ (/ (- 1.0 x) (+ 1.0 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) / (1.0 - x)) * ((1.0 - x) * (((1.0 - x) / (1.0 + 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) / (1.0d0 - x)) * ((1.0d0 - x) * (((1.0d0 - x) / (1.0d0 + 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) / (1.0 - x)) * ((1.0 - x) * (((1.0 - x) / (1.0 + 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) / (1.0 - x)) * ((1.0 - x) * (((1.0 - x) / (1.0 + 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(sqrt(Float64(Float64(Float64(1.0 + x) / Float64(1.0 - x)) * Float64(Float64(1.0 - x) * Float64(Float64(Float64(1.0 - x) / Float64(1.0 + 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) / (1.0 - x)) * ((1.0 - x) * (((1.0 - x) / (1.0 + 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[Sqrt[N[(N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - x), $MachinePrecision] * N[(N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

3. Simplified 0.0

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

   [Start] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \frac{1}{\left(1 + x\right) \cdot \frac{\frac{1 + x}{1 - x}}{1 - x}}}\right) \]
   rational.json-simplify-46 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \color{blue}{\frac{\frac{1}{1 + x}}{\frac{\frac{1 + x}{1 - x}}{1 - x}}}}\right) \]
   metadata-eval [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \frac{\frac{\color{blue}{--1}}{1 + x}}{\frac{\frac{1 + x}{1 - x}}{1 - x}}}\right) \]
   rational.json-simplify-17 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \frac{\frac{--1}{\color{blue}{x - -1}}}{\frac{\frac{1 + x}{1 - x}}{1 - x}}}\right) \]
   rational.json-simplify-50 [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \frac{\color{blue}{\frac{-1}{-1 - x}}}{\frac{\frac{1 + x}{1 - x}}{1 - x}}}\right) \]
   rational.json-simplify-47 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \frac{\frac{-1}{-1 - x}}{\color{blue}{\frac{1 + x}{\left(1 - x\right) \cdot \left(1 - x\right)}}}}\right) \]
   rational.json-simplify-61 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \color{blue}{\frac{\left(1 - x\right) \cdot \left(1 - x\right)}{\frac{1 + x}{\frac{-1}{-1 - x}}}}}\right) \]
   rational.json-simplify-49 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \color{blue}{\left(\left(1 - x\right) \cdot \frac{1 - x}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}}\right) \]
   rational.json-simplify-7 [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\color{blue}{\frac{1 - x}{1}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   metadata-eval [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\frac{1 - x}{\color{blue}{\frac{1}{1}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   rational.json-simplify-61 [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\color{blue}{\frac{1}{\frac{1}{1 - x}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   rational.json-simplify-35 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\color{blue}{\frac{1 + 1}{\frac{1}{1 - x} + \frac{1}{1 - x}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   metadata-eval [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\frac{\color{blue}{2}}{\frac{1}{1 - x} + \frac{1}{1 - x}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   rational.json-simplify-29 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\frac{2}{\color{blue}{\frac{\left(1 - x\right) + \left(1 - x\right)}{\left(1 - x\right) \cdot \left(1 - x\right)}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   rational.json-simplify-61 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\color{blue}{\frac{\left(1 - x\right) \cdot \left(1 - x\right)}{\frac{\left(1 - x\right) + \left(1 - x\right)}{2}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   metadata-eval [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\frac{\left(1 - x\right) \cdot \left(1 - x\right)}{\frac{\left(1 - x\right) + \left(1 - x\right)}{\color{blue}{1 + 1}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   rational.json-simplify-35 [<=] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\frac{\left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{\frac{1 - x}{1}}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]
   rational.json-simplify-7 [=>] 0.0	\[ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 + x}{1 - x} \cdot \left(\left(1 - x\right) \cdot \frac{\frac{\left(1 - x\right) \cdot \left(1 - x\right)}{\color{blue}{1 - x}}}{\frac{1 + x}{\frac{-1}{-1 - x}}}\right)}\right) \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	13504

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

Alternative 2
Error	0.0
Cost	13376

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

Alternative 3
Error	0.6
Cost	6720

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

Alternative 4
Error	1.2
Cost	6592

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

Reproduce

herbie shell --seed 2023075 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))