Average Error: 0.0 → 0.0
Time: 7.8s
Precision: binary64
Cost: 13632
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{-1}{1 - x} \cdot \left(-1 - x\right)}}\right) \]
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (* 2.0 (atan (/ 1.0 (sqrt (* (/ -1.0 (- 1.0 x)) (- -1.0 x)))))))
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(((-1.0 / (1.0 - x)) * (-1.0 - x)))));
}

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((((-1.0d0) / (1.0d0 - x)) * ((-1.0d0) - x)))))
end function

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(((-1.0 / (1.0 - x)) * (-1.0 - x)))));
}

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(((-1.0 / (1.0 - x)) * (-1.0 - x)))))

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(-1.0 / Float64(1.0 - x)) * Float64(-1.0 - x))))))
end

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(((-1.0 / (1.0 - x)) * (-1.0 - x)))));
end

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[(-1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

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

  3. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\frac{1}{\sqrt{\frac{-1 - x}{x + -1}}}\right)} \]
    Proof
    (*.f64 2 (atan.f64 (/.f64 1 (sqrt.f64 (/.f64 (-.f64 -1 x) (+.f64 x -1)))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (atan.f64 (/.f64 1 (sqrt.f64 (/.f64 (-.f64 -1 x) (Rewrite<= +-commutative_binary64 (+.f64 -1 x))))))): 0 points increase in error, 0 points decrease in error

  4. Applied egg-rr0.0

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

  5. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\color{blue}{\frac{-1}{1 - x} \cdot \left(-1 - x\right)}}}\right) \]
    Proof
    (*.f64 2 (atan.f64 (/.f64 1 (sqrt.f64 (*.f64 (/.f64 -1 (-.f64 1 x)) (-.f64 -1 x)))))): 0 points increase in error, 0 points decrease in error
    (*.f64 2 (atan.f64 (/.f64 1 (sqrt.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 -1 x) (/.f64 -1 (-.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error

  6. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost13504
\[2 \cdot \tan^{-1} \left(\frac{1}{\sqrt{\frac{-1 - x}{-1 + x}}}\right) \]
Alternative 2
Error0.0
Cost13376
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
Alternative 3
Error0.4
Cost7360
\[2 \cdot \tan^{-1} \left(1 + \left(\left(x \cdot x\right) \cdot \left(0.5 + x \cdot -0.5\right) - x\right)\right) \]
Alternative 4
Error0.5
Cost7232
\[2 \cdot \tan^{-1} \left(-1 + \left(2 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right)\right) \]
Alternative 5
Error0.5
Cost7104
\[2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right) \]
Alternative 6
Error0.7
Cost6976
\[2 \cdot \tan^{-1} \left(1 + \left(1 + \left(-1 - x\right)\right)\right) \]
Alternative 7
Error0.7
Cost6720
\[2 \cdot \tan^{-1} \left(1 - x\right) \]
Alternative 8
Error1.3
Cost6592
\[2 \cdot \tan^{-1} 1 \]

Error

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))