Average Error: 0.0 → 0.0
Time: 3.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\log \left(e^{\frac{\sqrt{1 \cdot 1 - x \cdot x}}{\left|1 + x\right|}}\right)\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\log \left(e^{\frac{\sqrt{1 \cdot 1 - x \cdot x}}{\left|1 + x\right|}}\right)\right)
double code(double x) {
	return (2.0 * atan(sqrt(((1.0 - x) / (1.0 + x)))));
}
↓
double code(double x) {
	return (2.0 * atan(log(exp((sqrt(((1.0 * 1.0) - (x * x))) / fabs((1.0 + x)))))));
}

Error

Bits error versus x

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. Using strategy rm

  3. Applied flip--0.0

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

  4. Applied associate-/l/0.0

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

  5. Applied sqrt-div0.0

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

  6. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\frac{\sqrt{1 \cdot 1 - x \cdot x}}{\color{blue}{\left|1 + x\right|}}\right)\]
  7. Using strategy rm

  8. Applied add-log-exp0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))