Average Error: 0.0 → 0.0
Time: 3.2s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)}}\right) \]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)}}\right)

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (* 2.0 (atan (sqrt (exp (- (log1p (- x)) (log1p x)))))))
double code(double x) {
	return 2.0 * atan(sqrt((1.0 - x) / (1.0 + x)));
}
↓
double code(double x) {
	return 2.0 * atan(sqrt(exp(log1p(-x) - log1p(x))));
}

Error

Bits error versus x

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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 add-exp-log_binary640.0

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

  3. Applied add-exp-log_binary640.0

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

  4. Applied div-exp_binary640.0

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

  5. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{e^{\color{blue}{\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)}}}\right) \]

  6. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)}}\right) \]

Reproduce

herbie shell --seed 2021275 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))