Average Error: 0.0 → 0.0
Time: 3.8s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \log \left(e^{\sqrt{\frac{1 - x}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \log \left(e^{\sqrt{\frac{1 - x}{1 + x}}}\right)
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (* 2.0 (atan (log (exp (sqrt (/ (- 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(log(exp(sqrt((1.0 - x) / (1.0 + x)))));
}

Error

Bits error versus x

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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 add-log-exp_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020315 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))