Average Error: 0.0 → 0.0
Time: 5.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\frac{1 + x}{\sqrt{1 - x}}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\frac{1 + x}{\sqrt{1 - x}}}}\right)
double code(double x) {
	return ((double) (2.0 * ((double) atan(((double) sqrt(((double) (((double) (1.0 - x)) / ((double) (1.0 + x))))))))));
}
↓
double code(double x) {
	return ((double) (2.0 * ((double) atan(((double) (((double) sqrt(((double) sqrt(((double) (1.0 - x)))))) / ((double) sqrt(((double) (((double) (1.0 + x)) / ((double) sqrt(((double) (1.0 - 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. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied associate-/l*0.0

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

  5. Applied sqrt-div0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))