Average Error: 0.0 → 0.0
Time: 3.1s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 - x}{1 + x}\right)}^{3}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 - x}{1 + x}\right)}^{3}}}\right)
double code(double x) {
	return ((double) (2.0 * ((double) atan(((double) sqrt((((double) (1.0 - x)) / ((double) (1.0 + x)))))))));
}
↓
double code(double x) {
	return ((double) (2.0 * ((double) atan(((double) sqrt(((double) cbrt(((double) pow((((double) (1.0 - x)) / ((double) (1.0 + x))), 3.0))))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 0.0 bits

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

  3. Applied add-cbrt-cubeError: 0.0 bits

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}}}}\right)\]

  4. Applied add-cbrt-cubeError: 0.0 bits

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt[3]{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right) \cdot \left(1 - x\right)}}}{\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}}}\right)\]

  5. Applied cbrt-undivError: 0.0 bits

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{\frac{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right) \cdot \left(1 - x\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}}}}\right)\]

  6. SimplifiedError: 0.0 bits

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{1 - x}{1 + x}\right)}^{3}}}}\right)\]

  7. Final simplificationError: 0.0 bits

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 - x}{1 + x}\right)}^{3}}}\right)\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))