Average Error: 0.0 → 0.0
Time: 4.0s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x}}{\sqrt{1 + x}} \cdot \left(\sqrt[3]{1 - x} \cdot \frac{\sqrt[3]{1 - x}}{\sqrt{1 + x}}\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x}}{\sqrt{1 + x}} \cdot \left(\sqrt[3]{1 - x} \cdot \frac{\sqrt[3]{1 - x}}{\sqrt{1 + x}}\right)}\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) sqrt(((double) (((double) (((double) cbrt(((double) (1.0 - x)))) / ((double) sqrt(((double) (1.0 + x)))))) * ((double) (((double) cbrt(((double) (1.0 - x)))) * ((double) (((double) cbrt(((double) (1.0 - x)))) / ((double) sqrt(((double) (1.0 + x))))))))))))))));
}

Error

Bits error versus x

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Results

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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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))