Average Error: 15.4 → 14.9
Time: 3.5s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double code(double x) {
	return ((double) (1.0 - ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (1.0 / ((double) hypot(1.0, x))))))))))));
}
↓
double code(double x) {
	return ((double) (((double) exp(((double) log(((double) (((double) (1.0 * ((double) (1.0 - 0.5)))) - ((double) (0.5 * ((double) cbrt(((double) pow(((double) (1.0 / ((double) hypot(1.0, x)))), 3.0)))))))))))) / ((double) (1.0 + ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (1.0 / ((double) hypot(1.0, x))))))))))))));
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 15.4

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip--15.4

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  4. Simplified14.9

    \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube14.9

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  7. Applied add-cbrt-cube14.9

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  8. Applied cbrt-undiv14.9

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  9. Simplified14.9

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \sqrt[3]{\color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  10. Using strategy rm

  11. Applied add-exp-log14.9

    \[\leadsto \frac{\color{blue}{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  12. Final simplification14.9

    \[\leadsto \frac{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \sqrt[3]{{\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))