Average Error: 15.3 → 14.8
Time: 3.8s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\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) (1.0 * ((double) (0.5 - ((double) (0.5 / ((double) hypot(1.0, x)))))))) / ((double) (1.0 + ((double) sqrt(((double) (1.0 * ((double) (0.5 + ((double) (0.5 / ((double) hypot(1.0, x))))))))))))));
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 15.3

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

  3. Applied flip--15.3

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

  4. Simplified14.8

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

  5. Simplified14.8

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

  6. Taylor expanded around 0 14.8

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

  7. Simplified14.8

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

  8. Final simplification14.8

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))