Average Error: 15.2 → 14.7
Time: 4.0s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1} + 1}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{1 \cdot \left(\left(1 \cdot 1 - 0.5 \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 + 1\right)\right)}{\left(1 + 0.5\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\sqrt{0.5} \cdot \sqrt{1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} + 1} + 1}
double f(double x) {
        double r285509 = 1.0;
        double r285510 = 0.5;
        double r285511 = x;
        double r285512 = hypot(r285509, r285511);
        double r285513 = r285509 / r285512;
        double r285514 = r285509 + r285513;
        double r285515 = r285510 * r285514;
        double r285516 = sqrt(r285515);
        double r285517 = r285509 - r285516;
        return r285517;
}
↓
double f(double x) {
        double r285518 = 1.0;
        double r285519 = r285518 * r285518;
        double r285520 = 0.5;
        double r285521 = r285520 * r285520;
        double r285522 = r285519 - r285521;
        double r285523 = x;
        double r285524 = hypot(r285518, r285523);
        double r285525 = r285522 * r285524;
        double r285526 = r285520 + r285518;
        double r285527 = r285520 * r285526;
        double r285528 = r285525 - r285527;
        double r285529 = r285518 * r285528;
        double r285530 = r285518 + r285520;
        double r285531 = r285530 * r285524;
        double r285532 = r285529 / r285531;
        double r285533 = sqrt(r285520);
        double r285534 = 1.0;
        double r285535 = r285534 / r285524;
        double r285536 = r285518 * r285535;
        double r285537 = r285536 + r285518;
        double r285538 = sqrt(r285537);
        double r285539 = r285533 * r285538;
        double r285540 = r285539 + r285518;
        double r285541 = r285532 / r285540;
        return r285541;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

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

  3. Applied flip--15.2

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

  4. Simplified14.7

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

  6. Applied associate-*r/14.7

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

  7. Applied flip--14.7

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

  8. Applied associate-*r/14.7

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

  9. Applied frac-sub14.7

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

  10. Simplified14.7

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

  11. Taylor expanded around 0 14.7

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

  12. Final simplification14.7

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

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))