Average Error: 15.8 → 15.4
Time: 6.3s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\log \left(e^{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\log \left(e^{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \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)}}
double f(double x) {
        double r204489 = 1.0;
        double r204490 = 0.5;
        double r204491 = x;
        double r204492 = hypot(r204489, r204491);
        double r204493 = r204489 / r204492;
        double r204494 = r204489 + r204493;
        double r204495 = r204490 * r204494;
        double r204496 = sqrt(r204495);
        double r204497 = r204489 - r204496;
        return r204497;
}
↓
double f(double x) {
        double r204498 = 1.0;
        double r204499 = 0.5;
        double r204500 = r204498 - r204499;
        double r204501 = r204498 * r204500;
        double r204502 = x;
        double r204503 = hypot(r204498, r204502);
        double r204504 = r204498 / r204503;
        double r204505 = r204499 * r204504;
        double r204506 = r204501 - r204505;
        double r204507 = exp(r204506);
        double r204508 = log(r204507);
        double r204509 = r204498 + r204504;
        double r204510 = r204499 * r204509;
        double r204511 = sqrt(r204510);
        double r204512 = r204498 + r204511;
        double r204513 = r204508 / r204512;
        return r204513;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.8

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

  3. Applied flip--15.8

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

  4. Simplified15.3

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

  6. Applied add-log-exp15.4

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

  7. Applied add-log-exp15.4

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

  8. Applied diff-log15.4

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

  9. Simplified15.4

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

  10. Final simplification15.4

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))