Average Error: 15.3 → 14.8
Time: 18.0s
Precision: 64
\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}
double f(double x) {
        double r5462630 = 1.0;
        double r5462631 = 0.5;
        double r5462632 = x;
        double r5462633 = hypot(r5462630, r5462632);
        double r5462634 = r5462630 / r5462633;
        double r5462635 = r5462630 + r5462634;
        double r5462636 = r5462631 * r5462635;
        double r5462637 = sqrt(r5462636);
        double r5462638 = r5462630 - r5462637;
        return r5462638;
}
↓
double f(double x) {
        double r5462639 = 0.5;
        double r5462640 = 1.0;
        double r5462641 = x;
        double r5462642 = hypot(r5462640, r5462641);
        double r5462643 = r5462639 / r5462642;
        double r5462644 = r5462639 - r5462643;
        double r5462645 = r5462639 + r5462643;
        double r5462646 = sqrt(r5462645);
        double r5462647 = r5462646 + r5462640;
        double r5462648 = r5462644 / r5462647;
        return r5462648;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

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

  2. Simplified15.3

    \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  3. Using strategy rm

  4. Applied flip--15.3

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

  5. Simplified14.8

    \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity14.8

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

  8. Applied *-un-lft-identity14.8

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

  9. Applied distribute-lft-out14.8

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

  10. Applied associate-/r*14.8

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{1}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}}\]

  11. Simplified14.8

    \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}\]

  12. Final simplification14.8

    \[\leadsto \frac{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}} + 1}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))