Average Error: 15.4 → 14.9
Time: 2.9m
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 r85187735 = 1.0;
        double r85187736 = 0.5;
        double r85187737 = x;
        double r85187738 = hypot(r85187735, r85187737);
        double r85187739 = r85187735 / r85187738;
        double r85187740 = r85187735 + r85187739;
        double r85187741 = r85187736 * r85187740;
        double r85187742 = sqrt(r85187741);
        double r85187743 = r85187735 - r85187742;
        return r85187743;
}
↓
double f(double x) {
        double r85187744 = 0.5;
        double r85187745 = 1.0;
        double r85187746 = x;
        double r85187747 = hypot(r85187745, r85187746);
        double r85187748 = r85187744 / r85187747;
        double r85187749 = r85187744 - r85187748;
        double r85187750 = r85187744 + r85187748;
        double r85187751 = sqrt(r85187750);
        double r85187752 = r85187751 + r85187745;
        double r85187753 = r85187749 / r85187752;
        return r85187753;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.4

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

  2. Simplified15.4

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

  4. Applied flip--15.4

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

  5. Simplified14.9

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

  6. Final simplification14.9

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

Reproduce

herbie shell --seed 2019125 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))