Average Error: 15.5 → 15.0
Time: 44.5s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r330745 = 1.0;
        double r330746 = 0.5;
        double r330747 = x;
        double r330748 = hypot(r330745, r330747);
        double r330749 = r330745 / r330748;
        double r330750 = r330745 + r330749;
        double r330751 = r330746 * r330750;
        double r330752 = sqrt(r330751);
        double r330753 = r330745 - r330752;
        return r330753;
}
↓
double f(double x) {
        double r330754 = 1.0;
        double r330755 = r330754 * r330754;
        double r330756 = x;
        double r330757 = hypot(r330754, r330756);
        double r330758 = r330754 / r330757;
        double r330759 = r330754 + r330758;
        double r330760 = 0.5;
        double r330761 = r330759 * r330760;
        double r330762 = r330755 - r330761;
        double r330763 = r330760 * r330759;
        double r330764 = sqrt(r330763);
        double r330765 = r330754 + r330764;
        double r330766 = r330762 / r330765;
        return r330766;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.5

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

  3. Applied flip--15.5

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

  4. Simplified15.0

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

  5. Final simplification15.0

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))