Average Error: 15.1 → 14.6
Time: 1.9m
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{{\left({\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3}\right)}^{3} - {\left({\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3}\right)}^{3}}{{\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} \cdot {\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} + \left({\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3} \cdot {\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} + {\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3} \cdot {\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3}\right)}}{\mathsf{fma}\left(1 \cdot \left(1 \cdot 1\right), 1 \cdot \left(1 \cdot 1\right), \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}, 1 \cdot \left(1 \cdot 1\right)\right)\right) \cdot \mathsf{fma}\left(1, \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}, \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{{\left({\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3}\right)}^{3} - {\left({\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3}\right)}^{3}}{{\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} \cdot {\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} + \left({\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3} \cdot {\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} + {\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3} \cdot {\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3}\right)}}{\mathsf{fma}\left(1 \cdot \left(1 \cdot 1\right), 1 \cdot \left(1 \cdot 1\right), \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}, 1 \cdot \left(1 \cdot 1\right)\right)\right) \cdot \mathsf{fma}\left(1, \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}, \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}
double f(double x) {
        double r15353705 = 1.0;
        double r15353706 = 0.5;
        double r15353707 = x;
        double r15353708 = hypot(r15353705, r15353707);
        double r15353709 = r15353705 / r15353708;
        double r15353710 = r15353705 + r15353709;
        double r15353711 = r15353706 * r15353710;
        double r15353712 = sqrt(r15353711);
        double r15353713 = r15353705 - r15353712;
        return r15353713;
}
↓
double f(double x) {
        double r15353714 = 1.0;
        double r15353715 = r15353714 * r15353714;
        double r15353716 = r15353714 * r15353715;
        double r15353717 = 3.0;
        double r15353718 = pow(r15353716, r15353717);
        double r15353719 = pow(r15353718, r15353717);
        double r15353720 = x;
        double r15353721 = hypot(r15353714, r15353720);
        double r15353722 = r15353714 / r15353721;
        double r15353723 = r15353714 + r15353722;
        double r15353724 = 0.5;
        double r15353725 = r15353723 * r15353724;
        double r15353726 = sqrt(r15353725);
        double r15353727 = r15353725 * r15353726;
        double r15353728 = pow(r15353727, r15353717);
        double r15353729 = pow(r15353728, r15353717);
        double r15353730 = r15353719 - r15353729;
        double r15353731 = r15353718 * r15353718;
        double r15353732 = r15353728 * r15353718;
        double r15353733 = r15353728 * r15353728;
        double r15353734 = r15353732 + r15353733;
        double r15353735 = r15353731 + r15353734;
        double r15353736 = r15353730 / r15353735;
        double r15353737 = fma(r15353725, r15353726, r15353716);
        double r15353738 = r15353727 * r15353737;
        double r15353739 = fma(r15353716, r15353716, r15353738);
        double r15353740 = fma(r15353714, r15353714, r15353725);
        double r15353741 = fma(r15353714, r15353726, r15353740);
        double r15353742 = r15353739 * r15353741;
        double r15353743 = r15353736 / r15353742;
        return r15353743;
}

Error

Bits error versus x

Derivation

  1. Initial program 15.1

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

  3. Applied flip3--15.4

    \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} + 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}}\]

  4. Simplified15.1

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

  5. Simplified14.6

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

  7. Applied flip3--15.1

    \[\leadsto \frac{\color{blue}{\frac{{\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3} - {\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + \left(\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)}}}{\mathsf{fma}\left(1, \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right)}\]

  8. Applied associate-/l/15.1

    \[\leadsto \color{blue}{\frac{{\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3} - {\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{\mathsf{fma}\left(1, \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \left(\left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(1 \cdot 1\right) \cdot 1\right) + \left(\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \cdot \left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right) + \left(\left(1 \cdot 1\right) \cdot 1\right) \cdot \left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\right)}}\]

  9. Simplified14.6

    \[\leadsto \frac{{\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3} - {\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(1, \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \mathsf{fma}\left(\left(1 \cdot 1\right) \cdot 1, \left(1 \cdot 1\right) \cdot 1, \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right), \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}, \left(1 \cdot 1\right) \cdot 1\right)\right)}}\]
  10. Using strategy rm

  11. Applied flip3--14.6

    \[\leadsto \frac{\color{blue}{\frac{{\left({\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3}\right)}^{3} - {\left({\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}\right)}^{3}}{{\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3} \cdot {\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3} + \left({\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3} \cdot {\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3} + {\left(\left(1 \cdot 1\right) \cdot 1\right)}^{3} \cdot {\left(\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}\right)}}}{\mathsf{fma}\left(1, \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}, \mathsf{fma}\left(1, 1, 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \mathsf{fma}\left(\left(1 \cdot 1\right) \cdot 1, \left(1 \cdot 1\right) \cdot 1, \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)\right)\right) \cdot \mathsf{fma}\left(0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right), \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}, \left(1 \cdot 1\right) \cdot 1\right)\right)}\]

  12. Final simplification14.6

    \[\leadsto \frac{\frac{{\left({\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3}\right)}^{3} - {\left({\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3}\right)}^{3}}{{\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} \cdot {\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} + \left({\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3} \cdot {\left(1 \cdot \left(1 \cdot 1\right)\right)}^{3} + {\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3} \cdot {\left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right)}^{3}\right)}}{\mathsf{fma}\left(1 \cdot \left(1 \cdot 1\right), 1 \cdot \left(1 \cdot 1\right), \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}\right) \cdot \mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}, 1 \cdot \left(1 \cdot 1\right)\right)\right) \cdot \mathsf{fma}\left(1, \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}, \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))