Average Error: 15.5 → 15.0
Time: 19.8s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot 1 - 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} - 1 \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)\right)\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot 1 - 0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}{{1}^{3} + {\left(\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{3}} \cdot \left(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)\right)
double f(double x) {
        double r154210 = 1.0;
        double r154211 = 0.5;
        double r154212 = x;
        double r154213 = hypot(r154210, r154212);
        double r154214 = r154210 / r154213;
        double r154215 = r154210 + r154214;
        double r154216 = r154211 * r154215;
        double r154217 = sqrt(r154216);
        double r154218 = r154210 - r154217;
        return r154218;
}
↓
double f(double x) {
        double r154219 = 1.0;
        double r154220 = r154219 * r154219;
        double r154221 = 0.5;
        double r154222 = x;
        double r154223 = hypot(r154219, r154222);
        double r154224 = r154219 / r154223;
        double r154225 = r154219 + r154224;
        double r154226 = r154221 * r154225;
        double r154227 = r154220 - r154226;
        double r154228 = 3.0;
        double r154229 = pow(r154219, r154228);
        double r154230 = sqrt(r154226);
        double r154231 = pow(r154230, r154228);
        double r154232 = r154229 + r154231;
        double r154233 = r154227 / r154232;
        double r154234 = r154230 * r154230;
        double r154235 = r154219 * r154230;
        double r154236 = r154234 - r154235;
        double r154237 = r154220 + r154236;
        double r154238 = r154233 * r154237;
        return r154238;
}

Error

Bits error versus x

Try it out

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

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

  6. Applied flip3-+15.5

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

  7. Applied associate-/r/15.0

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

  8. Final simplification15.0

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

Reproduce

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