Average Error: 15.3 → 14.8
Time: 4.2s
Precision: binary64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{{\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{4} \cdot {\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2} - 0.5 \cdot 0.5}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{{\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{4} \cdot {\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2} - 0.5 \cdot 0.5}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}
double code(double x) {
	return ((double) (1.0 - ((double) sqrt(((double) (0.5 * ((double) (1.0 + (1.0 / ((double) hypot(1.0, x)))))))))));
}
↓
double code(double x) {
	return (((double) (1.0 * (((double) (((double) pow(((double) (1.0 - 0.5)), 3.0)) - ((double) pow((0.5 / ((double) hypot(1.0, x))), 3.0)))) / ((double) (((double) (((double) (1.0 - 0.5)) * ((double) (1.0 - 0.5)))) + ((double) ((0.5 / ((double) hypot(1.0, x))) * ((double) (1.0 + (((double) (((double) (((double) pow(((double) cbrt((0.5 / ((double) hypot(1.0, x))))), 4.0)) * ((double) pow(((double) cbrt((0.5 / ((double) hypot(1.0, x))))), 2.0)))) - ((double) (0.5 * 0.5)))) / ((double) (0.5 + (0.5 / ((double) hypot(1.0, x))))))))))))))) / ((double) (1.0 + ((double) sqrt(((double) (1.0 * ((double) (0.5 + (0.5 / ((double) hypot(1.0, x))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program Error: 15.3 bits

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

  3. Applied flip--Error: 15.3 bits

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

  4. SimplifiedError: 14.8 bits

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

  5. SimplifiedError: 14.8 bits

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

  7. Applied flip3--Error: 14.8 bits

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

  8. SimplifiedError: 14.8 bits

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

  10. Applied flip--Error: 14.8 bits

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

  11. SimplifiedError: 14.8 bits

    \[\leadsto \frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{\color{blue}{{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} - 0.5 \cdot 0.5}}{\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  12. SimplifiedError: 14.8 bits

    \[\leadsto \frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{{\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} - 0.5 \cdot 0.5}{\color{blue}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  13. Using strategy rm

  14. Applied add-cube-cbrtError: 14.8 bits

    \[\leadsto \frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{{\color{blue}{\left(\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}^{2} - 0.5 \cdot 0.5}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  15. Applied unpow-prod-downError: 14.8 bits

    \[\leadsto \frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{\color{blue}{{\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2} \cdot {\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2}} - 0.5 \cdot 0.5}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  16. SimplifiedError: 14.8 bits

    \[\leadsto \frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{\color{blue}{{\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{4}} \cdot {\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2} - 0.5 \cdot 0.5}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  17. Final simplificationError: 14.8 bits

    \[\leadsto \frac{1 \cdot \frac{{\left(1 - 0.5\right)}^{3} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1 - 0.5\right) \cdot \left(1 - 0.5\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(1 + \frac{{\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{4} \cdot {\left(\sqrt[3]{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2} - 0.5 \cdot 0.5}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))