Average Error: 15.3 → 14.9
Time: 27.2s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{1}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}} \cdot e^{\log \left(\frac{1 - 0.5}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}} \cdot \sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}} - \frac{\frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}}}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{1}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}} \cdot e^{\log \left(\frac{1 - 0.5}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}} \cdot \sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}} - \frac{\frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}}}{\sqrt[3]{1 + \sqrt{\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}}\right)}
double f(double x) {
        double r189949 = 1.0;
        double r189950 = 0.5;
        double r189951 = x;
        double r189952 = hypot(r189949, r189951);
        double r189953 = r189949 / r189952;
        double r189954 = r189949 + r189953;
        double r189955 = r189950 * r189954;
        double r189956 = sqrt(r189955);
        double r189957 = r189949 - r189956;
        return r189957;
}
↓
double f(double x) {
        double r189958 = 1.0;
        double r189959 = x;
        double r189960 = hypot(r189958, r189959);
        double r189961 = r189958 / r189960;
        double r189962 = r189958 + r189961;
        double r189963 = 0.5;
        double r189964 = r189962 * r189963;
        double r189965 = sqrt(r189964);
        double r189966 = r189958 + r189965;
        double r189967 = cbrt(r189966);
        double r189968 = r189958 / r189967;
        double r189969 = r189958 - r189963;
        double r189970 = r189967 * r189967;
        double r189971 = r189969 / r189970;
        double r189972 = r189963 / r189960;
        double r189973 = r189972 / r189967;
        double r189974 = r189973 / r189967;
        double r189975 = r189971 - r189974;
        double r189976 = log(r189975);
        double r189977 = exp(r189976);
        double r189978 = r189968 * r189977;
        return r189978;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

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

  3. Applied flip--15.3

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

  4. Simplified14.8

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

  5. Simplified14.8

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

  7. Applied div-sub14.8

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

  8. Simplified14.8

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

  9. Simplified14.8

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

  11. Applied add-cube-cbrt29.9

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

  12. Applied associate-/r/29.9

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

  13. Applied times-frac29.9

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

  14. Applied add-cube-cbrt29.9

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

  15. Applied times-frac14.9

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

  16. Applied distribute-rgt-out--14.9

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

  17. Simplified14.9

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

  19. Applied add-exp-log14.9

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

  20. Simplified14.9

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

  21. Final simplification14.9

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

Reproduce

herbie shell --seed 2019196 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))