Average Error: 15.6 → 15.2
Time: 6.2s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{\frac{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right)}^{3}}{\left(0.5 \cdot \left(\frac{\sqrt{1}}{{\left(\sqrt[3]{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right) \cdot \left(0.5 \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{\frac{{\left(1 \cdot \left(1 - 0.5\right)\right)}^{3} - {\left(0.5 \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right)}^{3}}{\left(0.5 \cdot \left(\frac{\sqrt{1}}{{\left(\sqrt[3]{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right) \cdot \left(0.5 \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right) + 1 \cdot \left(1 - 0.5\right)\right) + \left(1 \cdot \left(1 - 0.5\right)\right) \cdot \left(1 \cdot \left(1 - 0.5\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double f(double x) {
        double r282950 = 1.0;
        double r282951 = 0.5;
        double r282952 = x;
        double r282953 = hypot(r282950, r282952);
        double r282954 = r282950 / r282953;
        double r282955 = r282950 + r282954;
        double r282956 = r282951 * r282955;
        double r282957 = sqrt(r282956);
        double r282958 = r282950 - r282957;
        return r282958;
}
↓
double f(double x) {
        double r282959 = 1.0;
        double r282960 = 0.5;
        double r282961 = r282959 - r282960;
        double r282962 = r282959 * r282961;
        double r282963 = 3.0;
        double r282964 = pow(r282962, r282963);
        double r282965 = sqrt(r282959);
        double r282966 = x;
        double r282967 = hypot(r282959, r282966);
        double r282968 = cbrt(r282967);
        double r282969 = r282968 * r282968;
        double r282970 = r282965 / r282969;
        double r282971 = r282965 / r282968;
        double r282972 = r282970 * r282971;
        double r282973 = r282960 * r282972;
        double r282974 = pow(r282973, r282963);
        double r282975 = r282964 - r282974;
        double r282976 = cbrt(r282968);
        double r282977 = 4.0;
        double r282978 = pow(r282976, r282977);
        double r282979 = r282976 * r282976;
        double r282980 = r282978 * r282979;
        double r282981 = r282965 / r282980;
        double r282982 = r282981 * r282971;
        double r282983 = r282960 * r282982;
        double r282984 = r282973 + r282962;
        double r282985 = r282983 * r282984;
        double r282986 = r282962 * r282962;
        double r282987 = r282985 + r282986;
        double r282988 = r282975 / r282987;
        double r282989 = r282959 / r282967;
        double r282990 = r282959 + r282989;
        double r282991 = r282960 * r282990;
        double r282992 = sqrt(r282991);
        double r282993 = r282959 + r282992;
        double r282994 = r282988 / r282993;
        return r282994;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.6

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

  3. Applied flip--15.6

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

  4. Simplified15.1

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

  6. Applied add-cube-cbrt15.2

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

  7. Applied add-sqr-sqrt15.2

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

  8. Applied times-frac15.2

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

  10. Applied flip3--15.2

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

  11. Simplified15.2

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

  13. Applied add-cube-cbrt15.2

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

  14. Applied add-cube-cbrt15.2

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

  15. Applied swap-sqr15.2

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

  16. Simplified15.2

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

  17. Final simplification15.2

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

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))