Average Error: 15.6 → 0.3
Time: 10.2s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right) \cdot \frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} \cdot \frac{\sqrt{1}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\\ \end{array}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1:\\
\;\;\;\;\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right) \cdot \frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} \cdot \frac{\sqrt{1}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\\

\end{array}
double f(double x) {
        double r351925 = 1.0;
        double r351926 = 0.5;
        double r351927 = x;
        double r351928 = hypot(r351925, r351927);
        double r351929 = r351925 / r351928;
        double r351930 = r351925 + r351929;
        double r351931 = r351926 * r351930;
        double r351932 = sqrt(r351931);
        double r351933 = r351925 - r351932;
        return r351933;
}
↓
double f(double x) {
        double r351934 = 1.0;
        double r351935 = x;
        double r351936 = hypot(r351934, r351935);
        bool r351937 = r351936 <= r351934;
        double r351938 = 0.25;
        double r351939 = 2.0;
        double r351940 = pow(r351935, r351939);
        double r351941 = sqrt(r351934);
        double r351942 = 3.0;
        double r351943 = pow(r351941, r351942);
        double r351944 = r351940 / r351943;
        double r351945 = 0.5;
        double r351946 = 1.0;
        double r351947 = r351946 / r351941;
        double r351948 = 0.1875;
        double r351949 = 4.0;
        double r351950 = pow(r351935, r351949);
        double r351951 = 5.0;
        double r351952 = pow(r351941, r351951);
        double r351953 = r351950 / r351952;
        double r351954 = r351948 * r351953;
        double r351955 = fma(r351945, r351947, r351954);
        double r351956 = r351945 - r351955;
        double r351957 = fma(r351938, r351944, r351956);
        double r351958 = r351934 / r351936;
        double r351959 = r351934 + r351958;
        double r351960 = r351945 * r351959;
        double r351961 = sqrt(r351960);
        double r351962 = r351934 + r351961;
        double r351963 = r351946 / r351962;
        double r351964 = r351957 * r351963;
        double r351965 = -r351945;
        double r351966 = r351934 * r351934;
        double r351967 = fma(r351965, r351959, r351966);
        double r351968 = sqrt(r351962);
        double r351969 = r351967 / r351968;
        double r351970 = sqrt(r351946);
        double r351971 = r351970 / r351968;
        double r351972 = r351969 * r351971;
        double r351973 = r351937 ? r351964 : r351972;
        return r351973;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (hypot 1.0 x) < 1.0

    1. Initial program 30.2

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

    3. Applied flip--30.2

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

    4. Simplified30.2

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

    6. Applied div-inv30.2

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

    7. Taylor expanded around 0 30.2

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}} + 0.5\right) - \left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

    8. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

    if 1.0 < (hypot 1.0 x)

    1. Initial program 1.5

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

    3. Applied flip--1.5

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

    4. Simplified0.5

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

    6. Applied div-inv0.5

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

    8. Applied add-sqr-sqrt1.5

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

    9. Applied add-sqr-sqrt1.5

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

    10. Applied times-frac0.5

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

    11. Applied associate-*r*0.5

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

    12. Simplified0.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}} \cdot \frac{\sqrt{1}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \le 1:\\ \;\;\;\;\mathsf{fma}\left(0.25, \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.5 - \mathsf{fma}\left(0.5, \frac{1}{\sqrt{1}}, 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)\right) \cdot \frac{1}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, 1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}, 1 \cdot 1\right)}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}} \cdot \frac{\sqrt{1}}{\sqrt{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))