Average Error: 14.7 → 0.3
Time: 25.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a} + \frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a} + \frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)
double f(double a, double b) {
        double r65929 = atan2(1.0, 0.0);
        double r65930 = 2.0;
        double r65931 = r65929 / r65930;
        double r65932 = 1.0;
        double r65933 = b;
        double r65934 = r65933 * r65933;
        double r65935 = a;
        double r65936 = r65935 * r65935;
        double r65937 = r65934 - r65936;
        double r65938 = r65932 / r65937;
        double r65939 = r65931 * r65938;
        double r65940 = r65932 / r65935;
        double r65941 = r65932 / r65933;
        double r65942 = r65940 - r65941;
        double r65943 = r65939 * r65942;
        return r65943;
}
↓
double f(double a, double b) {
        double r65944 = 1.0;
        double r65945 = sqrt(r65944);
        double r65946 = a;
        double r65947 = r65945 / r65946;
        double r65948 = b;
        double r65949 = r65944 / r65948;
        double r65950 = -r65949;
        double r65951 = fma(r65945, r65947, r65950);
        double r65952 = atan2(1.0, 0.0);
        double r65953 = 2.0;
        double r65954 = r65952 / r65953;
        double r65955 = r65951 * r65954;
        double r65956 = r65948 + r65946;
        double r65957 = r65955 / r65956;
        double r65958 = r65948 - r65946;
        double r65959 = r65944 / r65958;
        double r65960 = r65957 * r65959;
        double r65961 = r65956 * r65953;
        double r65962 = r65952 / r65961;
        double r65963 = r65950 + r65949;
        double r65964 = r65959 * r65963;
        double r65965 = r65962 * r65964;
        double r65966 = r65960 + r65965;
        return r65966;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. Using strategy rm

  3. Applied difference-of-squares9.9

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\left(b + a\right) \cdot \left(b - a\right)}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  4. Applied *-un-lft-identity9.9

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\color{blue}{1 \cdot 1}}{\left(b + a\right) \cdot \left(b - a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied times-frac9.4

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{1}{b + a} \cdot \frac{1}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  6. Applied associate-*r*9.4

    \[\leadsto \color{blue}{\left(\left(\frac{\pi}{2} \cdot \frac{1}{b + a}\right) \cdot \frac{1}{b - a}\right)} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  7. Simplified9.3

    \[\leadsto \left(\color{blue}{\frac{\frac{\pi}{2}}{b + a}} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  8. Using strategy rm

  9. Applied add-cube-cbrt9.5

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{a} - \color{blue}{\left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}}\right)\]

  10. Applied *-un-lft-identity9.5

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{1}{\color{blue}{1 \cdot a}} - \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}\right)\]

  11. Applied add-sqr-sqrt9.5

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot a} - \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}\right)\]

  12. Applied times-frac9.5

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \left(\color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{a}} - \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \sqrt[3]{\frac{1}{b}}\right)\]

  13. Applied prod-diff9.5

    \[\leadsto \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\frac{\sqrt{1}}{1}, \frac{\sqrt{1}}{a}, -\sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right)\right)}\]

  14. Applied distribute-lft-in9.5

    \[\leadsto \color{blue}{\left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \mathsf{fma}\left(\frac{\sqrt{1}}{1}, \frac{\sqrt{1}}{a}, -\sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right) + \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right)}\]

  15. Simplified0.3

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b + a}\right) \cdot \frac{1}{b - a}} + \left(\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}, \sqrt[3]{\frac{1}{b}} \cdot \left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right)\right)\]

  16. Simplified0.3

    \[\leadsto \left(\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\frac{\pi}{2}}{b + a}\right) \cdot \frac{1}{b - a} + \color{blue}{\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)}\]
  17. Using strategy rm

  18. Applied associate-*r/0.3

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\pi}{2}}{b + a}} \cdot \frac{1}{b - a} + \frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)\]

  19. Final simplification0.3

    \[\leadsto \frac{\mathsf{fma}\left(\sqrt{1}, \frac{\sqrt{1}}{a}, -\frac{1}{b}\right) \cdot \frac{\pi}{2}}{b + a} \cdot \frac{1}{b - a} + \frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(\frac{1}{b - a} \cdot \left(\left(-\frac{1}{b}\right) + \frac{1}{b}\right)\right)\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))