Average Error: 14.8 → 0.3
Time: 21.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{\pi \cdot \frac{1 \cdot \frac{1}{a}}{b - a}}{\left(a + b\right) \cdot 2} + \frac{\pi}{\left(a + b\right) \cdot 2} \cdot \left(\left(-1\right) \cdot \frac{\frac{1}{b - a}}{b}\right)\right) + \frac{\frac{\pi}{2} \cdot \frac{1}{b - a}}{b + a} \cdot \left(\left(-\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b}\right) + \frac{{\left(\sqrt[3]{1}\right)}^{3}}{b}\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{\pi \cdot \frac{1 \cdot \frac{1}{a}}{b - a}}{\left(a + b\right) \cdot 2} + \frac{\pi}{\left(a + b\right) \cdot 2} \cdot \left(\left(-1\right) \cdot \frac{\frac{1}{b - a}}{b}\right)\right) + \frac{\frac{\pi}{2} \cdot \frac{1}{b - a}}{b + a} \cdot \left(\left(-\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b}\right) + \frac{{\left(\sqrt[3]{1}\right)}^{3}}{b}\right)
double f(double a, double b) {
        double r64882 = atan2(1.0, 0.0);
        double r64883 = 2.0;
        double r64884 = r64882 / r64883;
        double r64885 = 1.0;
        double r64886 = b;
        double r64887 = r64886 * r64886;
        double r64888 = a;
        double r64889 = r64888 * r64888;
        double r64890 = r64887 - r64889;
        double r64891 = r64885 / r64890;
        double r64892 = r64884 * r64891;
        double r64893 = r64885 / r64888;
        double r64894 = r64885 / r64886;
        double r64895 = r64893 - r64894;
        double r64896 = r64892 * r64895;
        return r64896;
}
↓
double f(double a, double b) {
        double r64897 = atan2(1.0, 0.0);
        double r64898 = 1.0;
        double r64899 = a;
        double r64900 = r64898 / r64899;
        double r64901 = r64898 * r64900;
        double r64902 = b;
        double r64903 = r64902 - r64899;
        double r64904 = r64901 / r64903;
        double r64905 = r64897 * r64904;
        double r64906 = r64899 + r64902;
        double r64907 = 2.0;
        double r64908 = r64906 * r64907;
        double r64909 = r64905 / r64908;
        double r64910 = r64897 / r64908;
        double r64911 = -r64898;
        double r64912 = r64898 / r64903;
        double r64913 = r64912 / r64902;
        double r64914 = r64911 * r64913;
        double r64915 = r64910 * r64914;
        double r64916 = r64909 + r64915;
        double r64917 = r64897 / r64907;
        double r64918 = r64917 * r64912;
        double r64919 = r64902 + r64899;
        double r64920 = r64918 / r64919;
        double r64921 = cbrt(r64898);
        double r64922 = 3.0;
        double r64923 = pow(r64921, r64922);
        double r64924 = r64923 / r64902;
        double r64925 = -r64924;
        double r64926 = r64925 + r64924;
        double r64927 = r64920 * r64926;
        double r64928 = r64916 + r64927;
        return r64928;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.8

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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-frac9.5

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

  6. Applied associate-*r*9.5

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

  7. Simplified9.4

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

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

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

  10. Applied add-cube-cbrt9.4

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

  11. Applied times-frac9.4

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

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

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

  13. Applied add-cube-cbrt9.4

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

  14. Applied times-frac9.4

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

  15. Applied prod-diff9.4

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

  16. Applied distribute-lft-in9.4

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

  17. Simplified0.3

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

  18. Simplified0.3

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

  20. Applied fma-udef0.3

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

  21. Applied distribute-lft-in0.3

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

  22. Applied distribute-lft-in0.3

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

  23. Simplified0.3

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

  24. Simplified0.3

    \[\leadsto \left(\frac{\pi \cdot \frac{1 \cdot \frac{1}{a}}{b - a}}{\left(a + b\right) \cdot 2} + \color{blue}{\frac{\pi}{\left(a + b\right) \cdot 2} \cdot \left(\left(-\frac{1}{b}\right) \cdot \frac{1}{b - a}\right)}\right) + \frac{\frac{\pi}{2} \cdot \frac{1}{b - a}}{b + a} \cdot \left(\left(-\frac{{\left(\sqrt[3]{1}\right)}^{3}}{b}\right) + \frac{{\left(\sqrt[3]{1}\right)}^{3}}{b}\right)\]
  25. Using strategy rm

  26. Applied div-inv0.3

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

  27. Applied distribute-lft-neg-in0.3

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

  28. Applied associate-*l*0.3

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

  29. Simplified0.3

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

  30. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 +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))))