Average Error: 14.5 → 0.3
Time: 17.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\sqrt{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\sqrt[3]{\pi}}}{a + b}\right)\right) \cdot \frac{\frac{\sqrt{\pi}}{a} - \frac{\sqrt{\pi}}{b}}{\frac{b - a}{\sqrt{\frac{1}{2}}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\sqrt{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \frac{\sqrt{\sqrt[3]{\pi}}}{a + b}\right)\right) \cdot \frac{\frac{\sqrt{\pi}}{a} - \frac{\sqrt{\pi}}{b}}{\frac{b - a}{\sqrt{\frac{1}{2}}}}
double f(double a, double b) {
        double r785177 = atan2(1.0, 0.0);
        double r785178 = 2.0;
        double r785179 = r785177 / r785178;
        double r785180 = 1.0;
        double r785181 = b;
        double r785182 = r785181 * r785181;
        double r785183 = a;
        double r785184 = r785183 * r785183;
        double r785185 = r785182 - r785184;
        double r785186 = r785180 / r785185;
        double r785187 = r785179 * r785186;
        double r785188 = r785180 / r785183;
        double r785189 = r785180 / r785181;
        double r785190 = r785188 - r785189;
        double r785191 = r785187 * r785190;
        return r785191;
}
↓
double f(double a, double b) {
        double r785192 = atan2(1.0, 0.0);
        double r785193 = cbrt(r785192);
        double r785194 = r785193 * r785193;
        double r785195 = sqrt(r785194);
        double r785196 = 0.5;
        double r785197 = sqrt(r785196);
        double r785198 = sqrt(r785193);
        double r785199 = a;
        double r785200 = b;
        double r785201 = r785199 + r785200;
        double r785202 = r785198 / r785201;
        double r785203 = r785197 * r785202;
        double r785204 = r785195 * r785203;
        double r785205 = sqrt(r785192);
        double r785206 = r785205 / r785199;
        double r785207 = r785205 / r785200;
        double r785208 = r785206 - r785207;
        double r785209 = r785200 - r785199;
        double r785210 = r785209 / r785197;
        double r785211 = r785208 / r785210;
        double r785212 = r785204 * r785211;
        return r785212;
}

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 14.5

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

  2. Simplified14.5

    \[\leadsto \color{blue}{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{\frac{b \cdot b - a \cdot a}{\frac{1}{2}}}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt14.8

    \[\leadsto \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\frac{b \cdot b - a \cdot a}{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}}\]

  5. Applied difference-of-squares10.1

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

  6. Applied times-frac9.9

    \[\leadsto \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\color{blue}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}}\]

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

    \[\leadsto \frac{\frac{\pi}{a} - \frac{\pi}{\color{blue}{1 \cdot b}}}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}\]

  8. Applied add-sqr-sqrt9.9

    \[\leadsto \frac{\frac{\pi}{a} - \frac{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}{1 \cdot b}}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}\]

  9. Applied times-frac9.9

    \[\leadsto \frac{\frac{\pi}{a} - \color{blue}{\frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{b}}}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}\]

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

    \[\leadsto \frac{\frac{\pi}{\color{blue}{1 \cdot a}} - \frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{b}}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}\]

  11. Applied add-sqr-sqrt9.9

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{\pi} \cdot \sqrt{\pi}}}{1 \cdot a} - \frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{b}}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}\]

  12. Applied times-frac9.9

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{a}} - \frac{\sqrt{\pi}}{1} \cdot \frac{\sqrt{\pi}}{b}}{\frac{b + a}{\sqrt{\frac{1}{2}}} \cdot \frac{b - a}{\sqrt{\frac{1}{2}}}}\]

  13. Applied distribute-lft-out--9.9

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

  14. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{\pi}}{1}}{\frac{b + a}{\sqrt{\frac{1}{2}}}} \cdot \frac{\frac{\sqrt{\pi}}{a} - \frac{\sqrt{\pi}}{b}}{\frac{b - a}{\sqrt{\frac{1}{2}}}}}\]

  15. Simplified0.3

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

  17. Applied *-un-lft-identity0.3

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

  18. Applied *-un-lft-identity0.3

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

  19. Applied distribute-lft-out0.3

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

  20. Applied add-cube-cbrt0.5

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

  21. Applied sqrt-prod0.3

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

  22. Applied times-frac0.4

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

  23. Applied associate-*l*0.3

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

  24. Final simplification0.3

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))