Average Error: 14.5 → 0.8
Time: 24.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}
double f(double a, double b) {
        double r49295 = atan2(1.0, 0.0);
        double r49296 = 2.0;
        double r49297 = r49295 / r49296;
        double r49298 = 1.0;
        double r49299 = b;
        double r49300 = r49299 * r49299;
        double r49301 = a;
        double r49302 = r49301 * r49301;
        double r49303 = r49300 - r49302;
        double r49304 = r49298 / r49303;
        double r49305 = r49297 * r49304;
        double r49306 = r49298 / r49301;
        double r49307 = r49298 / r49299;
        double r49308 = r49306 - r49307;
        double r49309 = r49305 * r49308;
        return r49309;
}
↓
double f(double a, double b) {
        double r49310 = 1.0;
        double r49311 = b;
        double r49312 = a;
        double r49313 = r49311 - r49312;
        double r49314 = r49310 * r49313;
        double r49315 = atan2(1.0, 0.0);
        double r49316 = 2.0;
        double r49317 = r49315 / r49316;
        double r49318 = r49311 + r49312;
        double r49319 = r49317 / r49318;
        double r49320 = r49319 * r49310;
        double r49321 = r49314 * r49320;
        double r49322 = expm1(r49321);
        double r49323 = log1p(r49322);
        double r49324 = r49312 * r49311;
        double r49325 = r49313 * r49324;
        double r49326 = r49323 / r49325;
        return r49326;
}

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. Using strategy rm

  3. Applied difference-of-squares9.8

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

  4. Applied *-un-lft-identity9.8

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

  5. Applied times-frac9.3

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

  6. Applied associate-*r*9.3

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

  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 associate-*r/9.2

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

  10. Applied associate-*l/0.3

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

  12. Applied frac-sub0.3

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

  13. Applied associate-*r/0.3

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

  14. Applied associate-/l/0.8

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

  16. Applied log1p-expm1-u0.8

    \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(1 \cdot b - a \cdot 1\right)\right)\right)}}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]

  17. Simplified0.8

    \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)}\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]

  18. Final simplification0.8

    \[\leadsto \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right)\right)\right)}{\left(b - a\right) \cdot \left(a \cdot b\right)}\]

Reproduce

herbie shell --seed 2019325 +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))))