Average Error: 14.6 → 0.2
Time: 2.4m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{\frac{b + a}{b \cdot \frac{1}{2} - \frac{1}{2} \cdot a}}}{a \cdot b}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\frac{\pi}{\frac{b + a}{b \cdot \frac{1}{2} - \frac{1}{2} \cdot a}}}{a \cdot b}}{b - a}
double f(double a, double b) {
        double r2735337 = atan2(1.0, 0.0);
        double r2735338 = 2.0;
        double r2735339 = r2735337 / r2735338;
        double r2735340 = 1.0;
        double r2735341 = b;
        double r2735342 = r2735341 * r2735341;
        double r2735343 = a;
        double r2735344 = r2735343 * r2735343;
        double r2735345 = r2735342 - r2735344;
        double r2735346 = r2735340 / r2735345;
        double r2735347 = r2735339 * r2735346;
        double r2735348 = r2735340 / r2735343;
        double r2735349 = r2735340 / r2735341;
        double r2735350 = r2735348 - r2735349;
        double r2735351 = r2735347 * r2735350;
        return r2735351;
}
↓
double f(double a, double b) {
        double r2735352 = atan2(1.0, 0.0);
        double r2735353 = b;
        double r2735354 = a;
        double r2735355 = r2735353 + r2735354;
        double r2735356 = 0.5;
        double r2735357 = r2735353 * r2735356;
        double r2735358 = r2735356 * r2735354;
        double r2735359 = r2735357 - r2735358;
        double r2735360 = r2735355 / r2735359;
        double r2735361 = r2735352 / r2735360;
        double r2735362 = r2735354 * r2735353;
        double r2735363 = r2735361 / r2735362;
        double r2735364 = r2735353 - r2735354;
        double r2735365 = r2735363 / r2735364;
        return r2735365;
}

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 14.6

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

  2. Simplified9.4

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

  4. Applied associate-/r*0.2

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{2}}{a} - \frac{\frac{\pi}{2}}{b}}{a + b}}{b - a}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.2

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

  7. Applied div-inv0.2

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

  8. Applied times-frac0.3

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

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

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

  10. Applied div-inv0.3

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

  11. Applied times-frac0.3

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

  12. Applied distribute-lft-out--0.3

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

  13. Applied associate-/l*0.3

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

  14. Simplified0.3

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

  16. Applied frac-sub0.3

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

  17. Applied associate-/r/0.2

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

  18. Applied associate-/r*0.2

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

  19. Simplified0.2

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

  20. Final simplification0.2

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

Reproduce

herbie shell --seed 2019149 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))