Average Error: 14.8 → 0.3
Time: 5.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{1}}{\frac{b + a}{\frac{\pi}{2}}} \cdot \left(1 \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1}{1}}{\frac{b + a}{\frac{\pi}{2}}} \cdot \left(1 \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\right)
double f(double a, double b) {
        double r48437 = atan2(1.0, 0.0);
        double r48438 = 2.0;
        double r48439 = r48437 / r48438;
        double r48440 = 1.0;
        double r48441 = b;
        double r48442 = r48441 * r48441;
        double r48443 = a;
        double r48444 = r48443 * r48443;
        double r48445 = r48442 - r48444;
        double r48446 = r48440 / r48445;
        double r48447 = r48439 * r48446;
        double r48448 = r48440 / r48443;
        double r48449 = r48440 / r48441;
        double r48450 = r48448 - r48449;
        double r48451 = r48447 * r48450;
        return r48451;
}
↓
double f(double a, double b) {
        double r48452 = 1.0;
        double r48453 = r48452 / r48452;
        double r48454 = b;
        double r48455 = a;
        double r48456 = r48454 + r48455;
        double r48457 = atan2(1.0, 0.0);
        double r48458 = 2.0;
        double r48459 = r48457 / r48458;
        double r48460 = r48456 / r48459;
        double r48461 = r48453 / r48460;
        double r48462 = 1.0;
        double r48463 = r48462 / r48455;
        double r48464 = r48462 / r48454;
        double r48465 = r48463 - r48464;
        double r48466 = r48454 - r48455;
        double r48467 = r48465 / r48466;
        double r48468 = r48462 * r48467;
        double r48469 = r48461 * r48468;
        return r48469;
}

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-squares9.8

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

  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-*l*0.3

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

  11. Applied div-inv0.3

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

  12. Applied associate-*l*0.3

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

  13. Simplified0.3

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

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

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

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

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

  17. Applied times-frac0.3

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

  18. Applied associate-/l*0.3

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

  19. Final simplification0.3

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

Reproduce

herbie shell --seed 2020059 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))