Average Error: 14.2 → 0.3
Time: 15.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{2} \cdot \frac{1 \cdot \frac{1}{a \cdot b}}{b + a} + \frac{\frac{\pi}{2}}{b + a} \cdot \left(\frac{1}{b - a} \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{1}}{b}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \frac{\sqrt[3]{1}}{b} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{2} \cdot \frac{1 \cdot \frac{1}{a \cdot b}}{b + a} + \frac{\frac{\pi}{2}}{b + a} \cdot \left(\frac{1}{b - a} \cdot \mathsf{fma}\left(-\frac{\sqrt[3]{1}}{b}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \frac{\sqrt[3]{1}}{b} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\right)
double f(double a, double b) {
        double r66371 = atan2(1.0, 0.0);
        double r66372 = 2.0;
        double r66373 = r66371 / r66372;
        double r66374 = 1.0;
        double r66375 = b;
        double r66376 = r66375 * r66375;
        double r66377 = a;
        double r66378 = r66377 * r66377;
        double r66379 = r66376 - r66378;
        double r66380 = r66374 / r66379;
        double r66381 = r66373 * r66380;
        double r66382 = r66374 / r66377;
        double r66383 = r66374 / r66375;
        double r66384 = r66382 - r66383;
        double r66385 = r66381 * r66384;
        return r66385;
}
↓
double f(double a, double b) {
        double r66386 = atan2(1.0, 0.0);
        double r66387 = 2.0;
        double r66388 = r66386 / r66387;
        double r66389 = 1.0;
        double r66390 = a;
        double r66391 = b;
        double r66392 = r66390 * r66391;
        double r66393 = r66389 / r66392;
        double r66394 = r66389 * r66393;
        double r66395 = r66391 + r66390;
        double r66396 = r66394 / r66395;
        double r66397 = r66388 * r66396;
        double r66398 = r66388 / r66395;
        double r66399 = r66391 - r66390;
        double r66400 = r66389 / r66399;
        double r66401 = cbrt(r66389);
        double r66402 = r66401 / r66391;
        double r66403 = -r66402;
        double r66404 = r66401 * r66401;
        double r66405 = r66402 * r66404;
        double r66406 = fma(r66403, r66404, r66405);
        double r66407 = r66400 * r66406;
        double r66408 = r66398 * r66407;
        double r66409 = r66397 + r66408;
        return r66409;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.2

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

  3. Applied difference-of-squares9.5

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

  4. Applied *-un-lft-identity9.5

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.0

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

  7. Simplified9.0

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

  9. Applied *-un-lft-identity9.0

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

  10. Applied add-cube-cbrt9.0

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

  11. Applied times-frac9.0

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

  12. Applied *-un-lft-identity9.0

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

  13. Applied add-sqr-sqrt9.0

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

  14. Applied times-frac9.0

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

  15. Applied prod-diff9.0

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

  16. Applied distribute-lft-in9.0

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

  17. Simplified0.3

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

  18. Simplified0.3

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

  19. Taylor expanded around inf 0.3

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

  20. Simplified0.3

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

  22. Applied div-inv0.3

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

  23. Applied associate-*l*0.3

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

  24. Simplified0.3

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

  25. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +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))))