Average Error: 14.4 → 4.7
Time: 1.1m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{(\left(\frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\frac{\pi}{a}}{b - a}}{b + a}\right))_*}{2}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{(\left(\frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\frac{\pi}{a}}{b - a}}{b + a}\right))_*}{2}
double f(double a, double b) {
        double r2813518 = atan2(1.0, 0.0);
        double r2813519 = 2.0;
        double r2813520 = r2813518 / r2813519;
        double r2813521 = 1.0;
        double r2813522 = b;
        double r2813523 = r2813522 * r2813522;
        double r2813524 = a;
        double r2813525 = r2813524 * r2813524;
        double r2813526 = r2813523 - r2813525;
        double r2813527 = r2813521 / r2813526;
        double r2813528 = r2813520 * r2813527;
        double r2813529 = r2813521 / r2813524;
        double r2813530 = r2813521 / r2813522;
        double r2813531 = r2813529 - r2813530;
        double r2813532 = r2813528 * r2813531;
        return r2813532;
}
↓
double f(double a, double b) {
        double r2813533 = atan2(1.0, 0.0);
        double r2813534 = b;
        double r2813535 = a;
        double r2813536 = r2813534 + r2813535;
        double r2813537 = r2813533 / r2813536;
        double r2813538 = r2813534 - r2813535;
        double r2813539 = r2813537 / r2813538;
        double r2813540 = -1.0;
        double r2813541 = r2813540 / r2813534;
        double r2813542 = r2813533 / r2813535;
        double r2813543 = r2813542 / r2813538;
        double r2813544 = r2813543 / r2813536;
        double r2813545 = fma(r2813539, r2813541, r2813544);
        double r2813546 = 2.0;
        double r2813547 = r2813545 / r2813546;
        return r2813547;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.4

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

  2. Simplified14.4

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

  4. Applied *-un-lft-identity14.4

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

  5. Applied difference-of-squares14.4

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

  6. Applied *-un-lft-identity14.4

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

  7. Applied times-frac14.1

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

  8. Applied times-frac9.8

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

  9. Simplified9.8

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

  11. Applied difference-of-squares5.0

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

  12. Applied associate-/r*4.7

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

  14. Applied associate-*l/4.7

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

  15. Simplified4.7

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

  17. Applied associate-/r*4.7

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

  18. Final simplification4.7

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))