Average Error: 14.4 → 4.5
Time: 58.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{(\left(\frac{\frac{1}{a + b} \cdot \pi}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{1}{a + b}}{\frac{a}{\pi} \cdot \left(b - a\right)}\right))_*}{2}\]
\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{1}{a + b} \cdot \pi}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{1}{a + b}}{\frac{a}{\pi} \cdot \left(b - a\right)}\right))_*}{2}
double f(double a, double b) {
        double r2060572 = atan2(1.0, 0.0);
        double r2060573 = 2.0;
        double r2060574 = r2060572 / r2060573;
        double r2060575 = 1.0;
        double r2060576 = b;
        double r2060577 = r2060576 * r2060576;
        double r2060578 = a;
        double r2060579 = r2060578 * r2060578;
        double r2060580 = r2060577 - r2060579;
        double r2060581 = r2060575 / r2060580;
        double r2060582 = r2060574 * r2060581;
        double r2060583 = r2060575 / r2060578;
        double r2060584 = r2060575 / r2060576;
        double r2060585 = r2060583 - r2060584;
        double r2060586 = r2060582 * r2060585;
        return r2060586;
}
↓
double f(double a, double b) {
        double r2060587 = 1.0;
        double r2060588 = a;
        double r2060589 = b;
        double r2060590 = r2060588 + r2060589;
        double r2060591 = r2060587 / r2060590;
        double r2060592 = atan2(1.0, 0.0);
        double r2060593 = r2060591 * r2060592;
        double r2060594 = r2060589 - r2060588;
        double r2060595 = r2060593 / r2060594;
        double r2060596 = -1.0;
        double r2060597 = r2060596 / r2060589;
        double r2060598 = r2060588 / r2060592;
        double r2060599 = r2060598 * r2060594;
        double r2060600 = r2060591 / r2060599;
        double r2060601 = fma(r2060595, r2060597, r2060600);
        double r2060602 = 2.0;
        double r2060603 = r2060601 / r2060602;
        return r2060603;
}

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. Simplified14.3

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

  4. Applied difference-of-squares14.3

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

  5. Applied *-un-lft-identity14.3

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

  6. Applied times-frac14.0

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

  7. Applied associate-/l*9.7

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

  9. Applied difference-of-squares4.8

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

  10. Applied associate-/r*4.5

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

  12. Applied associate-/r/4.5

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

  14. Applied div-inv4.5

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

  15. Final simplification4.5

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))