NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.2

Time: 24.6s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r73245 = atan2(1.0, 0.0);
        double r73246 = 2.0;
        double r73247 = r73245 / r73246;
        double r73248 = 1.0;
        double r73249 = b;
        double r73250 = r73249 * r73249;
        double r73251 = a;
        double r73252 = r73251 * r73251;
        double r73253 = r73250 - r73252;
        double r73254 = r73248 / r73253;
        double r73255 = r73247 * r73254;
        double r73256 = r73248 / r73251;
        double r73257 = r73248 / r73249;
        double r73258 = r73256 - r73257;
        double r73259 = r73255 * r73258;
        return r73259;
}

↓

double f(double a, double b) {
        double r73260 = atan2(1.0, 0.0);
        double r73261 = 2.0;
        double r73262 = r73260 / r73261;
        double r73263 = b;
        double r73264 = a;
        double r73265 = r73263 + r73264;
        double r73266 = r73262 / r73265;
        double r73267 = 1.0;
        double r73268 = r73266 * r73267;
        double r73269 = r73264 * r73263;
        double r73270 = r73267 * r73269;
        double r73271 = r73268 / r73270;
        return r73271;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.3

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

3. Applied difference-of-squares 9.4

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

4. Applied *-un-lft-identity 9.4

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

5. Applied times-frac 9.0

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

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. Simplified 8.9

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

9. Applied frac-sub 8.9

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

10. Applied associate-*r/ 8.9

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

11. Applied frac-times 0.8

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

13. Applied associate-/l* 0.7

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

14. Simplified 0.7

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

15. Taylor expanded around 0 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019323 +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))))