NMSE Section 6.1 mentioned, B

Average Error: 14.3 → 0.2

Time: 23.0s

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 r73204 = atan2(1.0, 0.0);
        double r73205 = 2.0;
        double r73206 = r73204 / r73205;
        double r73207 = 1.0;
        double r73208 = b;
        double r73209 = r73208 * r73208;
        double r73210 = a;
        double r73211 = r73210 * r73210;
        double r73212 = r73209 - r73211;
        double r73213 = r73207 / r73212;
        double r73214 = r73206 * r73213;
        double r73215 = r73207 / r73210;
        double r73216 = r73207 / r73208;
        double r73217 = r73215 - r73216;
        double r73218 = r73214 * r73217;
        return r73218;
}

↓

double f(double a, double b) {
        double r73219 = atan2(1.0, 0.0);
        double r73220 = 2.0;
        double r73221 = r73219 / r73220;
        double r73222 = b;
        double r73223 = a;
        double r73224 = r73222 + r73223;
        double r73225 = r73221 / r73224;
        double r73226 = 1.0;
        double r73227 = r73225 * r73226;
        double r73228 = r73223 * r73222;
        double r73229 = r73226 * r73228;
        double r73230 = r73227 / r73229;
        return r73230;
}

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))))