NMSE Section 6.1 mentioned, B

Average Error: 14.0 → 0.3

Time: 8.3s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r55155 = atan2(1.0, 0.0);
        double r55156 = 2.0;
        double r55157 = r55155 / r55156;
        double r55158 = 1.0;
        double r55159 = b;
        double r55160 = r55159 * r55159;
        double r55161 = a;
        double r55162 = r55161 * r55161;
        double r55163 = r55160 - r55162;
        double r55164 = r55158 / r55163;
        double r55165 = r55157 * r55164;
        double r55166 = r55158 / r55161;
        double r55167 = r55158 / r55159;
        double r55168 = r55166 - r55167;
        double r55169 = r55165 * r55168;
        return r55169;
}

↓

double f(double a, double b) {
        double r55170 = atan2(1.0, 0.0);
        double r55171 = 2.0;
        double r55172 = r55170 / r55171;
        double r55173 = b;
        double r55174 = a;
        double r55175 = r55173 + r55174;
        double r55176 = r55172 / r55175;
        double r55177 = 1.0;
        double r55178 = r55176 * r55177;
        double r55179 = r55174 * r55173;
        double r55180 = r55177 / r55179;
        double r55181 = r55178 * r55180;
        return r55181;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.0

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
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 associate-*r/ 8.9

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

10. Applied associate-*l/ 0.3

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

12. Applied *-un-lft-identity 0.3

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

13. Applied times-frac 0.3

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

14. Simplified 0.3

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

15. Taylor expanded around 0 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020049 +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))))