NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 0.3

Time: 41.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r1148135 = atan2(1.0, 0.0);
        double r1148136 = 2.0;
        double r1148137 = r1148135 / r1148136;
        double r1148138 = 1.0;
        double r1148139 = b;
        double r1148140 = r1148139 * r1148139;
        double r1148141 = a;
        double r1148142 = r1148141 * r1148141;
        double r1148143 = r1148140 - r1148142;
        double r1148144 = r1148138 / r1148143;
        double r1148145 = r1148137 * r1148144;
        double r1148146 = r1148138 / r1148141;
        double r1148147 = r1148138 / r1148139;
        double r1148148 = r1148146 - r1148147;
        double r1148149 = r1148145 * r1148148;
        return r1148149;
}

↓

double f(double a, double b) {
        double r1148150 = 1.0;
        double r1148151 = a;
        double r1148152 = r1148150 / r1148151;
        double r1148153 = b;
        double r1148154 = r1148150 / r1148153;
        double r1148155 = r1148152 - r1148154;
        double r1148156 = 2.0;
        double r1148157 = r1148151 + r1148153;
        double r1148158 = r1148156 * r1148157;
        double r1148159 = r1148155 / r1148158;
        double r1148160 = r1148153 - r1148151;
        double r1148161 = atan2(1.0, 0.0);
        double r1148162 = r1148160 / r1148161;
        double r1148163 = r1148159 / r1148162;
        return r1148163;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 14.2

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

2. Simplified 0.3

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

4. Applied *-un-lft-identity 0.3

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

5. Applied div-inv 0.3

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

6. Applied distribute-rgt-neg-in 0.3

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

7. Applied div-inv 0.3

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

8. Applied distribute-lft-out 0.3

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

9. Applied times-frac 0.3

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

10. Applied times-frac 0.3

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

11. Simplified 0.3

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

12. Simplified 0.3

    \[\leadsto \frac{\pi}{b - a} \cdot \color{blue}{\frac{\frac{\frac{1}{a} - \frac{1}{b}}{2}}{b + a}}\]
13. Using strategy rm

14. Applied clear-num 0.3

    \[\leadsto \color{blue}{\frac{1}{\frac{b - a}{\pi}}} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{2}}{b + a}\]
15. Using strategy rm

16. Applied associate-*l/ 0.3

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019141 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))