Average Error: 15.0 → 0.3
Time: 5.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi \cdot 1}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi \cdot 1}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}
double f(double a, double b) {
        double r39270 = atan2(1.0, 0.0);
        double r39271 = 2.0;
        double r39272 = r39270 / r39271;
        double r39273 = 1.0;
        double r39274 = b;
        double r39275 = r39274 * r39274;
        double r39276 = a;
        double r39277 = r39276 * r39276;
        double r39278 = r39275 - r39277;
        double r39279 = r39273 / r39278;
        double r39280 = r39272 * r39279;
        double r39281 = r39273 / r39276;
        double r39282 = r39273 / r39274;
        double r39283 = r39281 - r39282;
        double r39284 = r39280 * r39283;
        return r39284;
}
↓
double f(double a, double b) {
        double r39285 = atan2(1.0, 0.0);
        double r39286 = 1.0;
        double r39287 = r39285 * r39286;
        double r39288 = 2.0;
        double r39289 = r39287 / r39288;
        double r39290 = a;
        double r39291 = r39286 / r39290;
        double r39292 = b;
        double r39293 = r39286 / r39292;
        double r39294 = r39291 - r39293;
        double r39295 = r39292 + r39290;
        double r39296 = r39294 / r39295;
        double r39297 = r39292 - r39290;
        double r39298 = r39296 / r39297;
        double r39299 = r39289 * r39298;
        return r39299;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.0

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

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.4

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

  6. Applied associate-*r/9.3

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

  7. Applied associate-*l/0.3

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

  9. Applied frac-times0.3

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

  10. Applied associate-*l/0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Applied times-frac0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019353 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))