Average Error: 14.3 → 0.3
Time: 4.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{\frac{2}{1}} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]
\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}{\frac{2}{1}} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}
double f(double a, double b) {
        double r39321 = atan2(1.0, 0.0);
        double r39322 = 2.0;
        double r39323 = r39321 / r39322;
        double r39324 = 1.0;
        double r39325 = b;
        double r39326 = r39325 * r39325;
        double r39327 = a;
        double r39328 = r39327 * r39327;
        double r39329 = r39326 - r39328;
        double r39330 = r39324 / r39329;
        double r39331 = r39323 * r39330;
        double r39332 = r39324 / r39327;
        double r39333 = r39324 / r39325;
        double r39334 = r39332 - r39333;
        double r39335 = r39331 * r39334;
        return r39335;
}
↓
double f(double a, double b) {
        double r39336 = atan2(1.0, 0.0);
        double r39337 = 2.0;
        double r39338 = 1.0;
        double r39339 = r39337 / r39338;
        double r39340 = r39336 / r39339;
        double r39341 = a;
        double r39342 = r39338 / r39341;
        double r39343 = b;
        double r39344 = r39338 / r39343;
        double r39345 = r39342 - r39344;
        double r39346 = r39343 + r39341;
        double r39347 = r39345 / r39346;
        double r39348 = r39343 - r39341;
        double r39349 = r39347 / r39348;
        double r39350 = r39340 * r39349;
        return r39350;
}

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 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-squares9.5

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

  4. Applied associate-/r*9.0

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

  6. Applied frac-times9.0

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*l/0.3

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

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

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

  13. Applied times-frac0.3

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

  14. Applied times-frac0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020027 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))