Average Error: 14.7 → 0.2
Time: 4.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a \cdot b} \cdot 0.5}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\pi}{a \cdot b} \cdot 0.5}{b + a}
double f(double a, double b) {
        double r39283 = atan2(1.0, 0.0);
        double r39284 = 2.0;
        double r39285 = r39283 / r39284;
        double r39286 = 1.0;
        double r39287 = b;
        double r39288 = r39287 * r39287;
        double r39289 = a;
        double r39290 = r39289 * r39289;
        double r39291 = r39288 - r39290;
        double r39292 = r39286 / r39291;
        double r39293 = r39285 * r39292;
        double r39294 = r39286 / r39289;
        double r39295 = r39286 / r39287;
        double r39296 = r39294 - r39295;
        double r39297 = r39293 * r39296;
        return r39297;
}
↓
double f(double a, double b) {
        double r39298 = atan2(1.0, 0.0);
        double r39299 = a;
        double r39300 = b;
        double r39301 = r39299 * r39300;
        double r39302 = r39298 / r39301;
        double r39303 = 0.5;
        double r39304 = r39302 * r39303;
        double r39305 = r39300 + r39299;
        double r39306 = r39304 / r39305;
        return r39306;
}

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 14.7

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

  3. Applied difference-of-squares9.7

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

  4. Applied *-un-lft-identity9.7

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

  5. Applied times-frac9.2

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.1

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

  9. Applied associate-*l/9.1

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]
  12. Using strategy rm

  13. Applied *-commutative0.2

    \[\leadsto \frac{\color{blue}{\frac{\pi}{a \cdot b} \cdot 0.5}}{b + a}\]

  14. Final simplification0.2

    \[\leadsto \frac{\frac{\pi}{a \cdot b} \cdot 0.5}{b + a}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))