Average Error: 14.2 → 0.3
Time: 2.7m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{b + a}}{\frac{1}{\frac{\frac{1}{a} - \frac{1}{b}}{2}}}}{b - a}\]
\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{\frac{\pi}{b + a}}{\frac{1}{\frac{\frac{1}{a} - \frac{1}{b}}{2}}}}{b - a}
double f(double a, double b) {
        double r7769320 = atan2(1.0, 0.0);
        double r7769321 = 2.0;
        double r7769322 = r7769320 / r7769321;
        double r7769323 = 1.0;
        double r7769324 = b;
        double r7769325 = r7769324 * r7769324;
        double r7769326 = a;
        double r7769327 = r7769326 * r7769326;
        double r7769328 = r7769325 - r7769327;
        double r7769329 = r7769323 / r7769328;
        double r7769330 = r7769322 * r7769329;
        double r7769331 = r7769323 / r7769326;
        double r7769332 = r7769323 / r7769324;
        double r7769333 = r7769331 - r7769332;
        double r7769334 = r7769330 * r7769333;
        return r7769334;
}
↓
double f(double a, double b) {
        double r7769335 = atan2(1.0, 0.0);
        double r7769336 = b;
        double r7769337 = a;
        double r7769338 = r7769336 + r7769337;
        double r7769339 = r7769335 / r7769338;
        double r7769340 = 1.0;
        double r7769341 = r7769340 / r7769337;
        double r7769342 = r7769340 / r7769336;
        double r7769343 = r7769341 - r7769342;
        double r7769344 = 2.0;
        double r7769345 = r7769343 / r7769344;
        double r7769346 = r7769340 / r7769345;
        double r7769347 = r7769339 / r7769346;
        double r7769348 = r7769336 - r7769337;
        double r7769349 = r7769347 / r7769348;
        return r7769349;
}

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 14.2

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

  2. Simplified14.2

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

  4. Applied difference-of-squares9.8

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

  5. Applied associate-/r*0.3

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

  6. Taylor expanded around 0 0.3

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

  7. Simplified0.3

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

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

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

  10. Applied div-inv0.3

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

  11. Applied div-inv0.3

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

  12. Applied distribute-lft-out--0.3

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

  13. Applied times-frac0.3

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

  14. Applied associate-/l*0.3

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

  16. Applied div-inv0.3

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

  17. Applied associate-/r*0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))