Average Error: 14.2 → 0.3
Time: 9.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}\]
\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}{2}}{b + a} \cdot \frac{1}{a \cdot b}
double f(double a, double b) {
        double r44455 = atan2(1.0, 0.0);
        double r44456 = 2.0;
        double r44457 = r44455 / r44456;
        double r44458 = 1.0;
        double r44459 = b;
        double r44460 = r44459 * r44459;
        double r44461 = a;
        double r44462 = r44461 * r44461;
        double r44463 = r44460 - r44462;
        double r44464 = r44458 / r44463;
        double r44465 = r44457 * r44464;
        double r44466 = r44458 / r44461;
        double r44467 = r44458 / r44459;
        double r44468 = r44466 - r44467;
        double r44469 = r44465 * r44468;
        return r44469;
}
↓
double f(double a, double b) {
        double r44470 = atan2(1.0, 0.0);
        double r44471 = 2.0;
        double r44472 = r44470 / r44471;
        double r44473 = b;
        double r44474 = a;
        double r44475 = r44473 + r44474;
        double r44476 = r44472 / r44475;
        double r44477 = 1.0;
        double r44478 = r44474 * r44473;
        double r44479 = r44477 / r44478;
        double r44480 = r44476 * r44479;
        return r44480;
}

Error

Bits error versus a

Bits error versus b

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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. 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 *-un-lft-identity9.5

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.0

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

  7. Simplified9.0

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

  9. Applied associate-*l*0.3

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

  11. Applied associate-*l/0.3

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

  12. Taylor expanded around 0 0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))