Average Error: 14.7 → 0.2
Time: 5.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{1 \cdot \left(a \cdot b\right)}\]
\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}{2}}{b + a} \cdot 1}{1 \cdot \left(a \cdot b\right)}
double f(double a, double b) {
        double r55541 = atan2(1.0, 0.0);
        double r55542 = 2.0;
        double r55543 = r55541 / r55542;
        double r55544 = 1.0;
        double r55545 = b;
        double r55546 = r55545 * r55545;
        double r55547 = a;
        double r55548 = r55547 * r55547;
        double r55549 = r55546 - r55548;
        double r55550 = r55544 / r55549;
        double r55551 = r55543 * r55550;
        double r55552 = r55544 / r55547;
        double r55553 = r55544 / r55545;
        double r55554 = r55552 - r55553;
        double r55555 = r55551 * r55554;
        return r55555;
}
↓
double f(double a, double b) {
        double r55556 = atan2(1.0, 0.0);
        double r55557 = 2.0;
        double r55558 = r55556 / r55557;
        double r55559 = b;
        double r55560 = a;
        double r55561 = r55559 + r55560;
        double r55562 = r55558 / r55561;
        double r55563 = 1.0;
        double r55564 = r55562 * r55563;
        double r55565 = r55560 * r55559;
        double r55566 = r55563 * r55565;
        double r55567 = r55564 / r55566;
        return r55567;
}

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-squares9.6

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

  4. Applied *-un-lft-identity9.6

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

  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*9.1

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

  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-*r/9.0

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-/l*0.3

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

  13. Taylor expanded around 0 0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020033 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))