Average Error: 14.2 → 0.3
Time: 14.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot 1}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\]
\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}{2} \cdot 1}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}
double f(double a, double b) {
        double r51551 = atan2(1.0, 0.0);
        double r51552 = 2.0;
        double r51553 = r51551 / r51552;
        double r51554 = 1.0;
        double r51555 = b;
        double r51556 = r51555 * r51555;
        double r51557 = a;
        double r51558 = r51557 * r51557;
        double r51559 = r51556 - r51558;
        double r51560 = r51554 / r51559;
        double r51561 = r51553 * r51560;
        double r51562 = r51554 / r51557;
        double r51563 = r51554 / r51555;
        double r51564 = r51562 - r51563;
        double r51565 = r51561 * r51564;
        return r51565;
}
↓
double f(double a, double b) {
        double r51566 = atan2(1.0, 0.0);
        double r51567 = 2.0;
        double r51568 = r51566 / r51567;
        double r51569 = 1.0;
        double r51570 = r51568 * r51569;
        double r51571 = b;
        double r51572 = a;
        double r51573 = r51571 + r51572;
        double r51574 = r51570 / r51573;
        double r51575 = r51569 / r51572;
        double r51576 = r51569 / r51571;
        double r51577 = r51575 - r51576;
        double r51578 = r51571 - r51572;
        double r51579 = r51577 / r51578;
        double r51580 = r51574 * r51579;
        return r51580;
}

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. Using strategy rm

  3. Applied difference-of-squares9.9

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

  4. Applied associate-/r*9.3

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

  6. Applied associate-*r/9.3

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

  7. Applied associate-*l/0.3

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

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

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))