Average Error: 14.6 → 0.3
Time: 5.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\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} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r47567 = atan2(1.0, 0.0);
        double r47568 = 2.0;
        double r47569 = r47567 / r47568;
        double r47570 = 1.0;
        double r47571 = b;
        double r47572 = r47571 * r47571;
        double r47573 = a;
        double r47574 = r47573 * r47573;
        double r47575 = r47572 - r47574;
        double r47576 = r47570 / r47575;
        double r47577 = r47569 * r47576;
        double r47578 = r47570 / r47573;
        double r47579 = r47570 / r47571;
        double r47580 = r47578 - r47579;
        double r47581 = r47577 * r47580;
        return r47581;
}
↓
double f(double a, double b) {
        double r47582 = atan2(1.0, 0.0);
        double r47583 = 2.0;
        double r47584 = r47582 / r47583;
        double r47585 = 1.0;
        double r47586 = b;
        double r47587 = a;
        double r47588 = r47586 + r47587;
        double r47589 = r47585 / r47588;
        double r47590 = r47584 * r47589;
        double r47591 = r47586 - r47587;
        double r47592 = r47585 / r47587;
        double r47593 = r47585 / r47586;
        double r47594 = r47592 - r47593;
        double r47595 = r47591 / r47594;
        double r47596 = r47590 / r47595;
        return r47596;
}

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 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 associate-/r*9.1

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

  6. Applied associate-*r/9.1

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

  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 associate-/l*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020001 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))