Average Error: 14.1 → 0.3
Time: 6.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1 \cdot \frac{\pi}{2}}{b - a}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \frac{1 \cdot \frac{\pi}{2}}{b - a}}{b + a}
double f(double a, double b) {
        double r51517 = atan2(1.0, 0.0);
        double r51518 = 2.0;
        double r51519 = r51517 / r51518;
        double r51520 = 1.0;
        double r51521 = b;
        double r51522 = r51521 * r51521;
        double r51523 = a;
        double r51524 = r51523 * r51523;
        double r51525 = r51522 - r51524;
        double r51526 = r51520 / r51525;
        double r51527 = r51519 * r51526;
        double r51528 = r51520 / r51523;
        double r51529 = r51520 / r51521;
        double r51530 = r51528 - r51529;
        double r51531 = r51527 * r51530;
        return r51531;
}
↓
double f(double a, double b) {
        double r51532 = 1.0;
        double r51533 = a;
        double r51534 = r51532 / r51533;
        double r51535 = b;
        double r51536 = r51532 / r51535;
        double r51537 = r51534 - r51536;
        double r51538 = atan2(1.0, 0.0);
        double r51539 = 2.0;
        double r51540 = r51538 / r51539;
        double r51541 = r51532 * r51540;
        double r51542 = r51535 - r51533;
        double r51543 = r51541 / r51542;
        double r51544 = r51537 * r51543;
        double r51545 = r51535 + r51533;
        double r51546 = r51544 / r51545;
        return r51546;
}

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 14.1

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 add-sqr-sqrt9.5

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

  5. Applied times-frac9.0

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

  7. Applied associate-*l/9.0

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

  8. Applied associate-*r/9.0

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

  9. Applied associate-*l/0.3

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

  10. Simplified0.3

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

  12. Applied associate-*l/0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020035 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))