Average Error: 13.7 → 0.2
Time: 30.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{b + a}\]
\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{\frac{\pi}{a}}{b} \cdot 0.5}{b + a}
double f(double a, double b) {
        double r2446680 = atan2(1.0, 0.0);
        double r2446681 = 2.0;
        double r2446682 = r2446680 / r2446681;
        double r2446683 = 1.0;
        double r2446684 = b;
        double r2446685 = r2446684 * r2446684;
        double r2446686 = a;
        double r2446687 = r2446686 * r2446686;
        double r2446688 = r2446685 - r2446687;
        double r2446689 = r2446683 / r2446688;
        double r2446690 = r2446682 * r2446689;
        double r2446691 = r2446683 / r2446686;
        double r2446692 = r2446683 / r2446684;
        double r2446693 = r2446691 - r2446692;
        double r2446694 = r2446690 * r2446693;
        return r2446694;
}
↓
double f(double a, double b) {
        double r2446695 = atan2(1.0, 0.0);
        double r2446696 = a;
        double r2446697 = r2446695 / r2446696;
        double r2446698 = b;
        double r2446699 = r2446697 / r2446698;
        double r2446700 = 0.5;
        double r2446701 = r2446699 * r2446700;
        double r2446702 = r2446698 + r2446696;
        double r2446703 = r2446701 / r2446702;
        return r2446703;
}

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 13.7

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

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac8.8

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.7

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

  9. Applied associate-*l/8.7

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{a + b}\]
  12. Using strategy rm

  13. Applied associate-/r*0.2

    \[\leadsto \frac{0.5 \cdot \color{blue}{\frac{\frac{\pi}{a}}{b}}}{a + b}\]

  14. Final simplification0.2

    \[\leadsto \frac{\frac{\frac{\pi}{a}}{b} \cdot 0.5}{b + a}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))