Average Error: 14.1 → 0.5
Time: 20.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \frac{\left(\left(b - a\right) \cdot 1\right) \cdot \pi}{b + a}}{2 \cdot \left(a \cdot b\right)}}{b - a}\]
\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{1 \cdot \frac{\left(\left(b - a\right) \cdot 1\right) \cdot \pi}{b + a}}{2 \cdot \left(a \cdot b\right)}}{b - a}
double f(double a, double b) {
        double r2572661 = atan2(1.0, 0.0);
        double r2572662 = 2.0;
        double r2572663 = r2572661 / r2572662;
        double r2572664 = 1.0;
        double r2572665 = b;
        double r2572666 = r2572665 * r2572665;
        double r2572667 = a;
        double r2572668 = r2572667 * r2572667;
        double r2572669 = r2572666 - r2572668;
        double r2572670 = r2572664 / r2572669;
        double r2572671 = r2572663 * r2572670;
        double r2572672 = r2572664 / r2572667;
        double r2572673 = r2572664 / r2572665;
        double r2572674 = r2572672 - r2572673;
        double r2572675 = r2572671 * r2572674;
        return r2572675;
}
↓
double f(double a, double b) {
        double r2572676 = 1.0;
        double r2572677 = b;
        double r2572678 = a;
        double r2572679 = r2572677 - r2572678;
        double r2572680 = r2572679 * r2572676;
        double r2572681 = atan2(1.0, 0.0);
        double r2572682 = r2572680 * r2572681;
        double r2572683 = r2572677 + r2572678;
        double r2572684 = r2572682 / r2572683;
        double r2572685 = r2572676 * r2572684;
        double r2572686 = 2.0;
        double r2572687 = r2572678 * r2572677;
        double r2572688 = r2572686 * r2572687;
        double r2572689 = r2572685 / r2572688;
        double r2572690 = r2572689 / r2572679;
        return r2572690;
}

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-squares9.2

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

  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. Simplified8.8

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

  9. Applied associate-*r/8.7

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

  10. Applied associate-*l/0.3

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

  12. Applied frac-sub0.4

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

  13. Applied associate-*l/0.4

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

  14. Applied frac-times0.3

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

  15. Simplified0.3

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

  17. Applied associate-*r/0.5

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

  18. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(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))))