Average Error: 14.3 → 0.3
Time: 20.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(b - a\right) \cdot 1}{b + a} \cdot \left(\frac{1}{b - a} \cdot \pi\right)}{2 \cdot \left(a \cdot b\right)}\]
\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{\left(b - a\right) \cdot 1}{b + a} \cdot \left(\frac{1}{b - a} \cdot \pi\right)}{2 \cdot \left(a \cdot b\right)}
double f(double a, double b) {
        double r2958677 = atan2(1.0, 0.0);
        double r2958678 = 2.0;
        double r2958679 = r2958677 / r2958678;
        double r2958680 = 1.0;
        double r2958681 = b;
        double r2958682 = r2958681 * r2958681;
        double r2958683 = a;
        double r2958684 = r2958683 * r2958683;
        double r2958685 = r2958682 - r2958684;
        double r2958686 = r2958680 / r2958685;
        double r2958687 = r2958679 * r2958686;
        double r2958688 = r2958680 / r2958683;
        double r2958689 = r2958680 / r2958681;
        double r2958690 = r2958688 - r2958689;
        double r2958691 = r2958687 * r2958690;
        return r2958691;
}
↓
double f(double a, double b) {
        double r2958692 = b;
        double r2958693 = a;
        double r2958694 = r2958692 - r2958693;
        double r2958695 = 1.0;
        double r2958696 = r2958694 * r2958695;
        double r2958697 = r2958692 + r2958693;
        double r2958698 = r2958696 / r2958697;
        double r2958699 = r2958695 / r2958694;
        double r2958700 = atan2(1.0, 0.0);
        double r2958701 = r2958699 * r2958700;
        double r2958702 = r2958698 * r2958701;
        double r2958703 = 2.0;
        double r2958704 = r2958693 * r2958692;
        double r2958705 = r2958703 * r2958704;
        double r2958706 = r2958702 / r2958705;
        return r2958706;
}

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 14.3

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

  3. Applied difference-of-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.9

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

  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 frac-sub8.8

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

  10. Applied associate-*l/8.9

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

  11. Applied frac-times8.8

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

  12. Simplified0.2

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

  14. Applied *-un-lft-identity0.2

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

  15. Applied times-frac0.2

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

  16. Simplified0.3

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

  18. Applied associate-/r/0.3

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

  19. Final simplification0.3

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

Reproduce

herbie shell --seed 2019169 
(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))))