Average Error: 14.6 → 0.3
Time: 13.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}
double f(double a, double b) {
        double r74787 = atan2(1.0, 0.0);
        double r74788 = 2.0;
        double r74789 = r74787 / r74788;
        double r74790 = 1.0;
        double r74791 = b;
        double r74792 = r74791 * r74791;
        double r74793 = a;
        double r74794 = r74793 * r74793;
        double r74795 = r74792 - r74794;
        double r74796 = r74790 / r74795;
        double r74797 = r74789 * r74796;
        double r74798 = r74790 / r74793;
        double r74799 = r74790 / r74791;
        double r74800 = r74798 - r74799;
        double r74801 = r74797 * r74800;
        return r74801;
}
↓
double f(double a, double b) {
        double r74802 = atan2(1.0, 0.0);
        double r74803 = 2.0;
        double r74804 = r74802 / r74803;
        double r74805 = b;
        double r74806 = a;
        double r74807 = r74805 + r74806;
        double r74808 = r74804 / r74807;
        double r74809 = 1.0;
        double r74810 = r74808 * r74809;
        double r74811 = r74809 / r74806;
        double r74812 = r74809 / r74805;
        double r74813 = r74811 - r74812;
        double r74814 = r74810 * r74813;
        double r74815 = r74805 - r74806;
        double r74816 = r74814 / r74815;
        return r74816;
}

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-squares9.9

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

  4. Applied *-un-lft-identity9.9

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

  5. Applied times-frac9.4

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

  6. Applied associate-*r*9.4

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

  7. Simplified9.3

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

  9. Applied associate-*r/9.3

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

  10. Applied associate-*l/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))