Average Error: 14.7 → 0.3
Time: 12.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{b + a} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\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}{2}}{b + a} \cdot 1}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r71785 = atan2(1.0, 0.0);
        double r71786 = 2.0;
        double r71787 = r71785 / r71786;
        double r71788 = 1.0;
        double r71789 = b;
        double r71790 = r71789 * r71789;
        double r71791 = a;
        double r71792 = r71791 * r71791;
        double r71793 = r71790 - r71792;
        double r71794 = r71788 / r71793;
        double r71795 = r71787 * r71794;
        double r71796 = r71788 / r71791;
        double r71797 = r71788 / r71789;
        double r71798 = r71796 - r71797;
        double r71799 = r71795 * r71798;
        return r71799;
}
↓
double f(double a, double b) {
        double r71800 = atan2(1.0, 0.0);
        double r71801 = 2.0;
        double r71802 = r71800 / r71801;
        double r71803 = b;
        double r71804 = a;
        double r71805 = r71803 + r71804;
        double r71806 = r71802 / r71805;
        double r71807 = 1.0;
        double r71808 = r71806 * r71807;
        double r71809 = r71803 - r71804;
        double r71810 = r71807 / r71804;
        double r71811 = r71807 / r71803;
        double r71812 = r71810 - r71811;
        double r71813 = r71809 / r71812;
        double r71814 = r71808 / r71813;
        return r71814;
}

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 14.7

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

  3. Applied difference-of-squares9.7

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

  4. Applied *-un-lft-identity9.7

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

  5. Applied times-frac9.3

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

  6. Applied associate-*r*9.3

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

  7. Simplified9.2

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

  9. Applied associate-*r/9.2

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

  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. Using strategy rm

  12. Applied associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020002 +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))))