Average Error: 13.9 → 0.3
Time: 37.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a + b} \cdot \left(\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}\right)}{b - a}\]
\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{\pi}{a + b} \cdot \left(\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}\right)}{b - a}
double f(double a, double b) {
        double r1341804 = atan2(1.0, 0.0);
        double r1341805 = 2.0;
        double r1341806 = r1341804 / r1341805;
        double r1341807 = 1.0;
        double r1341808 = b;
        double r1341809 = r1341808 * r1341808;
        double r1341810 = a;
        double r1341811 = r1341810 * r1341810;
        double r1341812 = r1341809 - r1341811;
        double r1341813 = r1341807 / r1341812;
        double r1341814 = r1341806 * r1341813;
        double r1341815 = r1341807 / r1341810;
        double r1341816 = r1341807 / r1341808;
        double r1341817 = r1341815 - r1341816;
        double r1341818 = r1341814 * r1341817;
        return r1341818;
}
↓
double f(double a, double b) {
        double r1341819 = atan2(1.0, 0.0);
        double r1341820 = a;
        double r1341821 = b;
        double r1341822 = r1341820 + r1341821;
        double r1341823 = r1341819 / r1341822;
        double r1341824 = 0.5;
        double r1341825 = r1341824 / r1341820;
        double r1341826 = r1341824 / r1341821;
        double r1341827 = r1341825 - r1341826;
        double r1341828 = r1341823 * r1341827;
        double r1341829 = r1341821 - r1341820;
        double r1341830 = r1341828 / r1341829;
        return r1341830;
}

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 13.9

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

  2. Simplified9.4

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

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

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

  5. Applied div-inv9.4

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

  6. Applied times-frac9.4

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

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

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

  8. Applied div-inv9.4

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

  9. Applied times-frac9.4

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

  10. Applied distribute-lft-out--9.4

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

  11. Applied times-frac0.3

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

  12. Simplified0.3

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

  13. Simplified0.3

    \[\leadsto \frac{\pi}{a + b} \cdot \color{blue}{\frac{\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}}{b - a}}\]
  14. Using strategy rm

  15. Applied associate-*r/0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019143 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))