Average Error: 14.5 → 0.3
Time: 25.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a + b}}{\frac{\left(b - a\right) \cdot 2}{\frac{\pi}{a} - \frac{\pi}{b}}}\]
\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{1}{a + b}}{\frac{\left(b - a\right) \cdot 2}{\frac{\pi}{a} - \frac{\pi}{b}}}
double f(double a, double b) {
        double r1777800 = atan2(1.0, 0.0);
        double r1777801 = 2.0;
        double r1777802 = r1777800 / r1777801;
        double r1777803 = 1.0;
        double r1777804 = b;
        double r1777805 = r1777804 * r1777804;
        double r1777806 = a;
        double r1777807 = r1777806 * r1777806;
        double r1777808 = r1777805 - r1777807;
        double r1777809 = r1777803 / r1777808;
        double r1777810 = r1777802 * r1777809;
        double r1777811 = r1777803 / r1777806;
        double r1777812 = r1777803 / r1777804;
        double r1777813 = r1777811 - r1777812;
        double r1777814 = r1777810 * r1777813;
        return r1777814;
}
↓
double f(double a, double b) {
        double r1777815 = 1.0;
        double r1777816 = a;
        double r1777817 = b;
        double r1777818 = r1777816 + r1777817;
        double r1777819 = r1777815 / r1777818;
        double r1777820 = r1777817 - r1777816;
        double r1777821 = 2.0;
        double r1777822 = r1777820 * r1777821;
        double r1777823 = atan2(1.0, 0.0);
        double r1777824 = r1777823 / r1777816;
        double r1777825 = r1777823 / r1777817;
        double r1777826 = r1777824 - r1777825;
        double r1777827 = r1777822 / r1777826;
        double r1777828 = r1777819 / r1777827;
        return r1777828;
}

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 14.5

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

  2. Simplified14.4

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

  4. Applied *-un-lft-identity14.4

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

  5. Applied *-un-lft-identity14.4

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

  6. Applied times-frac14.4

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

  7. Applied *-un-lft-identity14.4

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

  8. Applied *-un-lft-identity14.4

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

  9. Applied times-frac14.4

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

  10. Applied distribute-lft-out--14.4

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

  11. Applied times-frac14.4

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

  12. Simplified9.2

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

  14. Applied frac-times0.3

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

  16. Applied associate-/l*0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019134 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))