Average Error: 13.9 → 0.3
Time: 8.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \frac{\frac{1 \cdot \left(b - a\right)}{a \cdot b}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \frac{\frac{1 \cdot \left(b - a\right)}{a \cdot b}}{b - a}
double code(double a, double b) {
	return (((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b)));
}
↓
double code(double a, double b) {
	return ((((((double) M_PI) / 2.0) / (b + a)) * 1.0) * (((1.0 * (b - a)) / (a * b)) / (b - a)));
}

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. Using strategy rm

  3. Applied difference-of-squares8.9

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

  4. Applied *-un-lft-identity8.9

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

  5. Applied times-frac8.6

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

  6. Applied associate-*r*8.6

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

  7. Simplified8.5

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

  9. Applied associate-*r/8.5

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

  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 *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  16. Applied frac-sub0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 +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))))