Average Error: 14.1 → 0.3
Time: 3.4s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(1 \cdot \frac{1}{b \cdot a}\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{\left(b + a\right) \cdot 2} \cdot \left(1 \cdot \frac{1}{b \cdot a}\right)
double code(double a, double b) {
	return ((double) (((double) ((((double) M_PI) / 2.0) * (1.0 / ((double) (((double) (b * b)) - ((double) (a * a))))))) * ((double) ((1.0 / a) - (1.0 / b)))));
}
↓
double code(double a, double b) {
	return ((double) ((((double) M_PI) / ((double) (((double) (b + a)) * 2.0))) * ((double) (1.0 * (1.0 / ((double) (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 Error: 14.1 bits

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

  3. Applied difference-of-squaresError: 9.4 bits

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

  4. Applied *-un-lft-identityError: 9.4 bits

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

  5. Applied times-fracError: 9.0 bits

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

  6. Applied associate-*r*Error: 8.9 bits

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

  7. SimplifiedError: 8.9 bits

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

  9. Applied associate-*r/Error: 8.8 bits

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

  10. Applied associate-*l/Error: 0.3 bits

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

  11. SimplifiedError: 0.3 bits

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

  13. Applied *-un-lft-identityError: 0.3 bits

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

  14. Applied times-fracError: 0.3 bits

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

  15. SimplifiedError: 0.3 bits

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

  16. SimplifiedError: 0.3 bits

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

  17. Taylor expanded around 0 Error: 0.3 bits

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

  18. SimplifiedError: 0.3 bits

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

  19. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))