Average Error: 14.1 → 0.3
Time: 3.7s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{2} \cdot \left(\frac{\pi}{b - a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b + a}\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1}{2} \cdot \left(\frac{\pi}{b - a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b + 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) ((1.0 / 2.0) * ((double) ((((double) M_PI) / ((double) (b - a))) * (((double) ((1.0 / a) - (1.0 / b))) / ((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 14.1

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

  3. Applied difference-of-squares9.5

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

  4. Applied associate-/r*9.1

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

  6. Applied frac-times9.1

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

  7. Applied associate-*l/0.3

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

  8. Simplified0.3

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

  10. Applied associate-*l/0.3

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

  11. Applied associate-*r/0.3

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

  12. Simplified0.3

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

  14. Applied *-un-lft-identity0.3

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

  15. Applied times-frac0.3

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

  16. Applied times-frac0.4

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

  17. Simplified0.4

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

  18. Simplified0.3

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

  19. Final simplification0.3

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

Reproduce

herbie shell --seed 2020199 
(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))))