Average Error: 14.3 → 0.3
Time: 1.0m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\sqrt{\frac{1}{2}}}{\frac{1}{\pi}} \cdot \frac{\sqrt{\frac{1}{2}}}{b + a}}{a \cdot b}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.3

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

  2. Applied simplify9.5

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

  4. Applied frac-sub9.5

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

  5. Applied associate-/r/9.5

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

  6. Applied associate-/r*9.5

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

  7. Applied simplify0.2

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

  9. Applied *-un-lft-identity0.2

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

  10. Applied div-inv0.2

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

  11. Applied times-frac0.3

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

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

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

  14. Applied add-sqr-sqrt0.6

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

  15. Applied times-frac0.4

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

  16. Applied associate-*r*0.3

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

  17. Applied simplify0.3

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))