Average Error: 14.5 → 0.3
Time: 4.6m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{a} - \frac{1}{b}}{\frac{b - a}{\frac{\pi}{2}}}}{b + a}\]

Error

Bits error versus a

Bits error versus b

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. Applied simplify9.7

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

  4. Applied div-inv9.7

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

  5. Applied *-un-lft-identity9.7

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

  6. Applied times-frac9.7

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

  7. Applied simplify0.3

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{\pi}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}\]
  8. Removed slow pow expressions.

  9. Applied simplify0.3

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

Runtime

Time bar (total: 4.6m)Debug log

herbie shell --seed '#(3622638036 3041702260 3649696288 21285302 1742518495 296600799)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))