Average Error: 14.1 → 0.3
Time: 3.2m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{2} \cdot \frac{\frac{\pi}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]

Error

Bits error versus a

Bits error versus b

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. 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 div-inv9.5

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

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

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

  6. Applied times-frac9.5

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

  7. Applied simplify0.3

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))