Average Error: 14.4 → 0.2
Time: 1.1m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{b + a}}{a \cdot b + a \cdot b}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.4

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

  2. Applied simplify9.4

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

  4. Applied *-un-lft-identity9.4

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

  5. Applied times-frac0.3

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

  6. Taylor expanded around 0 0.3

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

  7. Applied simplify0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))