Average Error: 14.2 → 0.2
Time: 36.6s
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{b \cdot a}\]

Error

Bits error versus a

Bits error versus b

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Results

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Derivation

  1. Initial program 14.2

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

  2. Initial simplification0.3

    \[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2}}{a + b}\]
  3. Using strategy rm

  4. Applied associate-*r/0.3

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

  5. Taylor expanded around inf 0.2

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

  7. Applied div-inv0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*l/0.3

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Runtime

Time bar (total: 36.6s)Debug logProfile

herbie shell --seed 2018250 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))