Average Error: 14.4 → 0.2
Time: 3.4m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{2} \cdot \frac{\frac{\pi}{b + a}}{a \cdot b}\]

Error

Bits error versus a

Bits error versus b

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Results

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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. Initial simplification9.3

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

  4. Applied div-inv9.3

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

  5. Applied *-un-lft-identity9.3

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

  6. Applied times-frac9.3

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

  7. Simplified0.3

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

  8. Taylor expanded around -inf 0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018230 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))