Average Error: 14.3 → 0.3
Time: 4.2m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{a + b} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{\left(b - a\right) \cdot 2}\]

Error

Bits error versus a

Bits error versus b

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Results

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Derivation

  1. Initial program 14.3

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

  2. Initial simplification9.1

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

  4. Applied *-un-lft-identity9.1

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

  5. Applied div-inv9.1

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))