Average Error: 14.4 → 0.2
Time: 56.0s
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a \cdot b} \cdot \frac{1}{2}}{b + a}\]

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 simplification0.3

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

  4. Applied pow10.3

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

  5. Applied pow10.3

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

  6. Applied pow-prod-down0.3

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

  7. Simplified0.3

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

  8. Taylor expanded around inf 0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 56.0s)Debug logProfile

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