Average Error: 14.3 → 0.3
Time: 1.0m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{\pi}{a} - \frac{\pi}{b}\right) \cdot \frac{1}{2}}{a + b}}{b - a}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

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 simplification9.4

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

  4. Applied associate-/r*0.3

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

  5. Taylor expanded around inf 0.3

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

  6. Simplified0.3

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

  7. Taylor expanded around -inf 0.3

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

  8. Final simplification0.3

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))