Average Error: 14.4 → 0.3
Time: 3.3m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{b - a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{\left(b + a\right) \cdot 2}\]

Error

Bits error versus a

Bits error versus b

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Results

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Derivation

  1. Initial program 14.4

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

  2. Applied simplify9.7

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

  4. Applied *-un-lft-identity9.7

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

  5. Applied times-frac0.3

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

  7. Applied associate-*l/0.3

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

  8. Applied frac-times0.3

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

  9. Applied simplify0.3

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))