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

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 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 *-un-lft-identity0.3

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

  5. Applied div-inv0.3

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

  6. Applied times-frac0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Runtime

Time bar (total: 1.0m)Debug logProfile

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