Average Error: 14.3 → 0.3
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{a} - \frac{1}{b}}{2 \cdot \left(a + b\right)}}{\frac{b - a}{\pi}}\]

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

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

  4. Applied associate-*r/0.3

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

  6. Applied associate-/l*0.3

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

  7. Final simplification0.3

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

Runtime

Time bar (total: 1.2m)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))))