Average Error: 14.3 → 0.3
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a + b}}{\frac{b \cdot a}{\pi} \cdot 2}\]

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 simplification0.3

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

  4. Applied associate-*l/0.3

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

  5. Simplified0.3

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

  7. Applied *-un-lft-identity0.3

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

  8. Applied times-frac0.3

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

  9. Applied associate-/l*0.3

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

  10. Taylor expanded around inf 0.3

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

  11. Final simplification0.3

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018257 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))