Average Error: 14.5 → 0.3
Time: 4.3m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{\pi}{2} \cdot \frac{b - a}{b + a}\right) \cdot \frac{\frac{1}{b - a}}{a \cdot b}\]

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 14.5

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

  2. Applied simplify9.0

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

  4. Applied frac-sub9.0

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

  5. Applied associate-/r/9.0

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

  6. Applied div-inv9.0

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

  7. Applied times-frac0.3

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

  8. Applied simplify0.3

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))