Average Error: 14.5 → 0.3
Time: 4.0m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\frac{\pi}{a} - \pi \cdot \frac{1}{b}}{2}}{a + b}}{b - a}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 simplify9.5

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

  4. Applied associate-/r*9.5

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

  5. Applied simplify9.5

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

  7. Applied associate-/r*0.3

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

  9. Applied div-inv0.3

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

Runtime

Time bar (total: 4.0m)Debug logProfile

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