Average Error: 14.5 → 0.8
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{\left(\left(a + b\right) \cdot 2\right) \cdot \left(a \cdot b\right)}\]

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 simplify0.3

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

  4. Applied frac-sub0.3

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

  5. Applied associate-/l/0.9

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

  7. Applied associate-*l/0.8

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

  8. Applied simplify0.8

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))