Average Error: 14.2 → 0.2
Time: 2.4m
Precision: 64
Internal Precision: 320
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a \cdot b}}{a + b}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.2

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

  2. Initial simplification0.3

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

  4. Applied frac-sub0.3

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

  5. Applied associate-*l/0.3

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))