Average Error: 14.0 → 0.2
Time: 1.9m
Precision: 64
Internal Precision: 320
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a + b}}{a \cdot b}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.0

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

  2. Applied simplify9.4

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

  4. Applied frac-sub9.5

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

  5. Applied associate-*l/9.5

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

  6. Applied simplify0.2

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))