Average Error: 14.2 → 0.2
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a + b}}{a \cdot b}\]

Error

Bits error versus a

Bits error versus b

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

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. 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/9.4

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

  6. Applied simplify0.2

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))