\[\left(\frac{\pi}{2} \cdot \frac{1}{{b}^2 - {a}^2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
Test:
NMSE Section 6.1 mentioned, B
Bits:
128 bits
Bits error versus a
Bits error versus b
Time: 11.7 s
Input Error: 7.4
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\frac{\frac{1}{2}}{b + a} \cdot \frac{\frac{\pi}{a}}{b}\)
  1. Started with
    \[\left(\frac{\pi}{2} \cdot \frac{1}{{b}^2 - {a}^2}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
    7.4
  2. Using strategy rm
    7.4
  3. Applied frac-sub to get
    \[\left(\frac{\pi}{2} \cdot \frac{1}{{b}^2 - {a}^2}\right) \cdot \color{red}{\left(\frac{1}{a} - \frac{1}{b}\right)} \leadsto \left(\frac{\pi}{2} \cdot \frac{1}{{b}^2 - {a}^2}\right) \cdot \color{blue}{\frac{1 \cdot b - a \cdot 1}{a \cdot b}}\]
    7.5
  4. Applied associate-*l/ to get
    \[\color{red}{\left(\frac{\pi}{2} \cdot \frac{1}{{b}^2 - {a}^2}\right)} \cdot \frac{1 \cdot b - a \cdot 1}{a \cdot b} \leadsto \color{blue}{\frac{\pi \cdot \frac{1}{{b}^2 - {a}^2}}{2}} \cdot \frac{1 \cdot b - a \cdot 1}{a \cdot b}\]
    7.5
  5. Applied frac-times to get
    \[\color{red}{\frac{\pi \cdot \frac{1}{{b}^2 - {a}^2}}{2} \cdot \frac{1 \cdot b - a \cdot 1}{a \cdot b}} \leadsto \color{blue}{\frac{\left(\pi \cdot \frac{1}{{b}^2 - {a}^2}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{2 \cdot \left(a \cdot b\right)}}\]
    7.5
  6. Applied simplify to get
    \[\frac{\color{red}{\left(\pi \cdot \frac{1}{{b}^2 - {a}^2}\right) \cdot \left(1 \cdot b - a \cdot 1\right)}}{2 \cdot \left(a \cdot b\right)} \leadsto \frac{\color{blue}{\frac{\pi}{a + b}}}{2 \cdot \left(a \cdot b\right)}\]
    0.3
  7. Using strategy rm
    0.3
  8. Applied add-cube-cbrt to get
    \[\color{red}{\frac{\frac{\pi}{a + b}}{2 \cdot \left(a \cdot b\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{\frac{\pi}{a + b}}{2 \cdot \left(a \cdot b\right)}}\right)}^3}\]
    0.5
  9. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{\frac{\pi}{a + b}}{2 \cdot \left(a \cdot b\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{1}{2} \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}}\right)}^3\]
    3.3
  10. Taylor expanded around 0 to get
    \[{\color{red}{\left(\sqrt[3]{\frac{1}{2} \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{2} \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}}\right)}}^3\]
    3.3
  11. Applied simplify to get
    \[{\left(\sqrt[3]{\frac{1}{2} \cdot \frac{\pi}{b \cdot \left(a \cdot \left(b + a\right)\right)}}\right)}^3 \leadsto \frac{\frac{1}{2}}{b + a} \cdot \frac{\frac{\pi}{a}}{b}\]
    0.3
  12. Applied final simplification

Original test:


(lambda ((a default) (b default))
  #:name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (sqr b) (sqr a)))) (- (/ 1 a) (/ 1 b))))