Average Error: 14.6 → 0.3
Time: 3.2m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{b + a} \cdot \frac{\frac{\pi}{a} - \frac{\pi}{b}}{\left(b - a\right) \cdot 2}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.6

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

  2. Applied simplify9.6

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

  4. Applied *-un-lft-identity9.6

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  8. Applied frac-times0.4

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

  9. Applied simplify0.3

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))