Average Error: 14.2 → 0.3
Time: 25.3s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a} - \frac{1}{\frac{b}{\pi}}}{2 \cdot \left(a + b\right)}}{b - a}\]

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. Simplified0.3

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

  4. Applied associate-*l/0.3

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

  5. Simplified0.3

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

  7. Applied clear-num0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019021 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))