Average Error: 14.1 → 0.4
Time: 3.4m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{1}{a + b} \cdot \frac{b - a}{a \cdot b}\right) \cdot \frac{\frac{\pi}{2}}{b - a}\]

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 14.1

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

  2. Initial simplification14.1

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

  4. Applied difference-of-squares9.5

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

  5. Applied *-un-lft-identity9.5

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

  6. Applied times-frac9.0

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

  7. Applied associate-*r*0.4

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

  8. Final simplification0.4

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))