Average Error: 13.9 → 0.3
Time: 37.2s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{a + b} \cdot \left(\frac{\pi}{b \cdot a} \cdot \frac{1}{2}\right)\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.9

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

  2. Simplified9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac0.3

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

  6. Simplified0.3

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))