Average Error: 14.1 → 0.2
Time: 3.2m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a + b}}{a \cdot b}\]

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 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 simplification9.5

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

  4. Applied pow19.5

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

  5. Applied pow19.5

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

  6. Applied pow-prod-down9.5

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

  7. Simplified0.2

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

  8. Final simplification0.2

    \[\leadsto \frac{\frac{\frac{\pi}{2}}{a + b}}{a \cdot b}\]

Runtime

Time bar (total: 3.2m)Debug logProfile

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