Average Error: 52.5 → 36.1
Time: 5.0m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\frac{\left(\beta + \left(i + \alpha\right)\right) \cdot \frac{i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}}{\frac{2 \cdot i + \left(\alpha + \beta\right)}{\beta + i} \cdot \frac{2 \cdot i + \left(\alpha + \beta\right)}{i + \alpha}}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.5

    \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

  2. Initial simplification38.7

    \[\leadsto \frac{i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0} \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\left(\beta \cdot \alpha + \alpha \cdot i\right) + i \cdot \left(i + \beta\right)}}\]
  3. Using strategy rm

  4. Applied associate-*r/38.7

    \[\leadsto \color{blue}{\frac{\frac{i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0} \cdot \left(\left(\alpha + i\right) + \beta\right)}{\frac{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right)}{\left(\beta \cdot \alpha + \alpha \cdot i\right) + i \cdot \left(i + \beta\right)}}}\]

  5. Simplified36.1

    \[\leadsto \frac{\frac{i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0} \cdot \left(\left(\alpha + i\right) + \beta\right)}{\color{blue}{\frac{i \cdot 2 + \left(\alpha + \beta\right)}{i + \beta} \cdot \frac{i \cdot 2 + \left(\alpha + \beta\right)}{\alpha + i}}}\]

  6. Final simplification36.1

    \[\leadsto \frac{\left(\beta + \left(i + \alpha\right)\right) \cdot \frac{i}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}}{\frac{2 \cdot i + \left(\alpha + \beta\right)}{\beta + i} \cdot \frac{2 \cdot i + \left(\alpha + \beta\right)}{i + \alpha}}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018215 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))