Average Error: 52.6 → 12.7
Time: 2.8m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\alpha + i\right) \cdot \left(\beta + i\right)}{\frac{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{i}} \cdot \frac{\beta + \left(\alpha + i\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0} \le 0.0624999837307957:\\ \;\;\;\;\left(\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\left(\alpha + i\right) \cdot \left(\beta + i\right)}{i \cdot 2 + \left(\beta + \alpha\right)}\right) \cdot \frac{\beta + \left(\alpha + i\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \frac{1}{16}\\ \end{array}\]

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. Split input into 2 regimes

  2. if (* (/ (* (+ i beta) (+ alpha i)) (/ (* (+ (* 2 i) (+ alpha beta)) (+ (* 2 i) (+ alpha beta))) i)) (/ (+ (+ alpha i) beta) (- (* (+ (* 2 i) (+ alpha beta)) (+ (* 2 i) (+ alpha beta))) 1.0))) < 0.0624999837307957

    1. Initial program 49.7

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

    2. Initial simplification29.5

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

    4. Applied times-frac17.7

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

    5. Simplified17.7

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

    if 0.0624999837307957 < (* (/ (* (+ i beta) (+ alpha i)) (/ (* (+ (* 2 i) (+ alpha beta)) (+ (* 2 i) (+ alpha beta))) i)) (/ (+ (+ alpha i) beta) (- (* (+ (* 2 i) (+ alpha beta)) (+ (* 2 i) (+ alpha beta))) 1.0)))

    1. Initial program 53.0

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

    2. Initial simplification49.6

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

    3. Taylor expanded around 0 38.6

      \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot i\right)} \cdot \frac{\left(\alpha + i\right) + \beta}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}\]
    4. Using strategy rm

    5. Applied add-exp-log38.6

      \[\leadsto \color{blue}{e^{\log \left(\left(\frac{1}{4} \cdot i\right) \cdot \frac{\left(\alpha + i\right) + \beta}{\left(2 \cdot i + \left(\alpha + \beta\right)\right) \cdot \left(2 \cdot i + \left(\alpha + \beta\right)\right) - 1.0}\right)}}\]

    6. Taylor expanded around inf 12.0

      \[\leadsto \color{blue}{0.015625 \cdot \frac{1}{{i}^{2}} + \frac{1}{16}}\]

    7. Simplified12.0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{\frac{0.015625}{i}}{i}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\alpha + i\right) \cdot \left(\beta + i\right)}{\frac{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{i}} \cdot \frac{\beta + \left(\alpha + i\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0} \le 0.0624999837307957:\\ \;\;\;\;\left(\frac{i}{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \frac{\left(\alpha + i\right) \cdot \left(\beta + i\right)}{i \cdot 2 + \left(\beta + \alpha\right)}\right) \cdot \frac{\beta + \left(\alpha + i\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.015625}{i}}{i} + \frac{1}{16}\\ \end{array}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018217 
(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)))