Average Error: 52.7 → 36.6
Time: 3.4m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{{i}^{2} \cdot \frac{1}{4}}{\left(\beta + \left(i + \alpha\right)\right) \cdot \left(i \cdot 4\right) - 1.0} \le 5.997738384260425 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(i + \alpha\right) \cdot \left(i + \beta\right)}{(2 \cdot i + \beta)_* + \alpha} \cdot \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot i}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) \cdot \left(2 \cdot i + \left(\beta + \alpha\right)\right) - 1.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (* 1/4 (pow i 2)) (- (* (+ beta (+ i alpha)) (* 4 i)) 1.0)) < 5.997738384260425e-131

    1. Initial program 62.5

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

    2. Taylor expanded around inf 30.7

      \[\leadsto \color{blue}{0}\]

    if 5.997738384260425e-131 < (/ (* 1/4 (pow i 2)) (- (* (+ beta (+ i alpha)) (* 4 i)) 1.0))

    1. Initial program 51.2

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

    3. Applied times-frac37.5

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

    4. Simplified37.5

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \color{blue}{\frac{\left(\alpha + i\right) \cdot \left(i + \beta\right)}{(2 \cdot i + \beta)_* + \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification36.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{i}^{2} \cdot \frac{1}{4}}{\left(\beta + \left(i + \alpha\right)\right) \cdot \left(i \cdot 4\right) - 1.0} \le 5.997738384260425 \cdot 10^{-131}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(i + \alpha\right) \cdot \left(i + \beta\right)}{(2 \cdot i + \beta)_* + \alpha} \cdot \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot i}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) \cdot \left(2 \cdot i + \left(\beta + \alpha\right)\right) - 1.0}\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(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)))