Average Error: 52.3 → 36.3
Time: 6.3m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{(i \cdot \left(\alpha + \beta\right) + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\frac{(4 \cdot \left((i \cdot \left(\alpha + \beta\right) + \left(i \cdot i\right))_*\right) + \left(-1.0\right))_*}{\frac{\left(\alpha + i\right) \cdot \left(i + \beta\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}} \le 1.315722491495467 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{-i}{(2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{(\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{\beta + \left(\alpha + i\right)}}}{1.0 - \left((2 \cdot i + \alpha)_* + \beta\right) \cdot \left((2 \cdot i + \alpha)_* + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (fma i (+ alpha beta) (* i i)) (fma 2 i (+ alpha beta))) (/ (fma 4 (fma i (+ alpha beta) (* i i)) (- 1.0)) (/ (* (+ alpha i) (+ i beta)) (fma 2 i (+ alpha beta))))) < 1.315722491495467e+303

    1. Initial program 37.9

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

    3. Applied frac-2neg37.9

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

    4. Applied simplify5.3

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

    5. Applied simplify5.3

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

    if 1.315722491495467e+303 < (/ (/ (fma i (+ alpha beta) (* i i)) (fma 2 i (+ alpha beta))) (/ (fma 4 (fma i (+ alpha beta) (* i i)) (- 1.0)) (/ (* (+ alpha i) (+ i beta)) (fma 2 i (+ alpha beta)))))

    1. Initial program 62.2

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

    2. Taylor expanded around inf 57.8

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed 2018296 +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)))