Average Error: 52.3 → 35.5
Time: 3.3m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(-1.0\right) \cdot (i \cdot \left(\alpha + \beta\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{\left(\alpha + \beta\right) + i}} \le -3.8443177687147436 \cdot 10^{+296}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\left(\beta + \alpha\right) + i}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot \left(\sqrt{\frac{(\left(\left(\beta + \alpha\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}} \cdot \sqrt{\frac{(\left(\left(\beta + \alpha\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}}\right)\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* (- 1.0) (fma i (+ alpha beta) (* alpha beta))) (/ i (fma 2 i (+ alpha beta)))) (/ (fma 2 i (+ alpha beta)) (+ (+ alpha beta) i))) < -3.8443177687147436e+296

    1. Initial program 62.3

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

    2. Applied simplify62.3

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

    3. Taylor expanded around inf 54.0

      \[\leadsto \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \frac{\left(\beta + \alpha\right) + i}{(i \cdot 2 + \beta)_* + \alpha}\right) \cdot \color{blue}{0}\]

    4. Applied simplify54.0

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

    if -3.8443177687147436e+296 < (/ (* (* (- 1.0) (fma i (+ alpha beta) (* alpha beta))) (/ i (fma 2 i (+ alpha beta)))) (/ (fma 2 i (+ alpha beta)) (+ (+ alpha beta) i)))

    1. Initial program 47.9

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

    2. Applied simplify27.5

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

    4. Applied add-sqr-sqrt27.5

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2020178 +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)))