Average Error: 52.6 → 37.6
Time: 1.4m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\frac{\beta}{\frac{-1.0}{0.125}}\right) \cdot \left(\frac{\alpha}{-1.0}\right) + \left((\left(\frac{\alpha}{-1.0} \cdot \frac{\alpha}{-1.0}\right) \cdot \left(0.5625 + \frac{0.25}{-1.0}\right) + \left(\left(\beta \cdot \beta\right) \cdot 0.0625\right))_*\right))_* - \left(0.25 \cdot \alpha\right) \cdot \left(\beta + \alpha\right) \le 2.2924923524389303 \cdot 10^{+274}:\\ \;\;\;\;\frac{\frac{\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot (\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (- (fma (/ beta (/ -1.0 0.125)) (/ alpha -1.0) (fma (* (/ alpha -1.0) (/ alpha -1.0)) (+ 0.5625 (/ 0.25 -1.0)) (* (* beta beta) 0.0625))) (* (* 0.25 alpha) (+ beta alpha))) < 2.2924923524389303e+274

    1. Initial program 47.6

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

    2. Applied simplify31.5

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

    4. Applied add-sqr-sqrt31.6

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

    5. Applied associate-/r*31.6

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

    if 2.2924923524389303e+274 < (- (fma (/ beta (/ -1.0 0.125)) (/ alpha -1.0) (fma (* (/ alpha -1.0) (/ alpha -1.0)) (+ 0.5625 (/ 0.25 -1.0)) (* (* beta beta) 0.0625))) (* (* 0.25 alpha) (+ beta alpha)))

    1. Initial program 62.4

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

    2. Applied simplify54.7

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

    3. Taylor expanded around inf 49.3

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +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)))