Average Error: 24.2 → 12.5
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{(\left(\log_* (1 + (e^{\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}} - 1)^*)\right) \cdot \left(\log_* (1 + (e^{\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}} - 1)^*)\right) + 1.0)_*}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.2

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

  2. Applied simplify12.5

    \[\leadsto \color{blue}{\frac{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \left(\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}\right) + 1.0)_*}{2.0}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u12.5

    \[\leadsto \frac{(\left(\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}\right) \cdot \color{blue}{\left(\log_* (1 + (e^{\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}} - 1)^*)\right)} + 1.0)_*}{2.0}\]
  5. Using strategy rm

  6. Applied log1p-expm1-u12.5

    \[\leadsto \frac{(\color{blue}{\left(\log_* (1 + (e^{\frac{\beta + \alpha}{(i \cdot 2 + \beta)_* + \left(2.0 + \alpha\right)}} - 1)^*)\right)} \cdot \left(\log_* (1 + (e^{\frac{\beta - \alpha}{(2 \cdot i + \left(\beta + \alpha\right))_*}} - 1)^*)\right) + 1.0)_*}{2.0}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))