Average Error: 23.8 → 12.1
Time: 33.2s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log_* (1 + (e^{\log \left((\left(\frac{\beta - \alpha}{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}\right) \cdot \left(\frac{1}{\frac{\beta + (2 \cdot i + \alpha)_*}{\alpha + \beta}}\right) + 1.0)_*\right)} - 1)^*)}}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.8

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

  2. Initial simplification12.1

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

  4. Applied add-exp-log12.1

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

  6. Applied *-un-lft-identity12.1

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

  7. Applied associate-/l*12.1

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

  9. Applied log1p-expm1-u12.1

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

  10. Final simplification12.1

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

Runtime

Time bar (total: 33.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.112.111.90.30%
herbie shell --seed 2018353 +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))