Average Error: 23.9 → 12.8
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{(\left(\frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)} \cdot \sqrt{(2 \cdot i + \alpha)_* + \left(2.0 + \beta\right)}}\right) \cdot \left(\frac{1}{\beta + (2 \cdot i + \alpha)_*} \cdot \left(\alpha + \beta\right)\right) + 1.0)_*}{2.0}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.9

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

  2. Initial simplification12.7

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

  4. Applied add-sqr-sqrt12.8

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

  5. Applied associate-/r*12.8

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

  7. Applied associate-/l/12.8

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

  9. Applied div-inv12.8

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

  10. Final simplification12.8

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

Runtime

Time bar (total: 1.3m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.812.812.40.40%
herbie shell --seed 2018286 +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))