Average Error: 16.5 → 6.8
Time: 11.2s
Precision: binary64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.5233629444021591 \cdot 10^{36}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{2}{\alpha} \cdot -1\right)}{2}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 2.5233629444021591e36

    1. Initial program 1.8

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt1.8

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

    4. Applied *-un-lft-identity1.8

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

    5. Applied times-frac1.8

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

    if 2.5233629444021591e36 < alpha

    1. Initial program 50.9

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub50.8

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-49.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]

    5. Taylor expanded around inf 18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    6. Simplified18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{2}{\alpha} \cdot -1\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.5233629444021591 \cdot 10^{36}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\left(\alpha + \beta\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{2}{\alpha} \cdot -1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))