Average Error: 16.6 → 3.2
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \le 4.871513055717215 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \le 1.3731149631863883 \cdot 10^{-06}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 4.871513055717215e-304

    1. Initial program 0.5

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-inv0.5

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

    if 4.871513055717215e-304 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0) < 1.3731149631863883e-06

    1. Initial program 59.5

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub59.5

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-57.7

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    5. Taylor expanded around inf 11.0

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0}\]

    if 1.3731149631863883e-06 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* 4.0 (/ 1 (pow alpha 2))) (+ (* 8.0 (/ 1 (pow alpha 3))) (* 2.0 (/ 1 alpha))))) 2.0)

    1. Initial program 0.1

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-inv0.1

      \[\leadsto \frac{\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018198 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))