Average Error: 15.9 → 2.6
Time: 4.7m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha} + \left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right)}{\frac{\left(\beta + 2.0\right) + \alpha}{\beta}} - \frac{{\left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)}^{3} - {1.0}^{3}}{1}}{\left(\frac{\beta - \left(2.0 - \alpha\right)}{\beta - \left(2.0 - \alpha\right)} \cdot \left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right) - \left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right)\right) \cdot \left(\left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right) + \frac{\beta - \left(2.0 - \alpha\right)}{\beta - \left(2.0 - \alpha\right)} \cdot \left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)\right)} \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}{2.0} \le 2.1632339495852993 \cdot 10^{-112}:\\ \;\;\;\;\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{\frac{\frac{\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha} + \left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right)}{\frac{\left(\beta + 2.0\right) + \alpha}{\beta}} - \frac{{\left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)}^{3} - {1.0}^{3}}{1}}{\left(\frac{\beta - \left(2.0 - \alpha\right)}{\beta - \left(2.0 - \alpha\right)} \cdot \left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right) - \left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right)\right) \cdot \left(\left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right) + \frac{\beta - \left(2.0 - \alpha\right)}{\beta - \left(2.0 - \alpha\right)} \cdot \left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)\right)} \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (/ (- (/ (+ (* (/ alpha (+ (+ beta 2.0) alpha)) (/ alpha (+ (+ beta 2.0) alpha))) (+ (/ (* 1.0 alpha) (+ (+ beta 2.0) alpha)) (* 1.0 1.0))) (/ (+ (+ beta 2.0) alpha) beta)) (/ (- (pow (/ alpha (+ (+ beta 2.0) alpha)) 3) (pow 1.0 3)) 1)) (* (- (* (/ (- beta (- 2.0 alpha)) (- beta (- 2.0 alpha))) (* (/ alpha (+ (+ beta 2.0) alpha)) (/ alpha (+ (+ beta 2.0) alpha)))) (+ (/ (* 1.0 alpha) (+ (+ beta 2.0) alpha)) (* 1.0 1.0))) (+ (+ (/ (* 1.0 alpha) (+ (+ beta 2.0) alpha)) (* 1.0 1.0)) (* (/ (- beta (- 2.0 alpha)) (- beta (- 2.0 alpha))) (* (/ alpha (+ (+ beta 2.0) alpha)) (/ alpha (+ (+ beta 2.0) alpha))))))) (- (* (* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0)))) (- (+ alpha beta) 2.0)) (+ (* 1.0 1.0) (* (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))) 2.0) < 2.1632339495852993e-112

    1. Initial program 60.5

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

    3. Applied div-sub60.5

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

    4. Applied associate-+l-59.5

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

    5. Taylor expanded around inf 5.8

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

    if 2.1632339495852993e-112 < (/ (* (/ (- (/ (+ (* (/ alpha (+ (+ beta 2.0) alpha)) (/ alpha (+ (+ beta 2.0) alpha))) (+ (/ (* 1.0 alpha) (+ (+ beta 2.0) alpha)) (* 1.0 1.0))) (/ (+ (+ beta 2.0) alpha) beta)) (/ (- (pow (/ alpha (+ (+ beta 2.0) alpha)) 3) (pow 1.0 3)) 1)) (* (- (* (/ (- beta (- 2.0 alpha)) (- beta (- 2.0 alpha))) (* (/ alpha (+ (+ beta 2.0) alpha)) (/ alpha (+ (+ beta 2.0) alpha)))) (+ (/ (* 1.0 alpha) (+ (+ beta 2.0) alpha)) (* 1.0 1.0))) (+ (+ (/ (* 1.0 alpha) (+ (+ beta 2.0) alpha)) (* 1.0 1.0)) (* (/ (- beta (- 2.0 alpha)) (- beta (- 2.0 alpha))) (* (/ alpha (+ (+ beta 2.0) alpha)) (/ alpha (+ (+ beta 2.0) alpha))))))) (- (* (* (/ alpha (+ (+ alpha beta) 2.0)) (/ alpha (- (* (+ alpha beta) (+ alpha beta)) (* 2.0 2.0)))) (- (+ alpha beta) 2.0)) (+ (* 1.0 1.0) (* (/ alpha (+ (+ alpha beta) 2.0)) 1.0)))) 2.0)

    1. Initial program 2.8

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

    3. Applied div-sub2.8

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

    4. Applied associate-+l-2.4

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

    6. Applied flip3--2.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}{2.0}\]

    7. Applied frac-sub2.5

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

    9. Applied flip-+2.6

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}{\left(\alpha + \beta\right) - 2.0}}} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}}{2.0}\]

    10. Applied associate-/r/2.6

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \color{blue}{\left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0} \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right)} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}}{2.0}\]

    11. Applied associate-*r*2.6

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

    13. Applied flip-+2.6

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \color{blue}{\frac{\left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}}{2.0}\]

    14. Applied associate-*r/2.6

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\color{blue}{\frac{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)}}}}{2.0}\]

    15. Applied associate-/r/2.6

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right) \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right)\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right) \cdot \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)} \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}}{2.0}\]

    16. Applied simplify1.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha} + \left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right)}{\frac{\left(\beta + 2.0\right) + \alpha}{\beta}} - \frac{{\left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)}^{3} - {1.0}^{3}}{1}}{\left(\frac{\beta - \left(2.0 - \alpha\right)}{\beta - \left(2.0 - \alpha\right)} \cdot \left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right) - \left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right)\right) \cdot \left(\left(\frac{1.0 \cdot \alpha}{\left(\beta + 2.0\right) + \alpha} + 1.0 \cdot 1.0\right) + \frac{\beta - \left(2.0 - \alpha\right)}{\beta - \left(2.0 - \alpha\right)} \cdot \left(\frac{\alpha}{\left(\beta + 2.0\right) + \alpha} \cdot \frac{\alpha}{\left(\beta + 2.0\right) + \alpha}\right)\right)}} \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) - \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}{2.0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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))