Average Error: 23.4 → 6.2
Time: 6.6m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}} \cdot \sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}}}{\sqrt{1}} \cdot \frac{\sqrt[3]{\frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}}}}{\sqrt{\beta + \left(\left(2.0 + \alpha\right) + i \cdot 2\right)}} + 1.0\right)}^{3}}}{2.0} \le 3.3306690738754696 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (cbrt (pow (+ (* (/ (* (cbrt (/ (* (+ alpha beta) (/ (- beta alpha) (+ (* i 2) (+ alpha beta)))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (cbrt (/ (* (+ alpha beta) (/ (- beta alpha) (+ (* i 2) (+ alpha beta)))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))))) (sqrt 1)) (/ (cbrt (/ (* (+ alpha beta) (/ (- beta alpha) (+ (* i 2) (+ alpha beta)))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) 1.0) 3)) 2.0) < 3.3306690738754696e-16

    1. Initial program 62.6

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

    2. Taylor expanded around inf 28.2

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

    3. Applied simplify28.2

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}}\]

    if 3.3306690738754696e-16 < (/ (cbrt (pow (+ (* (/ (* (cbrt (/ (* (+ alpha beta) (/ (- beta alpha) (+ (* i 2) (+ alpha beta)))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (cbrt (/ (* (+ alpha beta) (/ (- beta alpha) (+ (* i 2) (+ alpha beta)))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2))))))) (sqrt 1)) (/ (cbrt (/ (* (+ alpha beta) (/ (- beta alpha) (+ (* i 2) (+ alpha beta)))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) (sqrt (+ beta (+ (+ 2.0 alpha) (* i 2)))))) 1.0) 3)) 2.0)

    1. Initial program 13.5

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

    3. Applied *-un-lft-identity13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied times-frac0.7

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

    6. Applied times-frac0.7

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

    7. Applied simplify0.7

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

Runtime

Time bar (total: 6.6m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(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))