Average Error: 52.4 → 11.7
Time: 7.5m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}\right)} \le 6.5685747469931 \cdot 10^{-116}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{i}{4 \cdot \left(\beta + \alpha\right) + i \cdot 8} \cdot \frac{\left(\beta + \alpha\right) + i}{\left(i + \beta\right) + \left(i + \alpha\right)}\right)}^{3}}\\ \mathbf{if}\;e^{\log \left(\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}\right)} \le 0.062499999999999126:\\ \;\;\;\;\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (exp (log (/ i (* (+ (* 4 (+ alpha beta)) (* 8 i)) (/ (+ (+ alpha beta) (+ i i)) (+ alpha (+ i beta))))))) < 6.5685747469931e-116

    1. Initial program 61.7

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

    3. Applied times-frac47.1

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

    4. Applied associate-/l*47.1

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

    5. Taylor expanded around 0 61.7

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

    6. Applied simplify60.3

      \[\leadsto \color{blue}{\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube30.3

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}} \cdot \frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}\right) \cdot \frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}}\]

    9. Applied simplify29.8

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

    if 6.5685747469931e-116 < (exp (log (/ i (* (+ (* 4 (+ alpha beta)) (* 8 i)) (/ (+ (+ alpha beta) (+ i i)) (+ alpha (+ i beta))))))) < 0.062499999999999126

    1. Initial program 52.1

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

    3. Applied times-frac40.9

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

    4. Applied associate-/l*40.9

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

    if 0.062499999999999126 < (exp (log (/ i (* (+ (* 4 (+ alpha beta)) (* 8 i)) (/ (+ (+ alpha beta) (+ i i)) (+ alpha (+ i beta)))))))

    1. Initial program 50.9

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

    3. Applied times-frac37.1

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

    4. Applied associate-/l*37.0

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

    5. Taylor expanded around 0 37.4

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

    6. Applied simplify0.5

      \[\leadsto \color{blue}{\frac{i}{\left(4 \cdot \left(\alpha + \beta\right) + 8 \cdot i\right) \cdot \frac{\left(\alpha + \beta\right) + \left(i + i\right)}{\alpha + \left(i + \beta\right)}}}\]
    7. Using strategy rm

    8. Applied add-exp-log0.5

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

Runtime

Time bar (total: 7.5m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))