Average Error: 52.3 → 37.9
Time: 1.9m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.1241950317340853 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{(\left(\beta + \alpha\right) \cdot i + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\left(i + \alpha\right) \cdot \frac{\beta + i}{(i \cdot 2 + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot i}{\sqrt{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\left(i + \alpha\right) \cdot \frac{\beta + i}{(i \cdot 2 + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.1241950317340853e+136

    1. Initial program 50.1

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

    3. Applied times-frac35.6

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

    4. Simplified35.6

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \color{blue}{\frac{\left(\alpha + i\right) \cdot \left(i + \beta\right)}{(2 \cdot i + \beta)_* + \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt35.6

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

    7. Applied times-frac35.6

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

    8. Simplified35.6

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

    9. Simplified35.5

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

    if 3.1241950317340853e+136 < alpha

    1. Initial program 62.4

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

    3. Applied times-frac54.6

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

    4. Simplified54.6

      \[\leadsto \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \color{blue}{\frac{\left(\alpha + i\right) \cdot \left(i + \beta\right)}{(2 \cdot i + \beta)_* + \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt54.6

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

    7. Applied times-frac54.6

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

    8. Simplified54.6

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

    9. Simplified53.8

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

    10. Taylor expanded around 0 48.3

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

  4. Final simplification37.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.1241950317340853 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{(\left(\beta + \alpha\right) \cdot i + \left(i \cdot i\right))_*}{(2 \cdot i + \left(\beta + \alpha\right))_*}}{\sqrt{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\left(i + \alpha\right) \cdot \frac{\beta + i}{(i \cdot 2 + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot i}{\sqrt{(\left((2 \cdot i + \left(\beta + \alpha\right))_*\right) \cdot \left((2 \cdot i + \left(\beta + \alpha\right))_*\right) + \left(-1.0\right))_*}} \cdot \frac{\left(i + \alpha\right) \cdot \frac{\beta + i}{(i \cdot 2 + \alpha)_* + \beta}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes38.937.936.32.640%
herbie shell --seed 2018297 +o rules:numerics
(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)))