Average Error: 52.7 → 11.5
Time: 7.1m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{1}{64} + \frac{0.01171875}{i \cdot i}}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{i} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{\left(\alpha + i\right) + \beta}} \le 1.859640338435923 \cdot 10^{-111}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{\sqrt[3]{\frac{1}{64} + \frac{0.01171875}{i \cdot i}}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{i} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{\left(\alpha + i\right) + \beta}} \le 0.062499999999999986:\\ \;\;\;\;(e^{\log_* (1 + \frac{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*})} - 1)^* \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right)\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (/ (cbrt (+ 1/64 (/ 0.01171875 (* i i)))) (* (/ (+ beta (fma i 2 alpha)) i) (/ (+ beta (fma i 2 alpha)) (+ (+ alpha i) beta)))) < 1.859640338435923e-111

    1. Initial program 62.1

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

    2. Applied simplify44.5

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

    3. Taylor expanded around inf 29.4

      \[\leadsto \color{blue}{0} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    4. Applied simplify29.4

      \[\leadsto \color{blue}{0}\]

    if 1.859640338435923e-111 < (/ (cbrt (+ 1/64 (/ 0.01171875 (* i i)))) (* (/ (+ beta (fma i 2 alpha)) i) (/ (+ beta (fma i 2 alpha)) (+ (+ alpha i) beta)))) < 0.062499999999999986

    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. Applied simplify40.8

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

    4. Applied expm1-log1p-u40.8

      \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*})} - 1)^*} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    if 0.062499999999999986 < (/ (cbrt (+ 1/64 (/ 0.01171875 (* i i)))) (* (/ (+ beta (fma i 2 alpha)) i) (/ (+ beta (fma i 2 alpha)) (+ (+ alpha i) beta))))

    1. Initial program 51.2

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

    2. Applied simplify36.8

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

    4. Applied add-cbrt-cube55.5

      \[\leadsto \frac{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\color{blue}{\sqrt[3]{\left((\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_* \cdot (\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*\right) \cdot (\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    5. Applied add-cbrt-cube55.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left((\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot (\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot (\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}}{\sqrt[3]{\left((\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_* \cdot (\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*\right) \cdot (\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    6. Applied cbrt-undiv55.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left((\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot (\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*\right) \cdot (\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\left((\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_* \cdot (\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*\right) \cdot (\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    7. Applied simplify37.2

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}\right)}^{3}}} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    8. Taylor expanded around inf 0.2

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\frac{1}{{i}^{2}} \cdot {\frac{1}{64}}^{\frac{1}{3}}\right) + {\left({\frac{1}{4}}^{3}\right)}^{\frac{1}{3}}\right)} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]

    9. Applied simplify0.3

      \[\leadsto \color{blue}{\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 7.1m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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)))