Average Error: 52.7 → 10.5
Time: 4.3m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \le 1.1970810262121994 \cdot 10^{-132}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \le 0.06149994495091202:\\ \;\;\;\;\frac{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right)\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (fma (cbrt 1/64) (/ (/ 0.25 i) i) 1/4) (+ (fma i 2 beta) alpha)) (* (/ i (+ (fma i 2 beta) alpha)) (+ (+ beta alpha) i))) < 1.1970810262121994e-132

    1. Initial program 62.2

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

    2. Applied simplify41.9

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

    3. Taylor expanded around inf 20.0

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

    4. Applied simplify20.0

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

    if 1.1970810262121994e-132 < (* (/ (fma (cbrt 1/64) (/ (/ 0.25 i) i) 1/4) (+ (fma i 2 beta) alpha)) (* (/ i (+ (fma i 2 beta) alpha)) (+ (+ beta alpha) i))) < 0.06149994495091202

    1. Initial program 52.9

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

    2. Applied simplify40.4

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

    if 0.06149994495091202 < (* (/ (fma (cbrt 1/64) (/ (/ 0.25 i) i) 1/4) (+ (fma i 2 beta) alpha)) (* (/ i (+ (fma i 2 beta) alpha)) (+ (+ beta alpha) i)))

    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 simplify37.3

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

    4. Applied add-cbrt-cube55.1

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

    5. Applied add-cbrt-cube55.0

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

    6. Applied cbrt-undiv55.1

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

    7. Applied simplify37.6

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

    8. Taylor expanded around inf 0.6

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

    9. Applied simplify0.8

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +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)))