Average Error: 53.0 → 10.8
Time: 5.9m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \le 1.9464838440783873 \cdot 10^{-134}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right) \le 0.06249999999999822:\\ \;\;\;\;\left(\frac{\sqrt{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}} \cdot \frac{\sqrt{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\right) \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right)\\ \end{array}\]

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))) < 1.9464838440783873e-134

    1. Initial program 62.6

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

    2. Applied simplify44.1

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

    3. Taylor expanded around inf 20.4

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

    4. Applied simplify20.4

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

    if 1.9464838440783873e-134 < (* (/ (fma (cbrt 1/64) (/ (/ 0.25 i) i) 1/4) (+ (fma i 2 beta) alpha)) (* (/ i (+ (fma i 2 beta) alpha)) (+ (+ beta alpha) i))) < 0.06249999999999822

    1. Initial program 53.5

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

    2. Applied simplify41.5

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

    4. Applied add-sqr-sqrt41.5

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

    5. Applied add-sqr-sqrt41.5

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

    6. Applied times-frac41.5

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

    if 0.06249999999999822 < (* (/ (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 51.6

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

    2. Applied simplify37.7

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

    4. Applied add-cbrt-cube55.6

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

    5. Applied add-cbrt-cube55.5

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

    6. Applied cbrt-undiv55.6

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

    7. Applied simplify38.1

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

    8. Taylor expanded around inf 0.3

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

    9. Applied simplify0.4

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

Runtime

Time bar (total: 5.9m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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)))