Average Error: 52.6 → 11.3
Time: 6.8m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{1}{64} + \frac{0.01171875}{i \cdot i}}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{i} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{\left(\alpha + i\right) + \beta}} \le 7.41335420571098 \cdot 10^{-102}:\\ \;\;\;\;0\\ \mathbf{if}\;\frac{\sqrt[3]{\frac{1}{64} + \frac{0.01171875}{i \cdot i}}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{i} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{\left(\alpha + i\right) + \beta}} \le 0.062499999999980876:\\ \;\;\;\;\frac{\frac{(\left(\left(\alpha + i\right) + \beta\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{1}{64}}\right) \cdot \left(\frac{\frac{0.25}{i}}{i}\right) + \frac{1}{4})_*}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\frac{i}{(i \cdot 2 + \beta)_* + \alpha} \cdot \left(\left(\beta + \alpha\right) + i\right)\right)\\ \end{array}\]

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)))) < 7.41335420571098e-102

    1. Initial program 61.1

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

    2. Applied simplify44.9

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

    3. Taylor expanded around inf 29.8

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

    4. Applied simplify29.8

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

    if 7.41335420571098e-102 < (/ (cbrt (+ 1/64 (/ 0.01171875 (* i i)))) (* (/ (+ beta (fma i 2 alpha)) i) (/ (+ beta (fma i 2 alpha)) (+ (+ alpha i) beta)))) < 0.062499999999980876

    1. Initial program 52.2

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

    2. Applied simplify40.3

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

    4. Applied add-sqr-sqrt40.3

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

    5. Applied associate-/r*40.3

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

    if 0.062499999999980876 < (/ (cbrt (+ 1/64 (/ 0.01171875 (* i i)))) (* (/ (+ beta (fma i 2 alpha)) i) (/ (+ beta (fma i 2 alpha)) (+ (+ 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. 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-cube55.2

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

    5. Applied add-cbrt-cube55.2

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

    6. Applied cbrt-undiv55.3

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

    7. Applied simplify37.7

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

    8. Taylor expanded around inf 0.1

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

    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: 6.8m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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)))