Average Error: 52.5 → 10.8
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}\right)}^{3}} \cdot \left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \le 0.06398106223002603:\\ \;\;\;\;\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \cdot \frac{(\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \log \left(e^{\frac{i}{(i \cdot 2 + \alpha)_* + \beta}}\right)\right) \cdot (\left(\frac{0.25}{i \cdot i}\right) \cdot \left(\sqrt[3]{\frac{1}{64}}\right) + \frac{1}{4})_*\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (* (cbrt (pow (/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0))) 3)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 alpha) beta)) (/ i (+ (fma i 2 alpha) beta)))) < 0.06398106223002603

    1. Initial program 39.3

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

    2. Initial simplification5.9

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

    if 0.06398106223002603 < (* (cbrt (pow (/ (fma (+ i (+ alpha beta)) i (* beta alpha)) (fma (+ (fma i 2 beta) alpha) (+ (fma i 2 beta) alpha) (- 1.0))) 3)) (* (/ (+ (+ alpha i) beta) (+ (fma i 2 alpha) beta)) (/ i (+ (fma i 2 alpha) beta))))

    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. Initial simplification62.1

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

    4. Applied add-cbrt-cube62.1

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

    5. Applied add-cbrt-cube62.1

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

    6. Applied cbrt-undiv62.1

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

    7. Simplified62.1

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

    8. Taylor expanded around inf 17.9

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

    9. Simplified17.9

      \[\leadsto \color{blue}{(\left(\frac{0.25}{i \cdot i}\right) \cdot \left(\sqrt[3]{\frac{1}{64}}\right) + \frac{1}{4})_*} \cdot \left(\frac{\left(\alpha + i\right) + \beta}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right)\]
    10. Using strategy rm

    11. Applied add-log-exp14.2

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left(\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \beta)_* + \alpha\right) \cdot \left((i \cdot 2 + \beta)_* + \alpha\right) + \left(-1.0\right))_*}\right)}^{3}} \cdot \left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \le 0.06398106223002603:\\ \;\;\;\;\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \cdot \frac{(\left(\beta + \left(\alpha + i\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\beta + \left(\alpha + i\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \log \left(e^{\frac{i}{(i \cdot 2 + \alpha)_* + \beta}}\right)\right) \cdot (\left(\frac{0.25}{i \cdot i}\right) \cdot \left(\sqrt[3]{\frac{1}{64}}\right) + \frac{1}{4})_*\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018215 +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)))