Average Error: 52.3 → 11.1
Time: 1.8m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\left(\sqrt[3]{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}} \cdot \sqrt[3]{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}\right) \cdot \sqrt[3]{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}} \le 2.8203040010432394 \cdot 10^{-139}:\\ \;\;\;\;\log \left(e^{\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}}\right)\\ \mathbf{if}\;\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\left(\sqrt[3]{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}} \cdot \sqrt[3]{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}\right) \cdot \sqrt[3]{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}} \le 1.1560519937171816 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \left(i + \beta\right)}{\beta + (i \cdot 2 + \alpha)_*}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{i}}}{\sqrt{(\left(\beta + (i \cdot 2 + \alpha)_*\right) \cdot \left(\beta + (i \cdot 2 + \alpha)_*\right) + \left(-1.0\right))_*}} \cdot \frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{\sqrt{(\left((i \cdot 2 + \alpha)_* + \beta\right) \cdot \left((i \cdot 2 + \alpha)_* + \beta\right) + \left(-1.0\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}\right)}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (/ (/ 0.0625 i) i) 1/4) (* (* (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i))) (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i)))) (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i))))) < 2.8203040010432394e-139

    1. Initial program 62.5

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

    2. Applied simplify44.0

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

    4. Applied add-sqr-sqrt44.0

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

    5. Taylor expanded around inf 60.2

      \[\leadsto \left(\frac{i + \left(\alpha + \beta\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \cdot \color{blue}{\left(\frac{1}{4} + 0.0625 \cdot \frac{1}{{i}^{2}}\right)}\]

    6. Applied simplify60.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}}\]
    7. Using strategy rm

    8. Applied add-log-exp18.6

      \[\leadsto \color{blue}{\log \left(e^{\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}}\right)}\]

    if 2.8203040010432394e-139 < (/ (+ (/ (/ 0.0625 i) i) 1/4) (* (* (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i))) (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i)))) (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i))))) < 1.1560519937171816e-13

    1. Initial program 53.7

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

    2. Applied simplify43.4

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

    4. Applied add-sqr-sqrt43.4

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

    5. Applied *-un-lft-identity43.4

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

    6. Applied times-frac43.4

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

    7. Applied associate-*r*43.4

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

    8. Applied simplify43.4

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

    if 1.1560519937171816e-13 < (/ (+ (/ (/ 0.0625 i) i) 1/4) (* (* (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i))) (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i)))) (cbrt (* (/ (+ beta (fma i 2 alpha)) (+ (+ i beta) alpha)) (/ (+ beta (fma i 2 alpha)) i)))))

    1. Initial program 50.6

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

    2. Applied simplify36.4

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

    4. Applied add-sqr-sqrt36.4

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

    5. Taylor expanded around inf 2.2

      \[\leadsto \left(\frac{i + \left(\alpha + \beta\right)}{(i \cdot 2 + \alpha)_* + \beta} \cdot \frac{i}{(i \cdot 2 + \alpha)_* + \beta}\right) \cdot \color{blue}{\left(\frac{1}{4} + 0.0625 \cdot \frac{1}{{i}^{2}}\right)}\]

    6. Applied simplify2.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}}\]
    7. Using strategy rm

    8. Applied add-exp-log2.2

      \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\frac{0.0625}{i}}{i} + \frac{1}{4}}{\frac{\beta + (i \cdot 2 + \alpha)_*}{\left(i + \beta\right) + \alpha} \cdot \frac{\beta + (i \cdot 2 + \alpha)_*}{i}}\right)}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))