Average Error: 23.4 → 6.3
Time: 3.9m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\frac{1}{\sqrt[3]{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \sqrt[3]{i \cdot 2 + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\sqrt[3]{i \cdot 2 + \left(\alpha + \beta\right)}}\right) \cdot \left(\alpha + \beta\right)}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} + 1.0}{2.0} \le 4.540638698369293 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^{3}}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\alpha + \beta}{\sqrt{i \cdot 2 + \left(2.0 + \left(\alpha + \beta\right)\right)}} + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (/ (* (+ beta alpha) (* (/ 1 (* (cbrt (+ (+ alpha beta) (* 2 i))) (cbrt (+ (+ alpha beta) (* 2 i))))) (/ (- beta alpha) (cbrt (+ (+ alpha beta) (* 2 i)))))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0) < 4.540638698369293e-16

    1. Initial program 62.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around inf 30.1

      \[\leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified30.1

      \[\leadsto \frac{\color{blue}{\left(\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^{3}}\right) - \frac{4.0}{\alpha \cdot \alpha}}}{2.0}\]

    if 4.540638698369293e-16 < (/ (+ (/ (* (+ beta alpha) (* (/ 1 (* (cbrt (+ (+ alpha beta) (* 2 i))) (cbrt (+ (+ alpha beta) (* 2 i))))) (/ (- beta alpha) (cbrt (+ (+ alpha beta) (* 2 i)))))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0)

    1. Initial program 13.9

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt13.9

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity13.9

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    5. Applied times-frac0.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    6. Applied times-frac0.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

    7. Simplified0.6

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{\sqrt{i \cdot 2 + \left(\left(\beta + \alpha\right) + 2.0\right)}}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\frac{1}{\sqrt[3]{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \sqrt[3]{i \cdot 2 + \left(\alpha + \beta\right)}} \cdot \frac{\beta - \alpha}{\sqrt[3]{i \cdot 2 + \left(\alpha + \beta\right)}}\right) \cdot \left(\alpha + \beta\right)}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} + 1.0}{2.0} \le 4.540638698369293 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^{3}}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\alpha + \beta}{\sqrt{i \cdot 2 + \left(2.0 + \left(\alpha + \beta\right)\right)}} + 1.0}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))