Average Error: 22.9 → 6.2
Time: 3.9m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\frac{\alpha + \beta}{\sqrt{\left(\alpha + \beta\right) + \left(i + i\right)}}}{\sqrt{\left(\alpha + \beta\right) + \left(\left(i + 2.0\right) + i\right)}}}{\left|\sqrt[3]{\left(\alpha + \beta\right) + \left(\left(i + 2.0\right) + i\right)}\right|} \cdot \frac{\frac{\beta - \alpha}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0} \le 3.3393426912553537 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\log \left(e^{\frac{\beta + \alpha}{\left(\left(i + 2.0\right) + i\right) + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + \left(i + i\right)} + 1.0}\right)\right)}^{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (/ (/ (/ (+ alpha beta) (sqrt (+ (+ alpha beta) (+ i i)))) (sqrt (+ (+ alpha beta) (+ (+ i 2.0) i)))) (fabs (cbrt (+ (+ alpha beta) (+ (+ i 2.0) i))))) (/ (/ (- beta alpha) (sqrt (+ (+ alpha beta) (* 2 i)))) (sqrt (cbrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) 1.0) 2.0) < 3.3393426912553537e-16

    1. Initial program 62.6

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

    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. Applied simplify30.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}}\]

    if 3.3393426912553537e-16 < (/ (+ (* (/ (/ (/ (+ alpha beta) (sqrt (+ (+ alpha beta) (+ i i)))) (sqrt (+ (+ alpha beta) (+ (+ i 2.0) i)))) (fabs (cbrt (+ (+ alpha beta) (+ (+ i 2.0) i))))) (/ (/ (- beta alpha) (sqrt (+ (+ alpha beta) (* 2 i)))) (sqrt (cbrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) 1.0) 2.0)

    1. Initial program 13.4

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

    3. Applied add-sqr-sqrt13.4

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

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

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

    5. Applied times-frac0.5

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

    6. Applied times-frac0.5

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

    7. Applied simplify0.5

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

    9. Applied add-cbrt-cube0.5

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

    10. Applied simplify0.5

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}\right)}^{3}}}}{2.0}\]
    11. Using strategy rm

    12. Applied add-log-exp0.5

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(\log \left(e^{1.0 + \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}} \cdot \frac{\alpha + \beta}{\sqrt{\left(i + \left(\alpha + \beta\right)\right) + \left(i + 2.0\right)}}}\right)\right)}}^{3}}}{2.0}\]

    13. Applied simplify0.5

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

Runtime

Time bar (total: 3.9m)Debug logProfile

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