Average Error: 24.1 → 6.2
Time: 7.9m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{\left(i + \beta\right) + \left(i + \alpha\right)}}{\sqrt{\left(\left(2.0 + \alpha\right) + \left(i + i\right)\right) + \beta}}}}{\frac{1}{\alpha + \beta}}}{\frac{\left|\sqrt[3]{\left(\left(2.0 + \alpha\right) + \left(i + i\right)\right) + \beta}\right|}{\sqrt[3]{\frac{\frac{\beta - \alpha}{\left(i + \beta\right) + \left(i + \alpha\right)}}{\sqrt{\left(\left(2.0 + \alpha\right) + \left(i + i\right)\right) + \beta}}}}} \cdot \frac{\sqrt[3]{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0} \le 2.5013384245820003 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \alpha}{1} \cdot \frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* (/ (/ (cbrt (/ (/ (- beta alpha) (+ (+ i beta) (+ i alpha))) (sqrt (+ (+ (+ 2.0 alpha) (+ i i)) beta)))) (/ 1 (+ alpha beta))) (/ (fabs (cbrt (+ (+ (+ 2.0 alpha) (+ i i)) beta))) (cbrt (/ (/ (- beta alpha) (+ (+ i beta) (+ i alpha))) (sqrt (+ (+ (+ 2.0 alpha) (+ i i)) beta)))))) (/ (cbrt (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) (sqrt (cbrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) 1.0) 2.0) < 2.5013384245820003e-10

    1. Initial program 62.2

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

    2. Taylor expanded around inf 28.8

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

    3. Applied simplify28.8

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

    if 2.5013384245820003e-10 < (/ (+ (* (/ (/ (cbrt (/ (/ (- beta alpha) (+ (+ i beta) (+ i alpha))) (sqrt (+ (+ (+ 2.0 alpha) (+ i i)) beta)))) (/ 1 (+ alpha beta))) (/ (fabs (cbrt (+ (+ (+ 2.0 alpha) (+ i i)) beta))) (cbrt (/ (/ (- beta alpha) (+ (+ i beta) (+ i alpha))) (sqrt (+ (+ (+ 2.0 alpha) (+ i i)) beta)))))) (/ (cbrt (/ (/ (- beta alpha) (+ (+ alpha beta) (* 2 i))) (sqrt (+ (+ (+ alpha beta) (* 2 i)) 2.0)))) (sqrt (cbrt (+ (+ (+ alpha beta) (* 2 i)) 2.0))))) 1.0) 2.0)

    1. Initial program 14.0

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

    3. Applied *-un-lft-identity14.0

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

    4. Applied *-un-lft-identity14.0

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

    5. Applied times-frac0.2

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

    6. Applied times-frac0.2

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

    7. Applied simplify0.2

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{1}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt0.2

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

    10. Applied associate-/r*0.2

      \[\leadsto \frac{\frac{\beta + \alpha}{1} \cdot \color{blue}{\frac{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 7.9m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))