Average Error: 23.9 → 6.3
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{\alpha + \beta}{\left(2.0 + \alpha\right) + (2 \cdot i + \beta)_*}\right) \cdot \left(\frac{1}{\sqrt{(2 \cdot i + \alpha)_* + \beta}} \cdot \frac{\beta - \alpha}{\sqrt{(2 \cdot i + \alpha)_* + \beta}}\right) + 1.0)_*}{2.0} \le 1.570757098534734 \cdot 10^{-11}:\\ \;\;\;\;\frac{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(\frac{8.0}{\alpha} - 4.0\right) + \left(\frac{2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{\alpha + \beta}{\left(2.0 + \alpha\right) + (2 \cdot i + \beta)_*}\right) \cdot \left(\left(\beta - \alpha\right) \cdot \frac{1}{(2 \cdot i + \alpha)_* + \beta}\right) + 1.0)_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (fma (/ (+ alpha beta) (+ (+ 2.0 alpha) (fma 2 i beta))) (* (/ 1 (sqrt (+ (fma 2 i alpha) beta))) (/ (- beta alpha) (sqrt (+ (fma 2 i alpha) beta)))) 1.0) 2.0) < 1.570757098534734e-11

    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. Applied simplify60.4

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

    3. Taylor expanded around inf 30.2

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

    4. Applied simplify30.2

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

    if 1.570757098534734e-11 < (/ (fma (/ (+ alpha beta) (+ (+ 2.0 alpha) (fma 2 i beta))) (* (/ 1 (sqrt (+ (fma 2 i alpha) beta))) (/ (- beta alpha) (sqrt (+ (fma 2 i alpha) beta)))) 1.0) 2.0)

    1. Initial program 14.3

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

    2. Applied simplify0.3

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

    4. Applied div-inv0.3

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(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))