Octave 3.8, jcobi/2

Average Error: 23.8 → 7.4

Time: 1.6m

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999998889776975374843459576368:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 \cdot \left(1 \cdot 1\right)} \cdot e^{\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)\right)}\right)}{\frac{1 \cdot 1 - \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}{\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} + 1} \cdot 1 + \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5532573 = alpha;
        double r5532574 = beta;
        double r5532575 = r5532573 + r5532574;
        double r5532576 = r5532574 - r5532573;
        double r5532577 = r5532575 * r5532576;
        double r5532578 = 2.0;
        double r5532579 = i;
        double r5532580 = r5532578 * r5532579;
        double r5532581 = r5532575 + r5532580;
        double r5532582 = r5532577 / r5532581;
        double r5532583 = r5532581 + r5532578;
        double r5532584 = r5532582 / r5532583;
        double r5532585 = 1.0;
        double r5532586 = r5532584 + r5532585;
        double r5532587 = r5532586 / r5532578;
        return r5532587;
}

↓

double f(double alpha, double beta, double i) {
        double r5532588 = beta;
        double r5532589 = alpha;
        double r5532590 = r5532588 + r5532589;
        double r5532591 = r5532588 - r5532589;
        double r5532592 = r5532590 * r5532591;
        double r5532593 = 2.0;
        double r5532594 = i;
        double r5532595 = r5532593 * r5532594;
        double r5532596 = r5532595 + r5532590;
        double r5532597 = r5532592 / r5532596;
        double r5532598 = r5532593 + r5532596;
        double r5532599 = r5532597 / r5532598;
        double r5532600 = -0.9999999999999999;
        bool r5532601 = r5532599 <= r5532600;
        double r5532602 = 8.0;
        double r5532603 = r5532589 * r5532589;
        double r5532604 = r5532603 * r5532589;
        double r5532605 = r5532602 / r5532604;
        double r5532606 = 4.0;
        double r5532607 = r5532606 / r5532603;
        double r5532608 = r5532605 - r5532607;
        double r5532609 = r5532593 / r5532589;
        double r5532610 = r5532608 + r5532609;
        double r5532611 = r5532610 / r5532593;
        double r5532612 = 1.0;
        double r5532613 = r5532612 * r5532612;
        double r5532614 = r5532612 * r5532613;
        double r5532615 = exp(r5532614);
        double r5532616 = r5532593 + r5532590;
        double r5532617 = r5532616 + r5532595;
        double r5532618 = r5532590 / r5532617;
        double r5532619 = r5532591 / r5532596;
        double r5532620 = r5532618 * r5532619;
        double r5532621 = r5532620 * r5532620;
        double r5532622 = r5532620 * r5532621;
        double r5532623 = exp(r5532622);
        double r5532624 = r5532615 * r5532623;
        double r5532625 = log(r5532624);
        double r5532626 = r5532613 - r5532621;
        double r5532627 = r5532620 + r5532612;
        double r5532628 = r5532626 / r5532627;
        double r5532629 = r5532628 * r5532612;
        double r5532630 = r5532629 + r5532621;
        double r5532631 = r5532625 / r5532630;
        double r5532632 = r5532631 / r5532593;
        double r5532633 = r5532601 ? r5532611 : r5532632;
        return r5532633;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) < -0.9999999999999999

   1. Initial program 63.3

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

   2. Taylor expanded around inf 32.7

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 32.7

      \[\leadsto \frac{\color{blue}{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}}{2}\]

   if -0.9999999999999999 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))

   1. Initial program 12.8

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
   2. Using strategy rm

   3. Applied flip3-+ 12.8

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

   4. Simplified 12.8

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

   5. Simplified 0.3

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

   7. Applied add-log-exp 0.3

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

   8. Applied add-log-exp 0.3

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

   9. Applied sum-log 0.3

      \[\leadsto \frac{\frac{\color{blue}{\log \left(e^{\left(1 \cdot 1\right) \cdot 1} \cdot e^{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\alpha + \beta}{\left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot i}\right) \cdot \left(\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\alpha + \beta}{\left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot i}\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\alpha + \beta}{\left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot i}\right)\right)}\right)}}{1 \cdot \left(1 - \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\alpha + \beta}{\left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot i}\right) + \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\alpha + \beta}{\left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot i}\right) \cdot \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\alpha + \beta}{\left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot i}\right)}}{2}\]
   10. Using strategy rm

   11. Applied flip-- 0.3

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999998889776975374843459576368:\\ \;\;\;\;\frac{\left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log \left(e^{1 \cdot \left(1 \cdot 1\right)} \cdot e^{\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)\right)}\right)}{\frac{1 \cdot 1 - \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}{\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)} + 1} \cdot 1 + \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\frac{\beta + \alpha}{\left(2 + \left(\beta + \alpha\right)\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))