Octave 3.8, jcobi/4

Average Error: 52.7 → 10.6

Time: 3.9m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 9.298324197453573 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \left(i \cdot \frac{\left(\alpha + \beta\right) + i}{(2 \cdot i + \left(\alpha + \beta\right))_*}\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{i}} \cdot \frac{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(i \cdot \frac{1}{2}\right))_*}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r23989535 = i;
        double r23989536 = alpha;
        double r23989537 = beta;
        double r23989538 = r23989536 + r23989537;
        double r23989539 = r23989538 + r23989535;
        double r23989540 = r23989535 * r23989539;
        double r23989541 = r23989537 * r23989536;
        double r23989542 = r23989541 + r23989540;
        double r23989543 = r23989540 * r23989542;
        double r23989544 = 2.0;
        double r23989545 = r23989544 * r23989535;
        double r23989546 = r23989538 + r23989545;
        double r23989547 = r23989546 * r23989546;
        double r23989548 = r23989543 / r23989547;
        double r23989549 = 1.0;
        double r23989550 = r23989547 - r23989549;
        double r23989551 = r23989548 / r23989550;
        return r23989551;
}

↓

double f(double alpha, double beta, double i) {
        double r23989552 = i;
        double r23989553 = 9.298324197453573e+133;
        bool r23989554 = r23989552 <= r23989553;
        double r23989555 = alpha;
        double r23989556 = beta;
        double r23989557 = r23989555 + r23989556;
        double r23989558 = r23989557 + r23989552;
        double r23989559 = r23989555 * r23989556;
        double r23989560 = fma(r23989558, r23989552, r23989559);
        double r23989561 = 2.0;
        double r23989562 = fma(r23989561, r23989552, r23989557);
        double r23989563 = r23989560 / r23989562;
        double r23989564 = 1.0;
        double r23989565 = sqrt(r23989564);
        double r23989566 = r23989565 + r23989562;
        double r23989567 = r23989563 / r23989566;
        double r23989568 = r23989558 / r23989562;
        double r23989569 = r23989552 * r23989568;
        double r23989570 = r23989567 * r23989569;
        double r23989571 = r23989562 - r23989565;
        double r23989572 = r23989570 / r23989571;
        double r23989573 = r23989562 / r23989552;
        double r23989574 = r23989558 / r23989573;
        double r23989575 = 0.25;
        double r23989576 = 0.5;
        double r23989577 = r23989552 * r23989576;
        double r23989578 = fma(r23989557, r23989575, r23989577);
        double r23989579 = r23989578 / r23989566;
        double r23989580 = r23989574 * r23989579;
        double r23989581 = r23989580 / r23989571;
        double r23989582 = r23989554 ? r23989572 : r23989581;
        return r23989582;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 9.298324197453573e+133

   1. Initial program 40.3

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

   2. Simplified 40.3

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

   4. Applied add-sqr-sqrt 40.3

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

   5. Applied difference-of-squares 40.3

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

   6. Applied times-frac 14.9

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

   7. Applied times-frac 10.4

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

   9. Applied associate-*r/ 10.4

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

   11. Applied associate-/l* 10.4

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

   13. Applied associate-/r/ 10.4

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

   if 9.298324197453573e+133 < i

   1. Initial program 62.1

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

   2. Simplified 62.1

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

   4. Applied add-sqr-sqrt 62.1

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

   5. Applied difference-of-squares 62.1

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

   6. Applied times-frac 56.7

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

   7. Applied times-frac 56.5

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

   9. Applied associate-*r/ 56.5

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

   11. Applied associate-/l* 56.5

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

   12. Taylor expanded around 0 10.8

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{i}}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}\]

   13. Simplified 10.8

       \[\leadsto \frac{\frac{\color{blue}{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(\frac{1}{2} \cdot i\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{i}}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 9.298324197453573 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \left(i \cdot \frac{\left(\alpha + \beta\right) + i}{(2 \cdot i + \left(\alpha + \beta\right))_*}\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) + i}{\frac{(2 \cdot i + \left(\alpha + \beta\right))_*}{i}} \cdot \frac{(\left(\alpha + \beta\right) \cdot \frac{1}{4} + \left(i \cdot \frac{1}{2}\right))_*}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))