Octave 3.8, jcobi/4

Average Error: 52.4 → 11.1

Time: 39.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 2.495505368754722 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, \alpha + \beta, i \cdot \frac{1}{2}\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r4310550 = i;
        double r4310551 = alpha;
        double r4310552 = beta;
        double r4310553 = r4310551 + r4310552;
        double r4310554 = r4310553 + r4310550;
        double r4310555 = r4310550 * r4310554;
        double r4310556 = r4310552 * r4310551;
        double r4310557 = r4310556 + r4310555;
        double r4310558 = r4310555 * r4310557;
        double r4310559 = 2.0;
        double r4310560 = r4310559 * r4310550;
        double r4310561 = r4310553 + r4310560;
        double r4310562 = r4310561 * r4310561;
        double r4310563 = r4310558 / r4310562;
        double r4310564 = 1.0;
        double r4310565 = r4310562 - r4310564;
        double r4310566 = r4310563 / r4310565;
        return r4310566;
}

↓

double f(double alpha, double beta, double i) {
        double r4310567 = i;
        double r4310568 = 2.495505368754722e+120;
        bool r4310569 = r4310567 <= r4310568;
        double r4310570 = alpha;
        double r4310571 = beta;
        double r4310572 = r4310570 + r4310571;
        double r4310573 = 2.0;
        double r4310574 = r4310567 * r4310573;
        double r4310575 = r4310572 + r4310574;
        double r4310576 = r4310572 + r4310567;
        double r4310577 = r4310575 / r4310576;
        double r4310578 = r4310567 / r4310577;
        double r4310579 = 1.0;
        double r4310580 = sqrt(r4310579);
        double r4310581 = r4310580 + r4310575;
        double r4310582 = r4310578 / r4310581;
        double r4310583 = r4310576 * r4310567;
        double r4310584 = r4310570 * r4310571;
        double r4310585 = r4310583 + r4310584;
        double r4310586 = r4310585 / r4310575;
        double r4310587 = r4310582 * r4310586;
        double r4310588 = r4310575 - r4310580;
        double r4310589 = r4310587 / r4310588;
        double r4310590 = 0.25;
        double r4310591 = 0.5;
        double r4310592 = r4310567 * r4310591;
        double r4310593 = fma(r4310590, r4310572, r4310592);
        double r4310594 = r4310593 / r4310588;
        double r4310595 = sqrt(r4310578);
        double r4310596 = sqrt(r4310581);
        double r4310597 = r4310595 / r4310596;
        double r4310598 = r4310597 * r4310597;
        double r4310599 = r4310594 * r4310598;
        double r4310600 = r4310569 ? r4310589 : r4310599;
        return r4310600;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 2.495505368754722e+120

   1. Initial program 37.3

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

   3. Applied add-sqr-sqrt 37.3

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

   4. Applied difference-of-squares 37.3

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

   5. Applied times-frac 14.7

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

   6. Applied times-frac 10.0

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

   8. Applied associate-/l* 10.0

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

   10. Applied associate-*r/ 9.9

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

   if 2.495505368754722e+120 < 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. Using strategy rm

   3. Applied add-sqr-sqrt 62.1

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

   4. Applied difference-of-squares 62.1

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

   5. Applied times-frac 54.9

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

   6. Applied times-frac 54.6

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

   8. Applied associate-/l* 54.6

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

   9. Taylor expanded around 0 11.8

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

   10. Simplified 11.8

       \[\leadsto \frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{4}, \alpha + \beta, i \cdot \frac{1}{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 12.3

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

   13. Applied add-sqr-sqrt 11.8

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

   14. Applied times-frac 11.8

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 2.495505368754722 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{4}, \alpha + \beta, i \cdot \frac{1}{2}\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \frac{\sqrt{\frac{i}{\frac{\left(\alpha + \beta\right) + i \cdot 2}{\left(\alpha + \beta\right) + i}}}}{\sqrt{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))