Octave 3.8, jcobi/4

Average Error: 54.7 → 36.8

Time: 18.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.791851745498054464731137749359772731396 \cdot 10^{221}:\\ \;\;\;\;\left(\sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}} \cdot \sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}\right) \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{i}{\frac{{\left(\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}^{\left(3 + 1\right)}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right)}^{\left(\frac{\alpha + \left(\beta + i\right)}{{\left(\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}^{\left(3 + 1\right)} - 1}\right)}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r106528 = i;
        double r106529 = alpha;
        double r106530 = beta;
        double r106531 = r106529 + r106530;
        double r106532 = r106531 + r106528;
        double r106533 = r106528 * r106532;
        double r106534 = r106530 * r106529;
        double r106535 = r106534 + r106533;
        double r106536 = r106533 * r106535;
        double r106537 = 2.0;
        double r106538 = r106537 * r106528;
        double r106539 = r106531 + r106538;
        double r106540 = r106539 * r106539;
        double r106541 = r106536 / r106540;
        double r106542 = 1.0;
        double r106543 = r106540 - r106542;
        double r106544 = r106541 / r106543;
        return r106544;
}

↓

double f(double alpha, double beta, double i) {
        double r106545 = alpha;
        double r106546 = 1.7918517454980545e+221;
        bool r106547 = r106545 <= r106546;
        double r106548 = i;
        double r106549 = 1.0;
        double r106550 = sqrt(r106549);
        double r106551 = 2.0;
        double r106552 = beta;
        double r106553 = r106545 + r106552;
        double r106554 = fma(r106551, r106548, r106553);
        double r106555 = r106550 + r106554;
        double r106556 = r106553 + r106548;
        double r106557 = r106548 * r106556;
        double r106558 = fma(r106552, r106545, r106557);
        double r106559 = sqrt(r106558);
        double r106560 = r106554 / r106559;
        double r106561 = r106555 * r106560;
        double r106562 = r106548 / r106561;
        double r106563 = sqrt(r106562);
        double r106564 = r106563 * r106563;
        double r106565 = r106556 / r106560;
        double r106566 = r106554 - r106550;
        double r106567 = r106565 / r106566;
        double r106568 = r106564 * r106567;
        double r106569 = sqrt(r106554);
        double r106570 = 3.0;
        double r106571 = 1.0;
        double r106572 = r106570 + r106571;
        double r106573 = pow(r106569, r106572);
        double r106574 = r106573 / r106558;
        double r106575 = r106548 / r106574;
        double r106576 = exp(r106575);
        double r106577 = r106552 + r106548;
        double r106578 = r106545 + r106577;
        double r106579 = r106573 - r106549;
        double r106580 = r106578 / r106579;
        double r106581 = pow(r106576, r106580);
        double r106582 = log(r106581);
        double r106583 = r106547 ? r106568 : r106582;
        return r106583;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.7918517454980545e+221

   1. Initial program 53.7

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

   3. Applied associate-/l* 38.8

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

   4. Simplified 38.8

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

   6. Applied add-sqr-sqrt 38.8

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

   7. Applied difference-of-squares 38.8

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

   8. Applied add-sqr-sqrt 38.8

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

   9. Applied times-frac 38.8

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

   10. Applied times-frac 38.8

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

   11. Applied times-frac 36.2

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

   12. Simplified 36.2

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

   13. Simplified 36.2

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

   15. Applied add-sqr-sqrt 36.2

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

   if 1.7918517454980545e+221 < alpha

   1. Initial program 64.0

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

   3. Applied associate-/l* 56.8

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

   4. Simplified 56.8

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

   6. Applied add-log-exp 56.8

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

   7. Simplified 42.4

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

4. Final simplification 36.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.791851745498054464731137749359772731396 \cdot 10^{221}:\\ \;\;\;\;\left(\sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}} \cdot \sqrt{\frac{i}{\left(\sqrt{1} + \mathsf{fma}\left(2, i, \alpha + \beta\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}\right) \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}}{\mathsf{fma}\left(2, i, \alpha + \beta\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{i}{\frac{{\left(\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}^{\left(3 + 1\right)}}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}}\right)}^{\left(\frac{\alpha + \left(\beta + i\right)}{{\left(\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)}^{\left(3 + 1\right)} - 1}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))