Average Error: 3.9 → 4.0
Time: 27.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\]
↓
\[\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}\]
\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}
↓
\frac{\frac{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\sqrt{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(1, 2, \alpha + \beta\right) + 1}{\frac{\sqrt{1 + \mathsf{fma}\left(\alpha, \beta, \alpha + \beta\right)}}{\mathsf{fma}\left(1, 2, \alpha + \beta\right)}}}
double f(double alpha, double beta) {
        double r162492 = alpha;
        double r162493 = beta;
        double r162494 = r162492 + r162493;
        double r162495 = r162493 * r162492;
        double r162496 = r162494 + r162495;
        double r162497 = 1.0;
        double r162498 = r162496 + r162497;
        double r162499 = 2.0;
        double r162500 = r162499 * r162497;
        double r162501 = r162494 + r162500;
        double r162502 = r162498 / r162501;
        double r162503 = r162502 / r162501;
        double r162504 = r162501 + r162497;
        double r162505 = r162503 / r162504;
        return r162505;
}
↓
double f(double alpha, double beta) {
        double r162506 = 1.0;
        double r162507 = alpha;
        double r162508 = beta;
        double r162509 = r162507 + r162508;
        double r162510 = fma(r162507, r162508, r162509);
        double r162511 = r162506 + r162510;
        double r162512 = sqrt(r162511);
        double r162513 = 2.0;
        double r162514 = fma(r162506, r162513, r162509);
        double r162515 = sqrt(r162514);
        double r162516 = r162512 / r162515;
        double r162517 = r162516 / r162515;
        double r162518 = r162514 + r162506;
        double r162519 = r162512 / r162514;
        double r162520 = r162518 / r162519;
        double r162521 = r162517 / r162520;
        return r162521;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 3.9

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

  2. Simplified3.9

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

  4. Applied add-sqr-sqrt4.4

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

  5. Applied add-sqr-sqrt4.8

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

  6. Applied add-sqr-sqrt4.7

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

  7. Applied times-frac4.7

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

  8. Applied times-frac4.5

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

  9. Applied associate-/l*4.5

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

  10. Simplified4.0

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

  11. Final simplification4.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))