Average Error: 3.9 → 4.0
Time: 26.5s
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 r84853 = alpha;
        double r84854 = beta;
        double r84855 = r84853 + r84854;
        double r84856 = r84854 * r84853;
        double r84857 = r84855 + r84856;
        double r84858 = 1.0;
        double r84859 = r84857 + r84858;
        double r84860 = 2.0;
        double r84861 = r84860 * r84858;
        double r84862 = r84855 + r84861;
        double r84863 = r84859 / r84862;
        double r84864 = r84863 / r84862;
        double r84865 = r84862 + r84858;
        double r84866 = r84864 / r84865;
        return r84866;
}
↓
double f(double alpha, double beta) {
        double r84867 = 1.0;
        double r84868 = alpha;
        double r84869 = beta;
        double r84870 = r84868 + r84869;
        double r84871 = fma(r84868, r84869, r84870);
        double r84872 = r84867 + r84871;
        double r84873 = sqrt(r84872);
        double r84874 = 2.0;
        double r84875 = fma(r84867, r84874, r84870);
        double r84876 = sqrt(r84875);
        double r84877 = r84873 / r84876;
        double r84878 = r84877 / r84876;
        double r84879 = r84875 + r84867;
        double r84880 = r84873 / r84875;
        double r84881 = r84879 / r84880;
        double r84882 = r84878 / r84881;
        return r84882;
}

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)))