Octave 3.8, jcobi/4

Average Error: 52.6 → 36.8

Time: 29.5s

Precision: 64

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r1772858 = i;
        double r1772859 = alpha;
        double r1772860 = beta;
        double r1772861 = r1772859 + r1772860;
        double r1772862 = r1772861 + r1772858;
        double r1772863 = r1772858 * r1772862;
        double r1772864 = r1772860 * r1772859;
        double r1772865 = r1772864 + r1772863;
        double r1772866 = r1772863 * r1772865;
        double r1772867 = 2.0;
        double r1772868 = r1772867 * r1772858;
        double r1772869 = r1772861 + r1772868;
        double r1772870 = r1772869 * r1772869;
        double r1772871 = r1772866 / r1772870;
        double r1772872 = 1.0;
        double r1772873 = r1772870 - r1772872;
        double r1772874 = r1772871 / r1772873;
        return r1772874;
}

↓

double f(double alpha, double beta, double i) {
        double r1772875 = i;
        double r1772876 = beta;
        double r1772877 = alpha;
        double r1772878 = r1772876 + r1772877;
        double r1772879 = r1772875 + r1772878;
        double r1772880 = r1772877 * r1772876;
        double r1772881 = fma(r1772879, r1772875, r1772880);
        double r1772882 = 1.0;
        double r1772883 = sqrt(r1772882);
        double r1772884 = 2.0;
        double r1772885 = fma(r1772884, r1772875, r1772878);
        double r1772886 = r1772883 + r1772885;
        double r1772887 = r1772881 / r1772886;
        double r1772888 = sqrt(r1772887);
        double r1772889 = r1772885 / r1772888;
        double r1772890 = r1772888 / r1772889;
        double r1772891 = r1772875 * r1772879;
        double r1772892 = r1772885 - r1772883;
        double r1772893 = r1772891 / r1772892;
        double r1772894 = r1772893 / r1772885;
        double r1772895 = r1772890 * r1772894;
        return r1772895;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 52.6

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

2. Simplified 52.6

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

4. Applied add-sqr-sqrt 52.6

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

5. Applied difference-of-squares 52.6

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

6. Applied times-frac 38.7

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

7. Applied times-frac 36.7

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

9. Applied add-sqr-sqrt 36.8

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

10. Applied associate-/l* 36.8

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

11. Final simplification 36.8

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

Reproduce

herbie shell --seed 2019152 +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)))