Octave 3.8, jcobi/4, as called

Average Error: 45.8 → 0.2

Time: 21.7s

Precision: 64

\[i \gt 0\]

Math

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

↓

\[\frac{i}{\mathsf{fma}\left(\left(\frac{1.0}{i}\right), -4, \left(16 \cdot i\right)\right)}\]

C

double f(double i) {
        double r11975580 = i;
        double r11975581 = r11975580 * r11975580;
        double r11975582 = r11975581 * r11975581;
        double r11975583 = 2.0;
        double r11975584 = r11975583 * r11975580;
        double r11975585 = r11975584 * r11975584;
        double r11975586 = r11975582 / r11975585;
        double r11975587 = 1.0;
        double r11975588 = r11975585 - r11975587;
        double r11975589 = r11975586 / r11975588;
        return r11975589;
}

↓

double f(double i) {
        double r11975590 = i;
        double r11975591 = 1.0;
        double r11975592 = r11975591 / r11975590;
        double r11975593 = -4.0;
        double r11975594 = 16.0;
        double r11975595 = r11975594 * r11975590;
        double r11975596 = fma(r11975592, r11975593, r11975595);
        double r11975597 = r11975590 / r11975596;
        return r11975597;
}

Error

Bits error versus i

Derivation

1. Initial program 45.8

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

2. Simplified 15.1

   \[\leadsto \color{blue}{\frac{i \cdot i}{\left(4 \cdot \left(i \cdot i\right) - 1.0\right) \cdot 4}}\]
3. Using strategy rm

4. Applied times-frac 15.2

   \[\leadsto \color{blue}{\frac{i}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot \frac{i}{4}}\]
5. Using strategy rm

6. Applied pow1 15.2

   \[\leadsto \frac{i}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot \color{blue}{{\left(\frac{i}{4}\right)}^{1}}\]

7. Applied pow1 15.2

   \[\leadsto \color{blue}{{\left(\frac{i}{4 \cdot \left(i \cdot i\right) - 1.0}\right)}^{1}} \cdot {\left(\frac{i}{4}\right)}^{1}\]

8. Applied pow-prod-down 15.2

   \[\leadsto \color{blue}{{\left(\frac{i}{4 \cdot \left(i \cdot i\right) - 1.0} \cdot \frac{i}{4}\right)}^{1}}\]

9. Simplified 0.2

   \[\leadsto {\color{blue}{\left(\frac{i}{\mathsf{fma}\left(\left(\frac{1.0}{i}\right), -4, \left(16 \cdot i\right)\right)}\right)}}^{1}\]

10. Final simplification 0.2

    \[\leadsto \frac{i}{\mathsf{fma}\left(\left(\frac{1.0}{i}\right), -4, \left(16 \cdot i\right)\right)}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))