Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 23.0s

Precision: 64

Math

\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

↓

\[\frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]

C

double f(double a, double rand) {
        double r3830041 = a;
        double r3830042 = 1.0;
        double r3830043 = 3.0;
        double r3830044 = r3830042 / r3830043;
        double r3830045 = r3830041 - r3830044;
        double r3830046 = 1.0;
        double r3830047 = 9.0;
        double r3830048 = r3830047 * r3830045;
        double r3830049 = sqrt(r3830048);
        double r3830050 = r3830046 / r3830049;
        double r3830051 = rand;
        double r3830052 = r3830050 * r3830051;
        double r3830053 = r3830046 + r3830052;
        double r3830054 = r3830045 * r3830053;
        return r3830054;
}

↓

double f(double a, double rand) {
        double r3830055 = a;
        double r3830056 = 1.0;
        double r3830057 = 3.0;
        double r3830058 = r3830056 / r3830057;
        double r3830059 = r3830055 - r3830058;
        double r3830060 = sqrt(r3830059);
        double r3830061 = 3.0;
        double r3830062 = r3830060 / r3830061;
        double r3830063 = rand;
        double r3830064 = r3830062 * r3830063;
        double r3830065 = r3830064 + r3830059;
        return r3830065;
}

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
2. Using strategy rm

3. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)}\]

4. Simplified 0.1

   \[\leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right)} + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

5. Simplified 0.1

   \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)}\]
6. Using strategy rm

7. Applied div-inv 0.1

   \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\left(rand \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)} \cdot \left(a - \frac{1.0}{3.0}\right)\]

8. Applied associate-*l* 0.1

   \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{rand \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}\]

9. Simplified 0.1

   \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \color{blue}{\frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}}\]
10. Using strategy rm

11. Applied sqrt-prod 0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{a - \frac{1.0}{3.0}}{\color{blue}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9}}}\]

12. Applied add-sqr-sqrt 0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\color{blue}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{a - \frac{1.0}{3.0}}}}{\sqrt{a - \frac{1.0}{3.0}} \cdot \sqrt{9}}\]

13. Applied times-frac 0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \color{blue}{\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{9}}\right)}\]

14. Applied associate-*r* 0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\left(rand \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{9}}}\]

15. Simplified 0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{rand} \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{9}}\]

16. Final simplification 0.1

    \[\leadsto \frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))