Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.2

Time: 3.3m

Precision: 64

Math

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

↓

\[\left(\left(rand \cdot \frac{\frac{1}{3}}{\sqrt{\sqrt{a - \frac{1.0}{3.0}}}}\right) \cdot \frac{1}{\sqrt{\sqrt{a - \frac{1.0}{3.0}}}} + 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]

C

double f(double a, double rand) {
        double r9423075 = a;
        double r9423076 = 1.0;
        double r9423077 = 3.0;
        double r9423078 = r9423076 / r9423077;
        double r9423079 = r9423075 - r9423078;
        double r9423080 = 1.0;
        double r9423081 = 9.0;
        double r9423082 = r9423081 * r9423079;
        double r9423083 = sqrt(r9423082);
        double r9423084 = r9423080 / r9423083;
        double r9423085 = rand;
        double r9423086 = r9423084 * r9423085;
        double r9423087 = r9423080 + r9423086;
        double r9423088 = r9423079 * r9423087;
        return r9423088;
}

↓

double f(double a, double rand) {
        double r9423089 = rand;
        double r9423090 = 0.3333333333333333;
        double r9423091 = a;
        double r9423092 = 1.0;
        double r9423093 = 3.0;
        double r9423094 = r9423092 / r9423093;
        double r9423095 = r9423091 - r9423094;
        double r9423096 = sqrt(r9423095);
        double r9423097 = sqrt(r9423096);
        double r9423098 = r9423090 / r9423097;
        double r9423099 = r9423089 * r9423098;
        double r9423100 = 1.0;
        double r9423101 = r9423100 / r9423097;
        double r9423102 = r9423099 * r9423101;
        double r9423103 = r9423102 + r9423100;
        double r9423104 = r9423103 * r9423095;
        return r9423104;
}

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 sqrt-prod 0.2

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

4. Applied associate-/r* 0.2

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

6. Applied add-sqr-sqrt 0.2

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

7. Applied *-un-lft-identity 0.2

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

8. Applied sqrt-prod 0.2

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

9. Applied *-un-lft-identity 0.2

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

10. Applied times-frac 0.2

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

11. Applied times-frac 0.2

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

12. Applied associate-*l* 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

    \[\leadsto \left(\left(rand \cdot \frac{\frac{1}{3}}{\sqrt{\sqrt{a - \frac{1.0}{3.0}}}}\right) \cdot \frac{1}{\sqrt{\sqrt{a - \frac{1.0}{3.0}}}} + 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(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))))