Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 35.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]

↓

\[\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}\right) + \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(\frac{1}{\sqrt{3}} \cdot \left(\frac{-1}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right)\right)\]

C

double f(double a, double rand) {
        double r177167 = a;
        double r177168 = 1.0;
        double r177169 = 3.0;
        double r177170 = r177168 / r177169;
        double r177171 = r177167 - r177170;
        double r177172 = 9.0;
        double r177173 = r177172 * r177171;
        double r177174 = sqrt(r177173);
        double r177175 = r177168 / r177174;
        double r177176 = rand;
        double r177177 = r177175 * r177176;
        double r177178 = r177168 + r177177;
        double r177179 = r177171 * r177178;
        return r177179;
}

↓

double f(double a, double rand) {
        double r177180 = 1.0;
        double r177181 = 9.0;
        double r177182 = a;
        double r177183 = 3.0;
        double r177184 = r177180 / r177183;
        double r177185 = r177182 - r177184;
        double r177186 = r177181 * r177185;
        double r177187 = sqrt(r177186);
        double r177188 = r177180 / r177187;
        double r177189 = rand;
        double r177190 = fma(r177188, r177189, r177180);
        double r177191 = sqrt(r177183);
        double r177192 = r177180 / r177191;
        double r177193 = r177192 / r177191;
        double r177194 = r177182 - r177193;
        double r177195 = r177190 * r177194;
        double r177196 = -1.0;
        double r177197 = r177196 / r177191;
        double r177198 = 1.0;
        double r177199 = r177198 / r177191;
        double r177200 = r177197 + r177199;
        double r177201 = r177192 * r177200;
        double r177202 = r177190 * r177201;
        double r177203 = r177195 + r177202;
        return r177203;
}

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.1

   \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{1}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right)\]

5. Applied add-sqr-sqrt 0.1

   \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\sqrt{3} \cdot \sqrt{3}}\right)\]

6. Applied times-frac 0.1

   \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \color{blue}{\frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}}\right)\]

7. Applied add-sqr-sqrt 0.6

   \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(\color{blue}{\sqrt{a} \cdot \sqrt{a}} - \frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}\right)\]

8. Applied prod-diff 0.6

   \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{a}, \sqrt{a}, -\frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}\right) + \mathsf{fma}\left(-\frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}\right)\right)}\]

9. Applied distribute-lft-in 0.6

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \mathsf{fma}\left(\sqrt{a}, \sqrt{a}, -\frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}\right) + \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \mathsf{fma}\left(-\frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}\right)}\]

10. Simplified 0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}\right)} + \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \mathsf{fma}\left(-\frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{1}}{\sqrt{3}} \cdot \frac{\sqrt{1}}{\sqrt{3}}\right)\]

11. Simplified 0.1

    \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}\right) + \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(\frac{1}{\sqrt{3}} \cdot \left(\frac{-1}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right)\right)}\]

12. Final simplification 0.1

    \[\leadsto \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}\right) + \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(\frac{1}{\sqrt{3}} \cdot \left(\frac{-1}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right)\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))