Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 24.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r1451251 = a;
        double r1451252 = 1.0;
        double r1451253 = 3.0;
        double r1451254 = r1451252 / r1451253;
        double r1451255 = r1451251 - r1451254;
        double r1451256 = 1.0;
        double r1451257 = 9.0;
        double r1451258 = r1451257 * r1451255;
        double r1451259 = sqrt(r1451258);
        double r1451260 = r1451256 / r1451259;
        double r1451261 = rand;
        double r1451262 = r1451260 * r1451261;
        double r1451263 = r1451256 + r1451262;
        double r1451264 = r1451255 * r1451263;
        return r1451264;
}

↓

double f(double a, double rand) {
        double r1451265 = 1.0;
        double r1451266 = 3.0;
        double r1451267 = r1451265 / r1451266;
        double r1451268 = cbrt(r1451267);
        double r1451269 = -r1451268;
        double r1451270 = r1451268 * r1451268;
        double r1451271 = r1451270 * r1451268;
        double r1451272 = fma(r1451269, r1451270, r1451271);
        double r1451273 = rand;
        double r1451274 = a;
        double r1451275 = r1451274 - r1451267;
        double r1451276 = sqrt(r1451275);
        double r1451277 = 3.0;
        double r1451278 = r1451276 / r1451277;
        double r1451279 = fma(r1451273, r1451278, r1451275);
        double r1451280 = r1451272 + r1451279;
        return r1451280;
}

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. Simplified 0.1

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

4. Applied sqrt-prod 0.1

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

5. Applied add-sqr-sqrt 0.1

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

6. Applied times-frac 0.1

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

7. Simplified 0.1

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}} \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}, rand, a - \frac{1.0}{3.0}\right)\]

8. Simplified 0.1

   \[\leadsto \mathsf{fma}\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot \color{blue}{1}, rand, a - \frac{1.0}{3.0}\right)\]
9. Using strategy rm

10. Applied fma-udef 0.1

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

11. Simplified 0.1

    \[\leadsto \color{blue}{rand \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}} + \left(a - \frac{1.0}{3.0}\right)\]
12. Using strategy rm

13. Applied add-cube-cbrt 0.1

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

14. Applied add-sqr-sqrt 0.5

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

15. Applied prod-diff 0.5

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

16. Applied associate-+r+ 0.5

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019152 +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))))