Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 27.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r4888035 = a;
        double r4888036 = 1.0;
        double r4888037 = 3.0;
        double r4888038 = r4888036 / r4888037;
        double r4888039 = r4888035 - r4888038;
        double r4888040 = 1.0;
        double r4888041 = 9.0;
        double r4888042 = r4888041 * r4888039;
        double r4888043 = sqrt(r4888042);
        double r4888044 = r4888040 / r4888043;
        double r4888045 = rand;
        double r4888046 = r4888044 * r4888045;
        double r4888047 = r4888040 + r4888046;
        double r4888048 = r4888039 * r4888047;
        return r4888048;
}

↓

double f(double a, double rand) {
        double r4888049 = a;
        double r4888050 = sqrt(r4888049);
        double r4888051 = 1.0;
        double r4888052 = 3.0;
        double r4888053 = r4888051 / r4888052;
        double r4888054 = sqrt(r4888053);
        double r4888055 = r4888050 - r4888054;
        double r4888056 = r4888049 - r4888053;
        double r4888057 = sqrt(r4888056);
        double r4888058 = r4888055 / r4888057;
        double r4888059 = r4888050 + r4888054;
        double r4888060 = 3.0;
        double r4888061 = r4888059 / r4888060;
        double r4888062 = r4888058 * r4888061;
        double r4888063 = rand;
        double r4888064 = fma(r4888062, r4888063, r4888056);
        return r4888064;
}

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

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

6. Applied add-sqr-sqrt 0.2

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

7. Applied difference-of-squares 0.1

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

8. Applied times-frac 0.1

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

9. Simplified 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019162 +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))))