Average Error: 0.2 → 0.2
Time: 27.9s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[a \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right) + \left(-\frac{1}{3} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right)\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
a \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right) + \left(-\frac{1}{3} \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right)\right)
double f(double a, double rand) {
        double r92144 = a;
        double r92145 = 1.0;
        double r92146 = 3.0;
        double r92147 = r92145 / r92146;
        double r92148 = r92144 - r92147;
        double r92149 = 9.0;
        double r92150 = r92149 * r92148;
        double r92151 = sqrt(r92150);
        double r92152 = r92145 / r92151;
        double r92153 = rand;
        double r92154 = r92152 * r92153;
        double r92155 = r92145 + r92154;
        double r92156 = r92148 * r92155;
        return r92156;
}
↓
double f(double a, double rand) {
        double r92157 = a;
        double r92158 = 1.0;
        double r92159 = 9.0;
        double r92160 = sqrt(r92159);
        double r92161 = r92158 / r92160;
        double r92162 = 3.0;
        double r92163 = r92158 / r92162;
        double r92164 = r92157 - r92163;
        double r92165 = sqrt(r92164);
        double r92166 = r92161 / r92165;
        double r92167 = rand;
        double r92168 = fma(r92166, r92167, r92158);
        double r92169 = r92157 * r92168;
        double r92170 = r92159 * r92164;
        double r92171 = sqrt(r92170);
        double r92172 = r92158 / r92171;
        double r92173 = fma(r92172, r92167, r92158);
        double r92174 = r92163 * r92173;
        double r92175 = -r92174;
        double r92176 = r92169 + r92175;
        return r92176;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  4. Applied sub-neg0.2

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

  5. Applied distribute-lft-in0.2

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  9. Applied sqrt-prod0.2

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

  10. Applied associate-/r*0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019306 +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))))