Average Error: 0.1 → 0.1
Time: 56.1s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\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{\frac{1}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r134119 = a;
        double r134120 = 1.0;
        double r134121 = 3.0;
        double r134122 = r134120 / r134121;
        double r134123 = r134119 - r134122;
        double r134124 = 9.0;
        double r134125 = r134124 * r134123;
        double r134126 = sqrt(r134125);
        double r134127 = r134120 / r134126;
        double r134128 = rand;
        double r134129 = r134127 * r134128;
        double r134130 = r134120 + r134129;
        double r134131 = r134123 * r134130;
        return r134131;
}
↓
double f(double a, double rand) {
        double r134132 = 1.0;
        double r134133 = a;
        double r134134 = 3.0;
        double r134135 = r134132 / r134134;
        double r134136 = r134133 - r134135;
        double r134137 = sqrt(r134136);
        double r134138 = r134132 / r134137;
        double r134139 = 9.0;
        double r134140 = sqrt(r134139);
        double r134141 = r134138 / r134140;
        double r134142 = rand;
        double r134143 = fma(r134141, r134142, r134132);
        double r134144 = r134143 * r134136;
        return r134144;
}

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. Simplified0.1

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

  4. Applied sqrt-prod0.1

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

  5. Applied *-un-lft-identity0.1

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

  6. Applied times-frac0.2

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

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019326 +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))))