Average Error: 0.1 → 0.2
Time: 33.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right)
double f(double a, double rand) {
        double r8933180 = a;
        double r8933181 = 1.0;
        double r8933182 = 3.0;
        double r8933183 = r8933181 / r8933182;
        double r8933184 = r8933180 - r8933183;
        double r8933185 = 9.0;
        double r8933186 = r8933185 * r8933184;
        double r8933187 = sqrt(r8933186);
        double r8933188 = r8933181 / r8933187;
        double r8933189 = rand;
        double r8933190 = r8933188 * r8933189;
        double r8933191 = r8933181 + r8933190;
        double r8933192 = r8933184 * r8933191;
        return r8933192;
}
↓
double f(double a, double rand) {
        double r8933193 = a;
        double r8933194 = 1.0;
        double r8933195 = 3.0;
        double r8933196 = r8933194 / r8933195;
        double r8933197 = r8933193 - r8933196;
        double r8933198 = 9.0;
        double r8933199 = sqrt(r8933198);
        double r8933200 = r8933194 / r8933199;
        double r8933201 = sqrt(r8933197);
        double r8933202 = r8933200 / r8933201;
        double r8933203 = rand;
        double r8933204 = fma(r8933202, r8933203, r8933194);
        double r8933205 = r8933197 * r8933204;
        return r8933205;
}

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 *-commutative0.1

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

  6. Applied sqrt-prod0.2

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

  7. Applied associate-/r*0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))