Average Error: 0.1 → 0.2
Time: 26.0s
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 r97265 = a;
        double r97266 = 1.0;
        double r97267 = 3.0;
        double r97268 = r97266 / r97267;
        double r97269 = r97265 - r97268;
        double r97270 = 9.0;
        double r97271 = r97270 * r97269;
        double r97272 = sqrt(r97271);
        double r97273 = r97266 / r97272;
        double r97274 = rand;
        double r97275 = r97273 * r97274;
        double r97276 = r97266 + r97275;
        double r97277 = r97269 * r97276;
        return r97277;
}
↓
double f(double a, double rand) {
        double r97278 = 1.0;
        double r97279 = a;
        double r97280 = 3.0;
        double r97281 = r97278 / r97280;
        double r97282 = r97279 - r97281;
        double r97283 = sqrt(r97282);
        double r97284 = r97278 / r97283;
        double r97285 = 9.0;
        double r97286 = sqrt(r97285);
        double r97287 = r97284 / r97286;
        double r97288 = rand;
        double r97289 = fma(r97287, r97288, r97278);
        double r97290 = r97289 * r97282;
        return r97290;
}

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

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

  4. Applied sqrt-prod0.2

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

  5. Applied associate-/r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +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))))