Average Error: 0.1 → 0.1
Time: 32.2s
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 r75469 = a;
        double r75470 = 1.0;
        double r75471 = 3.0;
        double r75472 = r75470 / r75471;
        double r75473 = r75469 - r75472;
        double r75474 = 9.0;
        double r75475 = r75474 * r75473;
        double r75476 = sqrt(r75475);
        double r75477 = r75470 / r75476;
        double r75478 = rand;
        double r75479 = r75477 * r75478;
        double r75480 = r75470 + r75479;
        double r75481 = r75473 * r75480;
        return r75481;
}
↓
double f(double a, double rand) {
        double r75482 = 1.0;
        double r75483 = a;
        double r75484 = 3.0;
        double r75485 = r75482 / r75484;
        double r75486 = r75483 - r75485;
        double r75487 = sqrt(r75486);
        double r75488 = r75482 / r75487;
        double r75489 = 9.0;
        double r75490 = sqrt(r75489);
        double r75491 = r75488 / r75490;
        double r75492 = rand;
        double r75493 = fma(r75491, r75492, r75482);
        double r75494 = r75493 * r75486;
        return r75494;
}

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))))