Average Error: 0.1 → 0.1
Time: 4.4m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)
double f(double a, double rand) {
        double r13363524 = a;
        double r13363525 = 1.0;
        double r13363526 = 3.0;
        double r13363527 = r13363525 / r13363526;
        double r13363528 = r13363524 - r13363527;
        double r13363529 = 1.0;
        double r13363530 = 9.0;
        double r13363531 = r13363530 * r13363528;
        double r13363532 = sqrt(r13363531);
        double r13363533 = r13363529 / r13363532;
        double r13363534 = rand;
        double r13363535 = r13363533 * r13363534;
        double r13363536 = r13363529 + r13363535;
        double r13363537 = r13363528 * r13363536;
        return r13363537;
}
↓
double f(double a, double rand) {
        double r13363538 = rand;
        double r13363539 = 9.0;
        double r13363540 = cbrt(r13363539);
        double r13363541 = r13363540 * r13363540;
        double r13363542 = a;
        double r13363543 = 1.0;
        double r13363544 = 3.0;
        double r13363545 = r13363543 / r13363544;
        double r13363546 = r13363542 - r13363545;
        double r13363547 = r13363540 * r13363546;
        double r13363548 = r13363541 * r13363547;
        double r13363549 = sqrt(r13363548);
        double r13363550 = r13363538 / r13363549;
        double r13363551 = fma(r13363550, r13363546, r13363546);
        return r13363551;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

    \[\leadsto \mathsf{fma}\left(\left(\frac{rand}{\sqrt{\color{blue}{\left(\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \sqrt[3]{9}\right)} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]

  5. Applied associate-*l*0.1

    \[\leadsto \mathsf{fma}\left(\left(\frac{rand}{\sqrt{\color{blue}{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(\sqrt[3]{9} \cdot \sqrt[3]{9}\right) \cdot \left(\sqrt[3]{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))