Average Error: 0.1 → 0.1
Time: 30.2s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(\left(-\frac{1.0}{3.0}\right) \cdot \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} + \left(a - \frac{1.0}{3.0}\right)\right) + \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} \cdot a\]
\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)
↓
\left(\left(-\frac{1.0}{3.0}\right) \cdot \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} + \left(a - \frac{1.0}{3.0}\right)\right) + \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} \cdot a
double f(double a, double rand) {
        double r1680534 = a;
        double r1680535 = 1.0;
        double r1680536 = 3.0;
        double r1680537 = r1680535 / r1680536;
        double r1680538 = r1680534 - r1680537;
        double r1680539 = 1.0;
        double r1680540 = 9.0;
        double r1680541 = r1680540 * r1680538;
        double r1680542 = sqrt(r1680541);
        double r1680543 = r1680539 / r1680542;
        double r1680544 = rand;
        double r1680545 = r1680543 * r1680544;
        double r1680546 = r1680539 + r1680545;
        double r1680547 = r1680538 * r1680546;
        return r1680547;
}
↓
double f(double a, double rand) {
        double r1680548 = 1.0;
        double r1680549 = 3.0;
        double r1680550 = r1680548 / r1680549;
        double r1680551 = -r1680550;
        double r1680552 = rand;
        double r1680553 = a;
        double r1680554 = r1680553 - r1680550;
        double r1680555 = sqrt(r1680554);
        double r1680556 = 3.0;
        double r1680557 = r1680555 * r1680556;
        double r1680558 = r1680552 / r1680557;
        double r1680559 = r1680551 * r1680558;
        double r1680560 = r1680559 + r1680554;
        double r1680561 = r1680558 * r1680553;
        double r1680562 = r1680560 + r1680561;
        return r1680562;
}

Error

Bits error versus a

Bits error versus rand

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied fma-udef0.1

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

  6. Applied sqrt-prod0.1

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

  7. Simplified0.1

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

  9. Applied sub-neg0.1

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

  10. Applied distribute-rgt-in0.1

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

  11. Applied associate-+l+0.1

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

  12. Final simplification0.1

    \[\leadsto \left(\left(-\frac{1.0}{3.0}\right) \cdot \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} + \left(a - \frac{1.0}{3.0}\right)\right) + \frac{rand}{\sqrt{a - \frac{1.0}{3.0}} \cdot 3} \cdot a\]

Reproduce

herbie shell --seed 2019152 +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))))