Average Error: 0.1 → 0.2
Time: 29.6s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \left(\frac{1}{3} \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot rand\right)\]
\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)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \left(\frac{1}{3} \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot rand\right)
double f(double a, double rand) {
        double r2484645 = a;
        double r2484646 = 1.0;
        double r2484647 = 3.0;
        double r2484648 = r2484646 / r2484647;
        double r2484649 = r2484645 - r2484648;
        double r2484650 = 1.0;
        double r2484651 = 9.0;
        double r2484652 = r2484651 * r2484649;
        double r2484653 = sqrt(r2484652);
        double r2484654 = r2484650 / r2484653;
        double r2484655 = rand;
        double r2484656 = r2484654 * r2484655;
        double r2484657 = r2484650 + r2484656;
        double r2484658 = r2484649 * r2484657;
        return r2484658;
}
↓
double f(double a, double rand) {
        double r2484659 = a;
        double r2484660 = 1.0;
        double r2484661 = 3.0;
        double r2484662 = r2484660 / r2484661;
        double r2484663 = r2484659 - r2484662;
        double r2484664 = 1.0;
        double r2484665 = 0.3333333333333333;
        double r2484666 = sqrt(r2484663);
        double r2484667 = r2484664 / r2484666;
        double r2484668 = r2484665 * r2484667;
        double r2484669 = rand;
        double r2484670 = r2484668 * r2484669;
        double r2484671 = r2484664 + r2484670;
        double r2484672 = r2484663 * r2484671;
        return r2484672;
}

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. Using strategy rm

  3. Applied sqrt-prod0.2

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

  4. Applied add-cube-cbrt0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}} \cdot rand\right)\]

  5. Applied times-frac0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{9}} \cdot \frac{\sqrt[3]{1}}{\sqrt{a - \frac{1.0}{3.0}}}\right)} \cdot rand\right)\]

  6. Simplified0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \left(\color{blue}{\frac{1}{3}} \cdot \frac{\sqrt[3]{1}}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot rand\right)\]

  7. Simplified0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \left(\frac{1}{3} \cdot \color{blue}{\frac{1}{\sqrt{a - \frac{1.0}{3.0}}}}\right) \cdot rand\right)\]

  8. Final simplification0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \left(\frac{1}{3} \cdot \frac{1}{\sqrt{a - \frac{1.0}{3.0}}}\right) \cdot rand\right)\]

Reproduce

herbie shell --seed 2019146 +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))))