Average Error: 0.1 → 0.2
Time: 15.6s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \left(\frac{1}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{1}{3}\right) \cdot rand\right)\]
\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}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{1}{3}\right) \cdot rand\right)
double f(double a, double rand) {
        double r1585000 = a;
        double r1585001 = 1.0;
        double r1585002 = 3.0;
        double r1585003 = r1585001 / r1585002;
        double r1585004 = r1585000 - r1585003;
        double r1585005 = 1.0;
        double r1585006 = 9.0;
        double r1585007 = r1585006 * r1585004;
        double r1585008 = sqrt(r1585007);
        double r1585009 = r1585005 / r1585008;
        double r1585010 = rand;
        double r1585011 = r1585009 * r1585010;
        double r1585012 = r1585005 + r1585011;
        double r1585013 = r1585004 * r1585012;
        return r1585013;
}
↓
double f(double a, double rand) {
        double r1585014 = a;
        double r1585015 = 1.0;
        double r1585016 = 3.0;
        double r1585017 = r1585015 / r1585016;
        double r1585018 = r1585014 - r1585017;
        double r1585019 = 1.0;
        double r1585020 = sqrt(r1585018);
        double r1585021 = r1585019 / r1585020;
        double r1585022 = 0.3333333333333333;
        double r1585023 = r1585021 * r1585022;
        double r1585024 = rand;
        double r1585025 = r1585023 * r1585024;
        double r1585026 = r1585019 + r1585025;
        double r1585027 = r1585018 * r1585026;
        return r1585027;
}

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

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

Reproduce

herbie shell --seed 2019156 
(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))))