Average Error: 0.1 → 0.1
Time: 19.6s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{rand \cdot {\left(\sqrt[3]{1}\right)}^{3}}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{rand \cdot {\left(\sqrt[3]{1}\right)}^{3}}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}\right)
double f(double a, double rand) {
        double r99832 = a;
        double r99833 = 1.0;
        double r99834 = 3.0;
        double r99835 = r99833 / r99834;
        double r99836 = r99832 - r99835;
        double r99837 = 9.0;
        double r99838 = r99837 * r99836;
        double r99839 = sqrt(r99838);
        double r99840 = r99833 / r99839;
        double r99841 = rand;
        double r99842 = r99840 * r99841;
        double r99843 = r99833 + r99842;
        double r99844 = r99836 * r99843;
        return r99844;
}
↓
double f(double a, double rand) {
        double r99845 = a;
        double r99846 = 1.0;
        double r99847 = 3.0;
        double r99848 = r99846 / r99847;
        double r99849 = r99845 - r99848;
        double r99850 = rand;
        double r99851 = cbrt(r99846);
        double r99852 = 3.0;
        double r99853 = pow(r99851, r99852);
        double r99854 = r99850 * r99853;
        double r99855 = sqrt(r99849);
        double r99856 = r99854 / r99855;
        double r99857 = 9.0;
        double r99858 = sqrt(r99857);
        double r99859 = r99856 / r99858;
        double r99860 = r99846 + r99859;
        double r99861 = r99849 * r99860;
        return r99861;
}

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

    \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
  2. Using strategy rm

  3. Applied sqrt-prod0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{9}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt{a - \frac{1}{3}}} \cdot rand\right)}\right)\]
  7. Using strategy rm

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))