Average Error: 0.1 → 0.2
Time: 16.3s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{\left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9}}\]
\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 1 + \frac{\left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9}}
double f(double a, double rand) {
        double r88760 = a;
        double r88761 = 1.0;
        double r88762 = 3.0;
        double r88763 = r88761 / r88762;
        double r88764 = r88760 - r88763;
        double r88765 = 9.0;
        double r88766 = r88765 * r88764;
        double r88767 = sqrt(r88766);
        double r88768 = r88761 / r88767;
        double r88769 = rand;
        double r88770 = r88768 * r88769;
        double r88771 = r88761 + r88770;
        double r88772 = r88764 * r88771;
        return r88772;
}
↓
double f(double a, double rand) {
        double r88773 = a;
        double r88774 = 1.0;
        double r88775 = 3.0;
        double r88776 = r88774 / r88775;
        double r88777 = r88773 - r88776;
        double r88778 = r88777 * r88774;
        double r88779 = sqrt(r88777);
        double r88780 = r88774 / r88779;
        double r88781 = rand;
        double r88782 = r88780 * r88781;
        double r88783 = r88782 * r88777;
        double r88784 = 9.0;
        double r88785 = sqrt(r88784);
        double r88786 = r88783 / r88785;
        double r88787 = r88778 + r88786;
        return r88787;
}

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 distribute-lft-in0.1

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

  5. Applied sqrt-prod0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied times-frac0.2

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

  8. Applied associate-*l*0.2

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

  10. Applied associate-*l/0.1

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

  11. Applied associate-*r/0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))