Average Error: 0.1 → 0.1
Time: 7.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{\frac{1 \cdot rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}\]
\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 + \left(a - \frac{1}{3}\right) \cdot \frac{\frac{1 \cdot rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}
double f(double a, double rand) {
        double r97132 = a;
        double r97133 = 1.0;
        double r97134 = 3.0;
        double r97135 = r97133 / r97134;
        double r97136 = r97132 - r97135;
        double r97137 = 9.0;
        double r97138 = r97137 * r97136;
        double r97139 = sqrt(r97138);
        double r97140 = r97133 / r97139;
        double r97141 = rand;
        double r97142 = r97140 * r97141;
        double r97143 = r97133 + r97142;
        double r97144 = r97136 * r97143;
        return r97144;
}
↓
double f(double a, double rand) {
        double r97145 = a;
        double r97146 = 1.0;
        double r97147 = 3.0;
        double r97148 = r97146 / r97147;
        double r97149 = r97145 - r97148;
        double r97150 = r97149 * r97146;
        double r97151 = rand;
        double r97152 = r97146 * r97151;
        double r97153 = sqrt(r97149);
        double r97154 = r97152 / r97153;
        double r97155 = 9.0;
        double r97156 = sqrt(r97155);
        double r97157 = r97154 / r97156;
        double r97158 = r97149 * r97157;
        double r97159 = r97150 + r97158;
        return r97159;
}

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 *-un-lft-identity0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

  8. Applied distribute-lft-in0.2

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

  9. Simplified0.2

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

  11. Applied *-un-lft-identity0.2

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

  12. Applied sqrt-prod0.2

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

  13. Applied times-frac0.1

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

  14. Simplified0.1

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

  16. Applied associate-*l/0.1

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

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))