Average Error: 0.1 → 0.1
Time: 11.0s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(\frac{\frac{1}{\frac{\sqrt{9}}{rand}}}{\sqrt{a - \frac{1}{3}}} + 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(\frac{\frac{1}{\frac{\sqrt{9}}{rand}}}{\sqrt{a - \frac{1}{3}}} + 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r128620 = a;
        double r128621 = 1.0;
        double r128622 = 3.0;
        double r128623 = r128621 / r128622;
        double r128624 = r128620 - r128623;
        double r128625 = 9.0;
        double r128626 = r128625 * r128624;
        double r128627 = sqrt(r128626);
        double r128628 = r128621 / r128627;
        double r128629 = rand;
        double r128630 = r128628 * r128629;
        double r128631 = r128621 + r128630;
        double r128632 = r128624 * r128631;
        return r128632;
}
↓
double f(double a, double rand) {
        double r128633 = 1.0;
        double r128634 = 9.0;
        double r128635 = sqrt(r128634);
        double r128636 = rand;
        double r128637 = r128635 / r128636;
        double r128638 = r128633 / r128637;
        double r128639 = a;
        double r128640 = 3.0;
        double r128641 = r128633 / r128640;
        double r128642 = r128639 - r128641;
        double r128643 = sqrt(r128642);
        double r128644 = r128638 / r128643;
        double r128645 = r128644 + r128633;
        double r128646 = r128645 * r128642;
        return r128646;
}

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-sqr-sqrt0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

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

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

  9. Applied associate-*l*0.2

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))