Average Error: 0.1 → 0.1
Time: 10.9s
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 r73821 = a;
        double r73822 = 1.0;
        double r73823 = 3.0;
        double r73824 = r73822 / r73823;
        double r73825 = r73821 - r73824;
        double r73826 = 9.0;
        double r73827 = r73826 * r73825;
        double r73828 = sqrt(r73827);
        double r73829 = r73822 / r73828;
        double r73830 = rand;
        double r73831 = r73829 * r73830;
        double r73832 = r73822 + r73831;
        double r73833 = r73825 * r73832;
        return r73833;
}
↓
double f(double a, double rand) {
        double r73834 = 1.0;
        double r73835 = 9.0;
        double r73836 = sqrt(r73835);
        double r73837 = rand;
        double r73838 = r73836 / r73837;
        double r73839 = r73834 / r73838;
        double r73840 = a;
        double r73841 = 3.0;
        double r73842 = r73834 / r73841;
        double r73843 = r73840 - r73842;
        double r73844 = sqrt(r73843);
        double r73845 = r73839 / r73844;
        double r73846 = r73845 + r73834;
        double r73847 = r73846 * r73843;
        return r73847;
}

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