Average Error: 0.1 → 0.1
Time: 26.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(\left(\left(a + \left(-\frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}\right)\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(\frac{1}{\sqrt{3}} \cdot \left(\frac{-1}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right)\right)\right)\]
\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 1 + \left(\left(\left(a + \left(-\frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}\right)\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(\frac{1}{\sqrt{3}} \cdot \left(\frac{-1}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right)\right)\right)
double f(double a, double rand) {
        double r88974 = a;
        double r88975 = 1.0;
        double r88976 = 3.0;
        double r88977 = r88975 / r88976;
        double r88978 = r88974 - r88977;
        double r88979 = 9.0;
        double r88980 = r88979 * r88978;
        double r88981 = sqrt(r88980);
        double r88982 = r88975 / r88981;
        double r88983 = rand;
        double r88984 = r88982 * r88983;
        double r88985 = r88975 + r88984;
        double r88986 = r88978 * r88985;
        return r88986;
}
↓
double f(double a, double rand) {
        double r88987 = a;
        double r88988 = 1.0;
        double r88989 = 3.0;
        double r88990 = r88988 / r88989;
        double r88991 = r88987 - r88990;
        double r88992 = r88991 * r88988;
        double r88993 = sqrt(r88989);
        double r88994 = r88988 / r88993;
        double r88995 = r88994 / r88993;
        double r88996 = -r88995;
        double r88997 = r88987 + r88996;
        double r88998 = 9.0;
        double r88999 = r88998 * r88991;
        double r89000 = sqrt(r88999);
        double r89001 = r88988 / r89000;
        double r89002 = r88997 * r89001;
        double r89003 = rand;
        double r89004 = r89002 * r89003;
        double r89005 = r89001 * r89003;
        double r89006 = -1.0;
        double r89007 = r89006 / r88993;
        double r89008 = 1.0;
        double r89009 = r89008 / r88993;
        double r89010 = r89007 + r89009;
        double r89011 = r88994 * r89010;
        double r89012 = r89005 * r89011;
        double r89013 = r89004 + r89012;
        double r89014 = r88992 + r89013;
        return r89014;
}

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. Simplified0.1

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

  6. Applied add-sqr-sqrt0.1

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

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

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

  8. Applied times-frac0.1

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

  9. Applied add-sqr-sqrt0.2

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

  10. Applied prod-diff0.2

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

  11. Applied distribute-rgt-in0.2

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

  12. Simplified0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 +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))))