Average Error: 0.1 → 0.1
Time: 15.4s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(\frac{\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand}{\sqrt{9}} + 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\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}{\sqrt{a - \frac{1}{3}}} \cdot rand}{\sqrt{9}} + 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r113876 = a;
        double r113877 = 1.0;
        double r113878 = 3.0;
        double r113879 = r113877 / r113878;
        double r113880 = r113876 - r113879;
        double r113881 = 9.0;
        double r113882 = r113881 * r113880;
        double r113883 = sqrt(r113882);
        double r113884 = r113877 / r113883;
        double r113885 = rand;
        double r113886 = r113884 * r113885;
        double r113887 = r113877 + r113886;
        double r113888 = r113880 * r113887;
        return r113888;
}
↓
double f(double a, double rand) {
        double r113889 = 1.0;
        double r113890 = a;
        double r113891 = 3.0;
        double r113892 = r113889 / r113891;
        double r113893 = r113890 - r113892;
        double r113894 = sqrt(r113893);
        double r113895 = r113889 / r113894;
        double r113896 = rand;
        double r113897 = r113895 * r113896;
        double r113898 = 9.0;
        double r113899 = sqrt(r113898);
        double r113900 = r113897 / r113899;
        double r113901 = r113900 + r113889;
        double r113902 = r113901 * r113893;
        return r113902;
}

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. Simplified0.1

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

  4. Applied sqrt-prod0.2

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

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

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

  6. Applied times-frac0.2

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

  8. Applied fma-udef0.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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))))