Average Error: 0.1 → 0.1
Time: 34.9s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}} \cdot \left(\left(a - \frac{1}{3}\right) \cdot 1\right) + \left(a - \frac{1}{3}\right) \cdot 1\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\frac{\frac{rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}} \cdot \left(\left(a - \frac{1}{3}\right) \cdot 1\right) + \left(a - \frac{1}{3}\right) \cdot 1
double f(double a, double rand) {
        double r3908137 = a;
        double r3908138 = 1.0;
        double r3908139 = 3.0;
        double r3908140 = r3908138 / r3908139;
        double r3908141 = r3908137 - r3908140;
        double r3908142 = 9.0;
        double r3908143 = r3908142 * r3908141;
        double r3908144 = sqrt(r3908143);
        double r3908145 = r3908138 / r3908144;
        double r3908146 = rand;
        double r3908147 = r3908145 * r3908146;
        double r3908148 = r3908138 + r3908147;
        double r3908149 = r3908141 * r3908148;
        return r3908149;
}
↓
double f(double a, double rand) {
        double r3908150 = rand;
        double r3908151 = 9.0;
        double r3908152 = sqrt(r3908151);
        double r3908153 = r3908150 / r3908152;
        double r3908154 = a;
        double r3908155 = 1.0;
        double r3908156 = 3.0;
        double r3908157 = r3908155 / r3908156;
        double r3908158 = r3908154 - r3908157;
        double r3908159 = sqrt(r3908158);
        double r3908160 = r3908153 / r3908159;
        double r3908161 = r3908158 * r3908155;
        double r3908162 = r3908160 * r3908161;
        double r3908163 = r3908162 + r3908161;
        return r3908163;
}

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

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

  4. Applied sqrt-prod0.1

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

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

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

  6. Applied times-frac0.1

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

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  11. Applied fma-udef0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))