Average Error: 0.2 → 0.1
Time: 3.2m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{3 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot 3\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{3 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot 3\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r9123063 = a;
        double r9123064 = 1.0;
        double r9123065 = 3.0;
        double r9123066 = r9123064 / r9123065;
        double r9123067 = r9123063 - r9123066;
        double r9123068 = 1.0;
        double r9123069 = 9.0;
        double r9123070 = r9123069 * r9123067;
        double r9123071 = sqrt(r9123070);
        double r9123072 = r9123068 / r9123071;
        double r9123073 = rand;
        double r9123074 = r9123072 * r9123073;
        double r9123075 = r9123068 + r9123074;
        double r9123076 = r9123067 * r9123075;
        return r9123076;
}
↓
double f(double a, double rand) {
        double r9123077 = a;
        double r9123078 = 1.0;
        double r9123079 = 3.0;
        double r9123080 = r9123078 / r9123079;
        double r9123081 = r9123077 - r9123080;
        double r9123082 = rand;
        double r9123083 = 3.0;
        double r9123084 = r9123081 * r9123083;
        double r9123085 = r9123083 * r9123084;
        double r9123086 = sqrt(r9123085);
        double r9123087 = r9123082 / r9123086;
        double r9123088 = r9123087 * r9123081;
        double r9123089 = r9123081 + r9123088;
        return r9123089;
}

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 0.2

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.1

    \[\leadsto \mathsf{fma}\left(\left(\frac{rand}{\sqrt{\color{blue}{\left(\sqrt{9} \cdot \sqrt{9}\right)} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]

  5. Applied associate-*l*0.1

    \[\leadsto \mathsf{fma}\left(\left(\frac{rand}{\sqrt{\color{blue}{\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]
  6. Using strategy rm

  7. Applied fma-udef0.1

    \[\leadsto \color{blue}{\frac{rand}{\sqrt{\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)}\]

  8. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{3 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot 3\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))