Average Error: 0.1 → 0.1
Time: 24.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}} \cdot \left(1 \cdot \left(a - \frac{1}{3}\right)\right) + 1 \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}} \cdot \left(1 \cdot \left(a - \frac{1}{3}\right)\right) + 1 \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r3472133 = a;
        double r3472134 = 1.0;
        double r3472135 = 3.0;
        double r3472136 = r3472134 / r3472135;
        double r3472137 = r3472133 - r3472136;
        double r3472138 = 9.0;
        double r3472139 = r3472138 * r3472137;
        double r3472140 = sqrt(r3472139);
        double r3472141 = r3472134 / r3472140;
        double r3472142 = rand;
        double r3472143 = r3472141 * r3472142;
        double r3472144 = r3472134 + r3472143;
        double r3472145 = r3472137 * r3472144;
        return r3472145;
}
↓
double f(double a, double rand) {
        double r3472146 = rand;
        double r3472147 = a;
        double r3472148 = 1.0;
        double r3472149 = 3.0;
        double r3472150 = r3472148 / r3472149;
        double r3472151 = r3472147 - r3472150;
        double r3472152 = 9.0;
        double r3472153 = r3472151 * r3472152;
        double r3472154 = sqrt(r3472153);
        double r3472155 = r3472146 / r3472154;
        double r3472156 = r3472148 * r3472151;
        double r3472157 = r3472155 * r3472156;
        double r3472158 = r3472157 + r3472156;
        return r3472158;
}

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 fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +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))))