Average Error: 0.1 → 0.1
Time: 7.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\left(1 \cdot rand\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)\]
\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(a - \frac{1}{3}\right) \cdot \left(\left(1 \cdot rand\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)
double f(double a, double rand) {
        double r77136 = a;
        double r77137 = 1.0;
        double r77138 = 3.0;
        double r77139 = r77137 / r77138;
        double r77140 = r77136 - r77139;
        double r77141 = 9.0;
        double r77142 = r77141 * r77140;
        double r77143 = sqrt(r77142);
        double r77144 = r77137 / r77143;
        double r77145 = rand;
        double r77146 = r77144 * r77145;
        double r77147 = r77137 + r77146;
        double r77148 = r77140 * r77147;
        return r77148;
}
↓
double f(double a, double rand) {
        double r77149 = a;
        double r77150 = 1.0;
        double r77151 = 3.0;
        double r77152 = r77150 / r77151;
        double r77153 = r77149 - r77152;
        double r77154 = r77153 * r77150;
        double r77155 = rand;
        double r77156 = r77150 * r77155;
        double r77157 = 1.0;
        double r77158 = 9.0;
        double r77159 = r77158 * r77153;
        double r77160 = sqrt(r77159);
        double r77161 = r77157 / r77160;
        double r77162 = r77156 * r77161;
        double r77163 = r77153 * r77162;
        double r77164 = r77154 + r77163;
        return r77164;
}

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. Using strategy rm

  5. Applied associate-*l/0.1

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

  7. Applied div-inv0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020024 +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))))