Average Error: 0.1 → 0.1
Time: 10.1s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \sqrt{1} \cdot \frac{\frac{\sqrt{1}}{\sqrt{a - \frac{1}{3}}} \cdot rand}{\sqrt{9}}\right)\]
\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 \left(1 + \sqrt{1} \cdot \frac{\frac{\sqrt{1}}{\sqrt{a - \frac{1}{3}}} \cdot rand}{\sqrt{9}}\right)
double f(double a, double rand) {
        double r86183 = a;
        double r86184 = 1.0;
        double r86185 = 3.0;
        double r86186 = r86184 / r86185;
        double r86187 = r86183 - r86186;
        double r86188 = 9.0;
        double r86189 = r86188 * r86187;
        double r86190 = sqrt(r86189);
        double r86191 = r86184 / r86190;
        double r86192 = rand;
        double r86193 = r86191 * r86192;
        double r86194 = r86184 + r86193;
        double r86195 = r86187 * r86194;
        return r86195;
}
↓
double f(double a, double rand) {
        double r86196 = a;
        double r86197 = 1.0;
        double r86198 = 3.0;
        double r86199 = r86197 / r86198;
        double r86200 = r86196 - r86199;
        double r86201 = sqrt(r86197);
        double r86202 = sqrt(r86200);
        double r86203 = r86201 / r86202;
        double r86204 = rand;
        double r86205 = r86203 * r86204;
        double r86206 = 9.0;
        double r86207 = sqrt(r86206);
        double r86208 = r86205 / r86207;
        double r86209 = r86201 * r86208;
        double r86210 = r86197 + r86209;
        double r86211 = r86200 * r86210;
        return r86211;
}

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 sqrt-prod0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

  8. Applied div-inv0.2

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

  9. Applied associate-*l*0.2

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020089 +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))))