Average Error: 0.1 → 0.1
Time: 9.0s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}\right)
double f(double a, double rand) {
        double r82182 = a;
        double r82183 = 1.0;
        double r82184 = 3.0;
        double r82185 = r82183 / r82184;
        double r82186 = r82182 - r82185;
        double r82187 = 9.0;
        double r82188 = r82187 * r82186;
        double r82189 = sqrt(r82188);
        double r82190 = r82183 / r82189;
        double r82191 = rand;
        double r82192 = r82190 * r82191;
        double r82193 = r82183 + r82192;
        double r82194 = r82186 * r82193;
        return r82194;
}
↓
double f(double a, double rand) {
        double r82195 = a;
        double r82196 = 1.0;
        double r82197 = 3.0;
        double r82198 = r82196 / r82197;
        double r82199 = r82195 - r82198;
        double r82200 = rand;
        double r82201 = r82200 * r82196;
        double r82202 = 9.0;
        double r82203 = sqrt(r82202);
        double r82204 = sqrt(r82199);
        double r82205 = r82203 * r82204;
        double r82206 = r82201 / r82205;
        double r82207 = r82196 + r82206;
        double r82208 = r82199 * r82207;
        return r82208;
}

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-prod0.1

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

  4. Applied add-sqr-sqrt0.1

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

  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 associate-*l/0.1

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

  9. Applied frac-times0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020021 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))