Average Error: 0.1 → 0.2
Time: 15.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{\left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9}}\]
\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 + \frac{\left(\frac{1}{\sqrt{a - \frac{1}{3}}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}{\sqrt{9}}
double f(double a, double rand) {
        double r92289 = a;
        double r92290 = 1.0;
        double r92291 = 3.0;
        double r92292 = r92290 / r92291;
        double r92293 = r92289 - r92292;
        double r92294 = 9.0;
        double r92295 = r92294 * r92293;
        double r92296 = sqrt(r92295);
        double r92297 = r92290 / r92296;
        double r92298 = rand;
        double r92299 = r92297 * r92298;
        double r92300 = r92290 + r92299;
        double r92301 = r92293 * r92300;
        return r92301;
}
↓
double f(double a, double rand) {
        double r92302 = a;
        double r92303 = 1.0;
        double r92304 = 3.0;
        double r92305 = r92303 / r92304;
        double r92306 = r92302 - r92305;
        double r92307 = r92306 * r92303;
        double r92308 = sqrt(r92306);
        double r92309 = r92303 / r92308;
        double r92310 = rand;
        double r92311 = r92309 * r92310;
        double r92312 = r92311 * r92306;
        double r92313 = 9.0;
        double r92314 = sqrt(r92313);
        double r92315 = r92312 / r92314;
        double r92316 = r92307 + r92315;
        return r92316;
}

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 sqrt-prod0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied times-frac0.2

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

  8. Applied associate-*l*0.2

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

  10. Applied associate-*l/0.1

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

  11. Applied associate-*r/0.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))