Average Error: 0.1 → 0.1
Time: 15.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(\left(a - \frac{1}{3}\right) \cdot \frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right) \cdot rand\]
\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(\left(a - \frac{1}{3}\right) \cdot \frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right) \cdot rand
double f(double a, double rand) {
        double r310 = a;
        double r311 = 1.0;
        double r312 = 3.0;
        double r313 = r311 / r312;
        double r314 = r310 - r313;
        double r315 = 9.0;
        double r316 = r315 * r314;
        double r317 = sqrt(r316);
        double r318 = r311 / r317;
        double r319 = rand;
        double r320 = r318 * r319;
        double r321 = r311 + r320;
        double r322 = r314 * r321;
        return r322;
}
↓
double f(double a, double rand) {
        double r323 = a;
        double r324 = 1.0;
        double r325 = 3.0;
        double r326 = r324 / r325;
        double r327 = r323 - r326;
        double r328 = r327 * r324;
        double r329 = 9.0;
        double r330 = sqrt(r329);
        double r331 = r324 / r330;
        double r332 = sqrt(r327);
        double r333 = r331 / r332;
        double r334 = r327 * r333;
        double r335 = rand;
        double r336 = r334 * r335;
        double r337 = r328 + r336;
        return r337;
}

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-*r*0.1

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

  7. Applied sqrt-prod0.1

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

  8. Applied associate-/r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 +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))))