Average Error: 0.1 → 0.1
Time: 21.8s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}} + \left(a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}} + \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r3256307 = a;
        double r3256308 = 1.0;
        double r3256309 = 3.0;
        double r3256310 = r3256308 / r3256309;
        double r3256311 = r3256307 - r3256310;
        double r3256312 = 1.0;
        double r3256313 = 9.0;
        double r3256314 = r3256313 * r3256311;
        double r3256315 = sqrt(r3256314);
        double r3256316 = r3256312 / r3256315;
        double r3256317 = rand;
        double r3256318 = r3256316 * r3256317;
        double r3256319 = r3256312 + r3256318;
        double r3256320 = r3256311 * r3256319;
        return r3256320;
}
↓
double f(double a, double rand) {
        double r3256321 = rand;
        double r3256322 = 3.0;
        double r3256323 = r3256321 / r3256322;
        double r3256324 = a;
        double r3256325 = 1.0;
        double r3256326 = 3.0;
        double r3256327 = r3256325 / r3256326;
        double r3256328 = r3256324 - r3256327;
        double r3256329 = sqrt(r3256328);
        double r3256330 = r3256323 * r3256329;
        double r3256331 = r3256330 + r3256328;
        return r3256331;
}

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

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}, rand, a - \frac{1.0}{3.0}\right)}\]
  3. Using strategy rm

  4. Applied sqrt-prod0.1

    \[\leadsto \mathsf{fma}\left(\frac{a - \frac{1.0}{3.0}}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}}, rand, a - \frac{1.0}{3.0}\right)\]

  5. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\frac{a - \frac{1.0}{3.0}}{\color{blue}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}}, rand, a - \frac{1.0}{3.0}\right)\]
  6. Using strategy rm

  7. Applied fma-udef0.1

    \[\leadsto \color{blue}{\frac{a - \frac{1.0}{3.0}}{3 \cdot \sqrt{a - \frac{1.0}{3.0}}} \cdot rand + \left(a - \frac{1.0}{3.0}\right)}\]

  8. Simplified0.1

    \[\leadsto \color{blue}{\frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}} + \left(a - \frac{1.0}{3.0}\right)\]

  9. Final simplification0.1

    \[\leadsto \frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}} + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))