Average Error: 0.1 → 0.1
Time: 1.5m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{\left(\left(\sqrt{a} + \sqrt{\frac{1.0}{3.0}}\right) \cdot 9\right) \cdot \left(\sqrt{a} - \sqrt{\frac{1.0}{3.0}}\right)}} + \left(a - \frac{1.0}{3.0}\right)\]
\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)
↓
rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{\left(\left(\sqrt{a} + \sqrt{\frac{1.0}{3.0}}\right) \cdot 9\right) \cdot \left(\sqrt{a} - \sqrt{\frac{1.0}{3.0}}\right)}} + \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r12487468 = a;
        double r12487469 = 1.0;
        double r12487470 = 3.0;
        double r12487471 = r12487469 / r12487470;
        double r12487472 = r12487468 - r12487471;
        double r12487473 = 1.0;
        double r12487474 = 9.0;
        double r12487475 = r12487474 * r12487472;
        double r12487476 = sqrt(r12487475);
        double r12487477 = r12487473 / r12487476;
        double r12487478 = rand;
        double r12487479 = r12487477 * r12487478;
        double r12487480 = r12487473 + r12487479;
        double r12487481 = r12487472 * r12487480;
        return r12487481;
}
↓
double f(double a, double rand) {
        double r12487482 = rand;
        double r12487483 = a;
        double r12487484 = 1.0;
        double r12487485 = 3.0;
        double r12487486 = r12487484 / r12487485;
        double r12487487 = r12487483 - r12487486;
        double r12487488 = sqrt(r12487483);
        double r12487489 = sqrt(r12487486);
        double r12487490 = r12487488 + r12487489;
        double r12487491 = 9.0;
        double r12487492 = r12487490 * r12487491;
        double r12487493 = r12487488 - r12487489;
        double r12487494 = r12487492 * r12487493;
        double r12487495 = sqrt(r12487494);
        double r12487496 = r12487487 / r12487495;
        double r12487497 = r12487482 * r12487496;
        double r12487498 = r12487497 + r12487487;
        return r12487498;
}

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

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied difference-of-squares0.1

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

  7. Applied associate-*r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019119 
(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))))