Average Error: 0.1 → 0.1
Time: 23.7s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}\]
\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)
↓
\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}
double f(double a, double rand) {
        double r3591758 = a;
        double r3591759 = 1.0;
        double r3591760 = 3.0;
        double r3591761 = r3591759 / r3591760;
        double r3591762 = r3591758 - r3591761;
        double r3591763 = 1.0;
        double r3591764 = 9.0;
        double r3591765 = r3591764 * r3591762;
        double r3591766 = sqrt(r3591765);
        double r3591767 = r3591763 / r3591766;
        double r3591768 = rand;
        double r3591769 = r3591767 * r3591768;
        double r3591770 = r3591763 + r3591769;
        double r3591771 = r3591762 * r3591770;
        return r3591771;
}
↓
double f(double a, double rand) {
        double r3591772 = a;
        double r3591773 = 1.0;
        double r3591774 = 3.0;
        double r3591775 = r3591773 / r3591774;
        double r3591776 = r3591772 - r3591775;
        double r3591777 = rand;
        double r3591778 = 3.0;
        double r3591779 = r3591777 / r3591778;
        double r3591780 = sqrt(r3591776);
        double r3591781 = r3591779 * r3591780;
        double r3591782 = r3591776 + r3591781;
        return r3591782;
}

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

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

  4. Applied sqrt-prod0.1

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

  5. Simplified0.1

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

  7. Applied fma-udef0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

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))))