Average Error: 0.2 → 0.1
Time: 3.6m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{\frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}}}{3} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]
double f(double a, double rand) {
        double r11211614 = a;
        double r11211615 = 1.0;
        double r11211616 = 3.0;
        double r11211617 = r11211615 / r11211616;
        double r11211618 = r11211614 - r11211617;
        double r11211619 = 1.0;
        double r11211620 = 9.0;
        double r11211621 = r11211620 * r11211618;
        double r11211622 = sqrt(r11211621);
        double r11211623 = r11211619 / r11211622;
        double r11211624 = rand;
        double r11211625 = r11211623 * r11211624;
        double r11211626 = r11211619 + r11211625;
        double r11211627 = r11211618 * r11211626;
        return r11211627;
}
↓
double f(double a, double rand) {
        double r11211628 = a;
        double r11211629 = 1.0;
        double r11211630 = 3.0;
        double r11211631 = r11211629 / r11211630;
        double r11211632 = r11211628 - r11211631;
        double r11211633 = sqrt(r11211632);
        double r11211634 = r11211632 / r11211633;
        double r11211635 = 3.0;
        double r11211636 = r11211634 / r11211635;
        double r11211637 = rand;
        double r11211638 = r11211636 * r11211637;
        double r11211639 = r11211638 + r11211632;
        return r11211639;
}


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

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

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

  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 sqrt-prod0.1

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

  5. Applied associate-/r*0.1

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

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

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

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

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

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

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

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

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

  11. Applied distribute-lft-out--0.1

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

  12. Applied times-frac0.1

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

  13. Applied times-frac0.1

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

  14. Simplified0.1

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

  15. Simplified0.1

    \[\leadsto rand \cdot \left(1 \cdot \color{blue}{\frac{\frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}}}{3}}\right) + \left(a - \frac{1.0}{3.0}\right)\]

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019101 
(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))))