Average Error: 0.1 → 0.1
Time: 18.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(\frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}} + 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(\frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}} + 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r129806 = a;
        double r129807 = 1.0;
        double r129808 = 3.0;
        double r129809 = r129807 / r129808;
        double r129810 = r129806 - r129809;
        double r129811 = 9.0;
        double r129812 = r129811 * r129810;
        double r129813 = sqrt(r129812);
        double r129814 = r129807 / r129813;
        double r129815 = rand;
        double r129816 = r129814 * r129815;
        double r129817 = r129807 + r129816;
        double r129818 = r129810 * r129817;
        return r129818;
}
↓
double f(double a, double rand) {
        double r129819 = 1.0;
        double r129820 = rand;
        double r129821 = r129819 * r129820;
        double r129822 = 9.0;
        double r129823 = sqrt(r129822);
        double r129824 = r129821 / r129823;
        double r129825 = a;
        double r129826 = 3.0;
        double r129827 = r129819 / r129826;
        double r129828 = r129825 - r129827;
        double r129829 = sqrt(r129828);
        double r129830 = r129824 / r129829;
        double r129831 = r129830 + r129819;
        double r129832 = r129831 * r129828;
        return r129832;
}

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. Simplified0.1

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

  4. Applied sqrt-prod0.1

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

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

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

  6. Applied times-frac0.2

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

  8. Applied fma-udef0.2

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

  9. Simplified0.1

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

  11. Applied associate-*l/0.1

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

  12. Applied associate-/l/0.1

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

  14. Applied associate-/r*0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +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))))