Average Error: 0.1 → 0.1
Time: 20.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \sqrt{a - \frac{1}{3}} \cdot \left(\sqrt{a - \frac{1}{3}} \cdot \frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot 1 + \sqrt{a - \frac{1}{3}} \cdot \left(\sqrt{a - \frac{1}{3}} \cdot \frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right)
double f(double a, double rand) {
        double r98101 = a;
        double r98102 = 1.0;
        double r98103 = 3.0;
        double r98104 = r98102 / r98103;
        double r98105 = r98101 - r98104;
        double r98106 = 9.0;
        double r98107 = r98106 * r98105;
        double r98108 = sqrt(r98107);
        double r98109 = r98102 / r98108;
        double r98110 = rand;
        double r98111 = r98109 * r98110;
        double r98112 = r98102 + r98111;
        double r98113 = r98105 * r98112;
        return r98113;
}
↓
double f(double a, double rand) {
        double r98114 = a;
        double r98115 = 1.0;
        double r98116 = 3.0;
        double r98117 = r98115 / r98116;
        double r98118 = r98114 - r98117;
        double r98119 = r98118 * r98115;
        double r98120 = sqrt(r98118);
        double r98121 = rand;
        double r98122 = r98115 * r98121;
        double r98123 = 9.0;
        double r98124 = sqrt(r98123);
        double r98125 = r98122 / r98124;
        double r98126 = r98125 / r98120;
        double r98127 = r98120 * r98126;
        double r98128 = r98120 * r98127;
        double r98129 = r98119 + r98128;
        return r98129;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.1

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

  5. Applied sqrt-prod0.1

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

  6. Applied add-sqr-sqrt0.1

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

  7. Applied times-frac0.2

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

  8. Applied associate-*l*0.2

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

  10. Applied add-sqr-sqrt0.2

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

  11. Applied associate-*l*0.2

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))