Average Error: 0.1 → 0.2
Time: 4.2m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{\sqrt{a} - \sqrt{\frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{\left(\sqrt{a} + \sqrt{\frac{1.0}{3.0}}\right) \cdot rand}{3} + \left(a - \frac{1.0}{3.0}\right)\]
\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)
↓
\frac{\sqrt{a} - \sqrt{\frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{\left(\sqrt{a} + \sqrt{\frac{1.0}{3.0}}\right) \cdot rand}{3} + \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r21592216 = a;
        double r21592217 = 1.0;
        double r21592218 = 3.0;
        double r21592219 = r21592217 / r21592218;
        double r21592220 = r21592216 - r21592219;
        double r21592221 = 1.0;
        double r21592222 = 9.0;
        double r21592223 = r21592222 * r21592220;
        double r21592224 = sqrt(r21592223);
        double r21592225 = r21592221 / r21592224;
        double r21592226 = rand;
        double r21592227 = r21592225 * r21592226;
        double r21592228 = r21592221 + r21592227;
        double r21592229 = r21592220 * r21592228;
        return r21592229;
}
↓
double f(double a, double rand) {
        double r21592230 = a;
        double r21592231 = sqrt(r21592230);
        double r21592232 = 1.0;
        double r21592233 = 3.0;
        double r21592234 = r21592232 / r21592233;
        double r21592235 = sqrt(r21592234);
        double r21592236 = r21592231 - r21592235;
        double r21592237 = r21592230 - r21592234;
        double r21592238 = sqrt(r21592237);
        double r21592239 = r21592236 / r21592238;
        double r21592240 = r21592231 + r21592235;
        double r21592241 = rand;
        double r21592242 = r21592240 * r21592241;
        double r21592243 = 3.0;
        double r21592244 = r21592242 / r21592243;
        double r21592245 = r21592239 * r21592244;
        double r21592246 = r21592245 + r21592237;
        return r21592246;
}

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 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 add-sqr-sqrt0.1

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

  6. Applied add-sqr-sqrt0.1

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

  7. Applied difference-of-squares0.1

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

  8. Applied times-frac0.1

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

  9. Applied associate-*r*0.1

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

  10. Simplified0.1

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

  12. Applied associate-*l/0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(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))))