Average Error: 0.1 → 0.1
Time: 1.4m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(\frac{a - \frac{1.0}{3.0}}{3 \cdot \left(\sqrt{a} - \sqrt{\frac{1.0}{3.0}}\right)} \cdot rand\right) \cdot \frac{\sqrt{a} - \sqrt{\frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}} + \left(a - \frac{1.0}{3.0}\right)\]
\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(\frac{a - \frac{1.0}{3.0}}{3 \cdot \left(\sqrt{a} - \sqrt{\frac{1.0}{3.0}}\right)} \cdot rand\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)
double f(double a, double rand) {
        double r10038291 = a;
        double r10038292 = 1.0;
        double r10038293 = 3.0;
        double r10038294 = r10038292 / r10038293;
        double r10038295 = r10038291 - r10038294;
        double r10038296 = 1.0;
        double r10038297 = 9.0;
        double r10038298 = r10038297 * r10038295;
        double r10038299 = sqrt(r10038298);
        double r10038300 = r10038296 / r10038299;
        double r10038301 = rand;
        double r10038302 = r10038300 * r10038301;
        double r10038303 = r10038296 + r10038302;
        double r10038304 = r10038295 * r10038303;
        return r10038304;
}
↓
double f(double a, double rand) {
        double r10038305 = a;
        double r10038306 = 1.0;
        double r10038307 = 3.0;
        double r10038308 = r10038306 / r10038307;
        double r10038309 = r10038305 - r10038308;
        double r10038310 = 3.0;
        double r10038311 = sqrt(r10038305);
        double r10038312 = sqrt(r10038308);
        double r10038313 = r10038311 - r10038312;
        double r10038314 = r10038310 * r10038313;
        double r10038315 = r10038309 / r10038314;
        double r10038316 = rand;
        double r10038317 = r10038315 * r10038316;
        double r10038318 = sqrt(r10038309);
        double r10038319 = r10038313 / r10038318;
        double r10038320 = r10038317 * r10038319;
        double r10038321 = r10038320 + r10038309;
        return r10038321;
}

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-sqrt0.2

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

  7. Applied difference-of-squares0.2

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

  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. Using strategy rm

  11. Applied flip-+0.1

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

  12. Applied associate-/l/0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019121 
(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))))