Average Error: 0.1 → 0.1
Time: 8.3s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1 \cdot \frac{rand \cdot {\left(\sqrt[3]{1}\right)}^{3}}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right)\]
\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 \left(1 + \frac{1 \cdot \frac{rand \cdot {\left(\sqrt[3]{1}\right)}^{3}}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right)
double f(double a, double rand) {
        double r84179 = a;
        double r84180 = 1.0;
        double r84181 = 3.0;
        double r84182 = r84180 / r84181;
        double r84183 = r84179 - r84182;
        double r84184 = 9.0;
        double r84185 = r84184 * r84183;
        double r84186 = sqrt(r84185);
        double r84187 = r84180 / r84186;
        double r84188 = rand;
        double r84189 = r84187 * r84188;
        double r84190 = r84180 + r84189;
        double r84191 = r84183 * r84190;
        return r84191;
}
↓
double f(double a, double rand) {
        double r84192 = a;
        double r84193 = 1.0;
        double r84194 = 3.0;
        double r84195 = r84193 / r84194;
        double r84196 = r84192 - r84195;
        double r84197 = 1.0;
        double r84198 = rand;
        double r84199 = cbrt(r84193);
        double r84200 = 3.0;
        double r84201 = pow(r84199, r84200);
        double r84202 = r84198 * r84201;
        double r84203 = 9.0;
        double r84204 = sqrt(r84203);
        double r84205 = r84202 / r84204;
        double r84206 = r84197 * r84205;
        double r84207 = sqrt(r84196);
        double r84208 = r84206 / r84207;
        double r84209 = r84193 + r84208;
        double r84210 = r84196 * r84209;
        return r84210;
}

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 sqrt-prod0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

  8. Applied associate-*l/0.2

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

  9. Applied associate-*r/0.2

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

  11. Applied *-un-lft-identity0.2

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

  12. Applied associate-*l*0.2

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020020 +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))))