Average Error: 0.2 → 0.3
Time: 4.1s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{1}{2} \cdot \left(x + \left(y \cdot {\left(\frac{1}{z}\right)}^{\frac{-1}{4}}\right) \cdot {z}^{\frac{1}{4}}\right)\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{1}{2} \cdot \left(x + \left(y \cdot {\left(\frac{1}{z}\right)}^{\frac{-1}{4}}\right) \cdot {z}^{\frac{1}{4}}\right)
double f(double x, double y, double z) {
        double r194920 = 1.0;
        double r194921 = 2.0;
        double r194922 = r194920 / r194921;
        double r194923 = x;
        double r194924 = y;
        double r194925 = z;
        double r194926 = sqrt(r194925);
        double r194927 = r194924 * r194926;
        double r194928 = r194923 + r194927;
        double r194929 = r194922 * r194928;
        return r194929;
}
↓
double f(double x, double y, double z) {
        double r194930 = 1.0;
        double r194931 = 2.0;
        double r194932 = r194930 / r194931;
        double r194933 = x;
        double r194934 = y;
        double r194935 = 1.0;
        double r194936 = z;
        double r194937 = r194935 / r194936;
        double r194938 = -0.25;
        double r194939 = pow(r194937, r194938);
        double r194940 = r194934 * r194939;
        double r194941 = 0.25;
        double r194942 = pow(r194936, r194941);
        double r194943 = r194940 * r194942;
        double r194944 = r194933 + r194943;
        double r194945 = r194932 * r194944;
        return r194945;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{1}{2} \cdot \left(x + y \cdot \sqrt{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}\right)\]

  4. Applied sqrt-prod0.3

    \[\leadsto \frac{1}{2} \cdot \left(x + y \cdot \color{blue}{\left(\sqrt{\sqrt{z}} \cdot \sqrt{\sqrt{z}}\right)}\right)\]

  5. Applied associate-*r*0.3

    \[\leadsto \frac{1}{2} \cdot \left(x + \color{blue}{\left(y \cdot \sqrt{\sqrt{z}}\right) \cdot \sqrt{\sqrt{z}}}\right)\]

  6. Taylor expanded around inf 0.3

    \[\leadsto \frac{1}{2} \cdot \left(x + \left(y \cdot \color{blue}{{\left(\frac{1}{z}\right)}^{\frac{-1}{4}}}\right) \cdot \sqrt{\sqrt{z}}\right)\]
  7. Using strategy rm

  8. Applied pow1/20.3

    \[\leadsto \frac{1}{2} \cdot \left(x + \left(y \cdot {\left(\frac{1}{z}\right)}^{\frac{-1}{4}}\right) \cdot \sqrt{\color{blue}{{z}^{\frac{1}{2}}}}\right)\]

  9. Applied sqrt-pow10.3

    \[\leadsto \frac{1}{2} \cdot \left(x + \left(y \cdot {\left(\frac{1}{z}\right)}^{\frac{-1}{4}}\right) \cdot \color{blue}{{z}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right)\]

  10. Simplified0.3

    \[\leadsto \frac{1}{2} \cdot \left(x + \left(y \cdot {\left(\frac{1}{z}\right)}^{\frac{-1}{4}}\right) \cdot {z}^{\color{blue}{\frac{1}{4}}}\right)\]

  11. Final simplification0.3

    \[\leadsto \frac{1}{2} \cdot \left(x + \left(y \cdot {\left(\frac{1}{z}\right)}^{\frac{-1}{4}}\right) \cdot {z}^{\frac{1}{4}}\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1 2) (+ x (* y (sqrt z)))))