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 code(double x, double y, double z) {
	return ((1.0 / 2.0) * (x + (y * sqrt(z))));
}
↓
double code(double x, double y, double z) {
	return ((1.0 / 2.0) * (x + ((y * pow((1.0 / z), -0.25)) * pow(z, 0.25))));
}

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)))))