Average Error: 38.3 → 0.5
Time: 3.6s
Precision: binary64
\[[x, y, z] = \mathsf{sort}([x, y, z]) \\]
\[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z} \]
↓
\[\mathsf{hypot}\left(x, z\right) \]
\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z}
↓
\mathsf{hypot}\left(x, z\right)

(FPCore (x y z) :precision binary64 (sqrt (+ (+ (* x x) (* y y)) (* z z))))
↓
(FPCore (x y z) :precision binary64 (hypot x z))
double code(double x, double y, double z) {
	return sqrt(((x * x) + (y * y)) + (z * z));
}
↓
double code(double x, double y, double z) {
	return hypot(x, z);
}

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

Target

Original38.3
Target19.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z < -6.396479394109776 \cdot 10^{+136}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z < 7.320293694404182 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{\left(z \cdot z + x \cdot x\right) + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Derivation

  1. Initial program 38.3

    \[\sqrt{\left(x \cdot x + y \cdot y\right) + z \cdot z} \]

  2. Simplified38.3

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, x, y \cdot y\right)\right)}} \]

  3. Taylor expanded in y around 0 38.5

    \[\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}}} \]

  4. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{hypot}\left(x, z\right)} \]

  5. Final simplification0.5

    \[\leadsto \mathsf{hypot}\left(x, z\right) \]

Reproduce

herbie shell --seed 2022076 
(FPCore (x y z)
  :name "FRP.Yampa.Vector3:vector3Rho from Yampa-0.10.2"
  :precision binary64

  :herbie-target
  (if (< z -6.396479394109776e+136) (- z) (if (< z 7.320293694404182e+117) (sqrt (+ (+ (* z z) (* x x)) (* y y))) z))

  (sqrt (+ (+ (* x x) (* y y)) (* z z))))