Average Error: 20.0 → 20.0
Time: 4.5s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[2 \cdot \sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
2 \cdot \sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}
double f(double x, double y, double z) {
        double r783436 = 2.0;
        double r783437 = x;
        double r783438 = y;
        double r783439 = r783437 * r783438;
        double r783440 = z;
        double r783441 = r783437 * r783440;
        double r783442 = r783439 + r783441;
        double r783443 = r783438 * r783440;
        double r783444 = r783442 + r783443;
        double r783445 = sqrt(r783444);
        double r783446 = r783436 * r783445;
        return r783446;
}
↓
double f(double x, double y, double z) {
        double r783447 = 2.0;
        double r783448 = x;
        double r783449 = y;
        double r783450 = z;
        double r783451 = r783449 + r783450;
        double r783452 = r783449 * r783450;
        double r783453 = fma(r783448, r783451, r783452);
        double r783454 = sqrt(r783453);
        double r783455 = r783447 * r783454;
        return r783455;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
Target19.4
Herbie20.0
\[\begin{array}{l} \mathbf{if}\;z \lt 7.6369500905736745 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 20.0

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

  3. Applied distribute-lft-out20.0

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

  4. Applied fma-def20.0

    \[\leadsto 2 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}\]

  5. Final simplification20.0

    \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2))

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