Average Error: 19.7 → 19.7
Time: 19.4s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[\sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)} \cdot 2\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
\sqrt{\mathsf{fma}\left(x, z, y \cdot \left(z + x\right)\right)} \cdot 2
double f(double x, double y, double z) {
        double r466423 = 2.0;
        double r466424 = x;
        double r466425 = y;
        double r466426 = r466424 * r466425;
        double r466427 = z;
        double r466428 = r466424 * r466427;
        double r466429 = r466426 + r466428;
        double r466430 = r466425 * r466427;
        double r466431 = r466429 + r466430;
        double r466432 = sqrt(r466431);
        double r466433 = r466423 * r466432;
        return r466433;
}
↓
double f(double x, double y, double z) {
        double r466434 = x;
        double r466435 = z;
        double r466436 = y;
        double r466437 = r466435 + r466434;
        double r466438 = r466436 * r466437;
        double r466439 = fma(r466434, r466435, r466438);
        double r466440 = sqrt(r466439);
        double r466441 = 2.0;
        double r466442 = r466440 * r466441;
        return r466442;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.7
Target18.7
Herbie19.7
\[\begin{array}{l} \mathbf{if}\;z \lt 7.636950090573674520215292914121377944071 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 19.7

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

  2. Simplified19.7

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

  3. Taylor expanded around 0 19.7

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

  4. Simplified19.7

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

  5. Final simplification19.7

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

Reproduce

herbie shell --seed 2019306 +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 7.6369500905736745e176) (* 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)))))