Average Error: 20.2 → 20.2
Time: 14.0s
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 r399276 = 2.0;
        double r399277 = x;
        double r399278 = y;
        double r399279 = r399277 * r399278;
        double r399280 = z;
        double r399281 = r399277 * r399280;
        double r399282 = r399279 + r399281;
        double r399283 = r399278 * r399280;
        double r399284 = r399282 + r399283;
        double r399285 = sqrt(r399284);
        double r399286 = r399276 * r399285;
        return r399286;
}
↓
double f(double x, double y, double z) {
        double r399287 = x;
        double r399288 = z;
        double r399289 = y;
        double r399290 = r399288 + r399287;
        double r399291 = r399289 * r399290;
        double r399292 = fma(r399287, r399288, r399291);
        double r399293 = sqrt(r399292);
        double r399294 = 2.0;
        double r399295 = r399293 * r399294;
        return r399295;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.2
Target19.4
Herbie20.2
\[\begin{array}{l} \mathbf{if}\;z \lt 7.636950090573674520215292914121377944071 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 20.2

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

  2. Simplified20.2

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

  3. Taylor expanded around 0 20.2

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

  4. Simplified20.2

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

  5. Final simplification20.2

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

Reproduce

herbie shell --seed 2019304 +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)))))