Average Error: 19.5 → 19.5
Time: 14.5s
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 r452141 = 2.0;
        double r452142 = x;
        double r452143 = y;
        double r452144 = r452142 * r452143;
        double r452145 = z;
        double r452146 = r452142 * r452145;
        double r452147 = r452144 + r452146;
        double r452148 = r452143 * r452145;
        double r452149 = r452147 + r452148;
        double r452150 = sqrt(r452149);
        double r452151 = r452141 * r452150;
        return r452151;
}
↓
double f(double x, double y, double z) {
        double r452152 = x;
        double r452153 = z;
        double r452154 = y;
        double r452155 = r452153 + r452152;
        double r452156 = r452154 * r452155;
        double r452157 = fma(r452152, r452153, r452156);
        double r452158 = sqrt(r452157);
        double r452159 = 2.0;
        double r452160 = r452158 * r452159;
        return r452160;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.5
Target18.8
Herbie19.5
\[\begin{array}{l} \mathbf{if}\;z \lt 7.636950090573674520215292914121377944071 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 19.5

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

  2. Simplified19.5

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

  3. Taylor expanded around 0 19.5

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

  4. Simplified19.5

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

  5. Final simplification19.5

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

Reproduce

herbie shell --seed 2019325 +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)))))