Average Error: 19.6 → 19.6
Time: 17.3s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[2 \cdot \sqrt{\mathsf{fma}\left(y + x, z, x \cdot y\right)}\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
2 \cdot \sqrt{\mathsf{fma}\left(y + x, z, x \cdot y\right)}
double f(double x, double y, double z) {
        double r27200894 = 2.0;
        double r27200895 = x;
        double r27200896 = y;
        double r27200897 = r27200895 * r27200896;
        double r27200898 = z;
        double r27200899 = r27200895 * r27200898;
        double r27200900 = r27200897 + r27200899;
        double r27200901 = r27200896 * r27200898;
        double r27200902 = r27200900 + r27200901;
        double r27200903 = sqrt(r27200902);
        double r27200904 = r27200894 * r27200903;
        return r27200904;
}
↓
double f(double x, double y, double z) {
        double r27200905 = 2.0;
        double r27200906 = y;
        double r27200907 = x;
        double r27200908 = r27200906 + r27200907;
        double r27200909 = z;
        double r27200910 = r27200907 * r27200906;
        double r27200911 = fma(r27200908, r27200909, r27200910);
        double r27200912 = sqrt(r27200911);
        double r27200913 = r27200905 * r27200912;
        return r27200913;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.6
Target19.0
Herbie19.6
\[\begin{array}{l} \mathbf{if}\;z \lt 7.636950090573674520215292914121377944071 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 19.6

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

  2. Simplified19.6

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

  3. Final simplification19.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2.0 (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)))) 2.0))

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