Average Error: 20.0 → 20.0
Time: 14.3s
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 r28983696 = 2.0;
        double r28983697 = x;
        double r28983698 = y;
        double r28983699 = r28983697 * r28983698;
        double r28983700 = z;
        double r28983701 = r28983697 * r28983700;
        double r28983702 = r28983699 + r28983701;
        double r28983703 = r28983698 * r28983700;
        double r28983704 = r28983702 + r28983703;
        double r28983705 = sqrt(r28983704);
        double r28983706 = r28983696 * r28983705;
        return r28983706;
}
↓
double f(double x, double y, double z) {
        double r28983707 = 2.0;
        double r28983708 = y;
        double r28983709 = x;
        double r28983710 = r28983708 + r28983709;
        double r28983711 = z;
        double r28983712 = r28983709 * r28983708;
        double r28983713 = fma(r28983710, r28983711, r28983712);
        double r28983714 = sqrt(r28983713);
        double r28983715 = r28983707 * r28983714;
        return r28983715;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
Target19.0
Herbie20.0
\[\begin{array}{l} \mathbf{if}\;z \lt 7.636950090573674520215292914121377944071 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 20.0

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

  2. Simplified20.0

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

  3. Final simplification20.0

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

Reproduce

herbie shell --seed 2019174 +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)))))