Average Error: 19.7 → 19.7
Time: 17.9s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)} \cdot 2\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)} \cdot 2
double f(double x, double y, double z) {
        double r490242 = 2.0;
        double r490243 = x;
        double r490244 = y;
        double r490245 = r490243 * r490244;
        double r490246 = z;
        double r490247 = r490243 * r490246;
        double r490248 = r490245 + r490247;
        double r490249 = r490244 * r490246;
        double r490250 = r490248 + r490249;
        double r490251 = sqrt(r490250);
        double r490252 = r490242 * r490251;
        return r490252;
}
↓
double f(double x, double y, double z) {
        double r490253 = y;
        double r490254 = z;
        double r490255 = x;
        double r490256 = r490253 + r490254;
        double r490257 = r490255 * r490256;
        double r490258 = fma(r490253, r490254, r490257);
        double r490259 = sqrt(r490258);
        double r490260 = 2.0;
        double r490261 = r490259 * r490260;
        return r490261;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.7
Target18.9
Herbie19.7
\[\begin{array}{l} \mathbf{if}\;z \lt 7.6369500905736745 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

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. Final simplification19.7

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

Reproduce

herbie shell --seed 2019198 +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)))))