Average Error: 19.8 → 19.8
Time: 16.8s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[\sqrt{\mathsf{fma}\left(y, z, x \cdot y + x \cdot z\right)} \cdot 2\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
\sqrt{\mathsf{fma}\left(y, z, x \cdot y + x \cdot z\right)} \cdot 2
double f(double x, double y, double z) {
        double r456558 = 2.0;
        double r456559 = x;
        double r456560 = y;
        double r456561 = r456559 * r456560;
        double r456562 = z;
        double r456563 = r456559 * r456562;
        double r456564 = r456561 + r456563;
        double r456565 = r456560 * r456562;
        double r456566 = r456564 + r456565;
        double r456567 = sqrt(r456566);
        double r456568 = r456558 * r456567;
        return r456568;
}
↓
double f(double x, double y, double z) {
        double r456569 = y;
        double r456570 = z;
        double r456571 = x;
        double r456572 = r456571 * r456569;
        double r456573 = r456571 * r456570;
        double r456574 = r456572 + r456573;
        double r456575 = fma(r456569, r456570, r456574);
        double r456576 = sqrt(r456575);
        double r456577 = 2.0;
        double r456578 = r456576 * r456577;
        return r456578;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.8
Target19.2
Herbie19.8
\[\begin{array}{l} \mathbf{if}\;z \lt 7.636950090573674520215292914121377944071 \cdot 10^{176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 19.8

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

  2. Simplified19.8

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

  4. Applied distribute-lft-in19.8

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

  5. Final simplification19.8

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

Reproduce

herbie shell --seed 2019323 +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)))))