Average Error: 20.0 → 20.0
Time: 4.7s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[2 \cdot \sqrt{x \cdot y + \mathsf{fma}\left(x, z, y \cdot z\right)}\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
2 \cdot \sqrt{x \cdot y + \mathsf{fma}\left(x, z, y \cdot z\right)}
double f(double x, double y, double z) {
        double r736536 = 2.0;
        double r736537 = x;
        double r736538 = y;
        double r736539 = r736537 * r736538;
        double r736540 = z;
        double r736541 = r736537 * r736540;
        double r736542 = r736539 + r736541;
        double r736543 = r736538 * r736540;
        double r736544 = r736542 + r736543;
        double r736545 = sqrt(r736544);
        double r736546 = r736536 * r736545;
        return r736546;
}
↓
double f(double x, double y, double z) {
        double r736547 = 2.0;
        double r736548 = x;
        double r736549 = y;
        double r736550 = r736548 * r736549;
        double r736551 = z;
        double r736552 = r736549 * r736551;
        double r736553 = fma(r736548, r736551, r736552);
        double r736554 = r736550 + r736553;
        double r736555 = sqrt(r736554);
        double r736556 = r736547 * r736555;
        return r736556;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
Target19.3
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. Using strategy rm

  3. Applied associate-+l+20.0

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

  4. Simplified20.0

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

  5. Final simplification20.0

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

Reproduce

herbie shell --seed 2020002 +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)))))