Average Error: 20.0 → 20.0
Time: 12.5s
Precision: 64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}\]
↓
\[2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}\]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
2 \cdot \sqrt{\mathsf{fma}\left(z, x + y, y \cdot x\right)}
double f(double x, double y, double z) {
        double r26282749 = 2.0;
        double r26282750 = x;
        double r26282751 = y;
        double r26282752 = r26282750 * r26282751;
        double r26282753 = z;
        double r26282754 = r26282750 * r26282753;
        double r26282755 = r26282752 + r26282754;
        double r26282756 = r26282751 * r26282753;
        double r26282757 = r26282755 + r26282756;
        double r26282758 = sqrt(r26282757);
        double r26282759 = r26282749 * r26282758;
        return r26282759;
}
↓
double f(double x, double y, double z) {
        double r26282760 = 2.0;
        double r26282761 = z;
        double r26282762 = x;
        double r26282763 = y;
        double r26282764 = r26282762 + r26282763;
        double r26282765 = r26282763 * r26282762;
        double r26282766 = fma(r26282761, r26282764, r26282765);
        double r26282767 = sqrt(r26282766);
        double r26282768 = r26282760 * r26282767;
        return r26282768;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.0
Target19.1
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. Taylor expanded around 0 20.0

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

  4. Simplified20.0

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

  5. Final simplification20.0

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

Reproduce

herbie shell --seed 2019171 +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)))))