Average Error: 20.4 → 20.4
Time: 7.6s
Precision: binary64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
↓
\[2 \cdot \sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)} \]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
2 \cdot \sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))
↓
(FPCore (x y z) :precision binary64 (* 2.0 (sqrt (fma x (+ y z) (* y z)))))
double code(double x, double y, double z) {
	return 2.0 * sqrt(((x * y) + (x * z)) + (y * z));
}
↓
double code(double x, double y, double z) {
	return 2.0 * sqrt(fma(x, (y + z), (y * z)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original20.4
Target19.8
Herbie20.4
\[\begin{array}{l} \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array} \]

Derivation

  1. Initial program 20.4

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

  2. Simplified20.4

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

  3. Applied add-sqr-sqrt_binary6420.7

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

  4. Applied add-sqr-sqrt_binary6420.8

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

  5. Applied associate-*l*_binary6420.8

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

  6. Applied *-un-lft-identity_binary6420.8

    \[\leadsto 2 \cdot \left(\color{blue}{\left(1 \cdot \sqrt{\sqrt{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}}\right)} \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}\right)\right) \]

  7. Applied associate-*l*_binary6420.8

    \[\leadsto 2 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}} \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}\right)\right)\right)} \]

  8. Simplified20.4

    \[\leadsto 2 \cdot \left(1 \cdot \color{blue}{\sqrt{\mathsf{fma}\left(x, y + z, y \cdot z\right)}}\right) \]

  9. Final simplification20.4

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

Reproduce

herbie shell --seed 2022077 
(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) (* 2.0 (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)))))