Average Error: 19.8 → 19.8
Time: 9.0s
Precision: binary64
\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]
↓
\[2 \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.5} \]
2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z}
↓
2 \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.5}

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))
↓
(FPCore (x y z) :precision binary64 (* 2.0 (pow (fma y z (* x (+ y z))) 0.5)))
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 * pow(fma(y, z, (x * (y + z))), 0.5);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original19.8
Target19.1
Herbie19.8
\[\begin{array}{l} \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array} \]

Derivation

  1. Initial program 19.8

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

  2. Applied add-sqr-sqrt_binary6420.0

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

  3. Simplified20.0

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

  4. Simplified20.0

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

  5. Applied add-sqr-sqrt_binary6420.2

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

  6. Applied pow1/2_binary6420.2

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}}} \cdot \sqrt{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.5}}}}\right)\right) \]

  7. Applied sqrt-pow1_binary6420.1

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}}} \cdot \sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}}\right)\right) \]

  8. Applied sqrt-pow1_binary6420.1

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}}} \cdot \color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{\frac{0.5}{2}}{2}\right)}}\right)\right) \]

  9. Applied pow1/2_binary6420.1

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \left(\sqrt{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.5}}}} \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{\frac{0.5}{2}}{2}\right)}\right)\right) \]

  10. Applied sqrt-pow1_binary6420.1

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{\frac{0.5}{2}}{2}\right)}\right)\right) \]

  11. Applied sqrt-pow1_binary6420.1

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{\frac{0.5}{2}}{2}\right)}} \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{\frac{0.5}{2}}{2}\right)}\right)\right) \]

  12. Applied pow-sqr_binary6420.0

    \[\leadsto 2 \cdot \left(\sqrt{\sqrt{\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)}} \cdot \color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(2 \cdot \frac{\frac{0.5}{2}}{2}\right)}}\right) \]

  13. Applied pow1/2_binary6420.0

    \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.5}}} \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(2 \cdot \frac{\frac{0.5}{2}}{2}\right)}\right) \]

  14. Applied sqrt-pow1_binary6420.0

    \[\leadsto 2 \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}} \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(2 \cdot \frac{\frac{0.5}{2}}{2}\right)}\right) \]

  15. Applied pow-prod-up_binary6419.8

    \[\leadsto 2 \cdot \color{blue}{{\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{\left(\frac{0.5}{2} + 2 \cdot \frac{\frac{0.5}{2}}{2}\right)}} \]

  16. Final simplification19.8

    \[\leadsto 2 \cdot {\left(\mathsf{fma}\left(y, z, x \cdot \left(y + z\right)\right)\right)}^{0.5} \]

Reproduce

herbie shell --seed 2022067 
(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)))))