Average Error: 25.0 → 1.2
Time: 3.1s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z}\]
↓
\[\left(\left|\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}\right| \cdot x\right) \cdot \sqrt{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}\]
x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\left(\left|\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}\right| \cdot x\right) \cdot \sqrt{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}
(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (*
  (* (fabs (* (cbrt (+ y z)) (cbrt (- y z)))) x)
  (sqrt (* (cbrt (+ y z)) (cbrt (- y z))))))
double code(double x, double y, double z) {
	return x * sqrt((y * y) - (z * z));
}
↓
double code(double x, double y, double z) {
	return (fabs(cbrt(y + z) * cbrt(y - z)) * x) * sqrt(cbrt(y + z) * cbrt(y - z));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.0
Target0.5
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array}\]

Derivation

  1. Initial program 25.0

    \[x \cdot \sqrt{y \cdot y - z \cdot z}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt_binary6425.5

    \[\leadsto x \cdot \sqrt{\color{blue}{\left(\sqrt[3]{y \cdot y - z \cdot z} \cdot \sqrt[3]{y \cdot y - z \cdot z}\right) \cdot \sqrt[3]{y \cdot y - z \cdot z}}}\]

  4. Applied sqrt-prod_binary6425.5

    \[\leadsto x \cdot \color{blue}{\left(\sqrt{\sqrt[3]{y \cdot y - z \cdot z} \cdot \sqrt[3]{y \cdot y - z \cdot z}} \cdot \sqrt{\sqrt[3]{y \cdot y - z \cdot z}}\right)}\]

  5. Applied associate-*r*_binary6425.5

    \[\leadsto \color{blue}{\left(x \cdot \sqrt{\sqrt[3]{y \cdot y - z \cdot z} \cdot \sqrt[3]{y \cdot y - z \cdot z}}\right) \cdot \sqrt{\sqrt[3]{y \cdot y - z \cdot z}}}\]

  6. Simplified25.5

    \[\leadsto \color{blue}{\left(\left|\sqrt[3]{y \cdot y - z \cdot z}\right| \cdot x\right)} \cdot \sqrt{\sqrt[3]{y \cdot y - z \cdot z}}\]
  7. Using strategy rm

  8. Applied difference-of-squares_binary6425.5

    \[\leadsto \left(\left|\sqrt[3]{y \cdot y - z \cdot z}\right| \cdot x\right) \cdot \sqrt{\sqrt[3]{\color{blue}{\left(y + z\right) \cdot \left(y - z\right)}}}\]

  9. Applied cbrt-prod_binary6425.5

    \[\leadsto \left(\left|\sqrt[3]{y \cdot y - z \cdot z}\right| \cdot x\right) \cdot \sqrt{\color{blue}{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}}\]
  10. Using strategy rm

  11. Applied difference-of-squares_binary6425.5

    \[\leadsto \left(\left|\sqrt[3]{\color{blue}{\left(y + z\right) \cdot \left(y - z\right)}}\right| \cdot x\right) \cdot \sqrt{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}\]

  12. Applied cbrt-prod_binary641.2

    \[\leadsto \left(\left|\color{blue}{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}\right| \cdot x\right) \cdot \sqrt{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}\]

  13. Final simplification1.2

    \[\leadsto \left(\left|\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}\right| \cdot x\right) \cdot \sqrt{\sqrt[3]{y + z} \cdot \sqrt[3]{y - z}}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))