Average Error: 0.1 → 0.2
Time: 2.5s
Precision: binary64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
↓
\[\sqrt{x \cdot x + y \cdot \left(y + y\right)} \cdot \sqrt{x \cdot x + y \cdot \left(y + y\right)} + y \cdot y\]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\sqrt{x \cdot x + y \cdot \left(y + y\right)} \cdot \sqrt{x \cdot x + y \cdot \left(y + y\right)} + y \cdot y
(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))
↓
(FPCore (x y)
 :precision binary64
 (+
  (* (sqrt (+ (* x x) (* y (+ y y)))) (sqrt (+ (* x x) (* y (+ y y)))))
  (* y y)))
double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}
↓
double code(double x, double y) {
	return (sqrt((x * x) + (y * (y + y))) * sqrt((x * x) + (y * (y + y)))) + (y * y);
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.1
Target0.1
Herbie0.2
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right)\]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt_binary640.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020219 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :herbie-target
  (+ (* x x) (* y (+ y (+ y y))))

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))