Average Error: 0.1 → 0.1
Time: 2.7s
Precision: binary64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right) \]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)

(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma x x (* (* y y) 3.0)))
double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(x, x, ((y * y) * 3.0));
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

  2. Applied add-sqr-sqrt_binary640.2

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

  3. Applied fma-def_binary640.2

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

  4. Applied add-sqr-sqrt_binary640.5

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  7. Taylor expanded in x around 0 0.4

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

  8. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]

  9. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right) \]

Reproduce

herbie shell --seed 2022125 
(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)))