Linear.Quaternion:$c/ from linear-1.19.1.3, E

Average Error: 0.1 → 0.1

Time: 2.9s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(y, y + y, {\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}\right) \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))

↓

(FPCore (x y) :precision binary64 (fma y (+ y y) (pow (hypot x y) 2.0)))

C

double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}

↓

double code(double x, double y) {
	return fma(y, (y + y), pow(hypot(x, y), 2.0));
}

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) + Float64(y * y)) + Float64(y * y))
end

↓

function code(x, y)
	return fma(y, Float64(y + y), (hypot(x, y) ^ 2.0))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(y * N[(y + y), $MachinePrecision] + N[Power[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Target

Original	0.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022133 
(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)))