Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.5 → 0.0

Time: 1.4s

Precision: binary64

Math

\[\sqrt{x \cdot x + y} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -8.458628286260894 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.2426380335745907 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -8.458628286260894e+153)
   (- x)
   (if (<= x 2.2426380335745907e+152) (sqrt (fma x x y)) (fma 0.5 (/ y x) x))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (x <= -8.458628286260894e+153) {
		tmp = -x;
	} else if (x <= 2.2426380335745907e+152) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = fma(0.5, (y / x), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (x <= -8.458628286260894e+153)
		tmp = Float64(-x);
	elseif (x <= 2.2426380335745907e+152)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = fma(0.5, Float64(y / x), x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := If[LessEqual[x, -8.458628286260894e+153], (-x), If[LessEqual[x, 2.2426380335745907e+152], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(y / x), $MachinePrecision] + x), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Target

Original	21.5
Target	0.6
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-\left(0.5 \cdot \frac{y}{x} + x\right)\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{x} + x\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -8.45862828626089413e153

   1. Initial program 63.9

      \[\sqrt{x \cdot x + y} \]

   2. Simplified 63.9

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

   3. Taylor expanded in x around -inf 0.0

      \[\leadsto \color{blue}{-1 \cdot x} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{-x} \]

   if -8.45862828626089413e153 < x < 2.2426380335745907e152

   1. Initial program 0.0

      \[\sqrt{x \cdot x + y} \]

   2. Simplified 0.0

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

   if 2.2426380335745907e152 < x

   1. Initial program 63.2

      \[\sqrt{x \cdot x + y} \]

   2. Simplified 63.2

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

   3. Taylor expanded in x around inf 0

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{x} + x} \]

   4. Simplified 0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.458628286260894 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.2426380335745907 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< x -1.5097698010472593e+153) (- (+ (* 0.5 (/ y x)) x)) (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) (+ (* 0.5 (/ y x)) x)))

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