Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.7 → 0.1

Time: 1.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \mathbf{if}\;x \leq -5.298076076378501 \cdot 10^{+152}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;x \leq 5.945564708482161 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.5 (/ y x) x)))
   (if (<= x -5.298076076378501e+152)
     (- t_0)
     (if (<= x 5.945564708482161e+108) (sqrt (fma x x y)) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = fma(0.5, (y / x), x);
	double tmp;
	if (x <= -5.298076076378501e+152) {
		tmp = -t_0;
	} else if (x <= 5.945564708482161e+108) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

↓

function code(x, y)
	t_0 = fma(0.5, Float64(y / x), x)
	tmp = 0.0
	if (x <= -5.298076076378501e+152)
		tmp = Float64(-t_0);
	elseif (x <= 5.945564708482161e+108)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(0.5 * N[(y / x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -5.298076076378501e+152], (-t$95$0), If[LessEqual[x, 5.945564708482161e+108], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Target

Original	21.7
Target	0.5
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if x < -5.29807607637850107e152

   1. Initial program 63.5

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

   2. Simplified 63.5

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

   3. Taylor expanded in x around -inf 0

      \[\leadsto \color{blue}{-\left(0.5 \cdot \frac{y}{x} + x\right)} \]

   4. Simplified 0

      \[\leadsto \color{blue}{-\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)} \]

   if -5.29807607637850107e152 < x < 5.94556470848216066e108

   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 5.94556470848216066e108 < x

   1. Initial program 50.0

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

   2. Simplified 50.0

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

   3. Taylor expanded in x around inf 0.5

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

   4. Simplified 0.5

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.298076076378501 \cdot 10^{+152}:\\ \;\;\;\;-\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \mathbf{elif}\;x \leq 5.945564708482161 \cdot 10^{+108}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{y}{x}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))