Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.1 → 1.2

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -8.33118696796323 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 225767106306.02066:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -8.33118696796323e+153)
   (- x)
   (if (<= x 225767106306.02066) (sqrt (fma x x y)) x)))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (x <= -8.33118696796323e+153) {
		tmp = -x;
	} else if (x <= 225767106306.02066) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = 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 <= -8.33118696796323e+153)
		tmp = Float64(-x);
	elseif (x <= 225767106306.02066)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = 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, -8.33118696796323e+153], (-x), If[LessEqual[x, 225767106306.02066], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], x]]

Error

Bits error versus x

Bits error versus y

Target

Original	21.1
Target	0.6
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if x < -8.33118696796323e153

   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 -8.33118696796323e153 < x < 225767106306.02066

   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 225767106306.02066 < x

   1. Initial program 35.3

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

   2. Simplified 35.2

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

   3. Taylor expanded in x around inf 4.1

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.33118696796323 \cdot 10^{+153}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 225767106306.02066:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))