Linear.Quaternion:$clog from linear-1.19.1.3

Average Accuracy: 66.4% → 99.7%

Time: 4.0s

Precision: binary64

Cost: 13256

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+156}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+156) (- x) (if (<= x 2e+148) (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 <= -2e+156) {
		tmp = -x;
	} else if (x <= 2e+148) {
		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 <= -2e+156)
		tmp = Float64(-x);
	elseif (x <= 2e+148)
		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, -2e+156], (-x), If[LessEqual[x, 2e+148], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], x]]

Target

Original	66.4%
Target	99.3%
Herbie	99.7%

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

Derivation

1. Split input into 3 regimes

2. if x < -2e156

   1. Initial program 0.0%

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

   2. Taylor expanded in x around -inf 100.0%

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

   3. Simplified 100.0%

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

      [Start] 100.0	\[ -1 \cdot x \]
      mul-1-neg [=>] 100.0	\[ \color{blue}{-x} \]

   if -2e156 < x < 2.0000000000000001e148

   1. Initial program 99.5%

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

   2. Simplified 99.5%

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

      [Start] 99.5	\[ \sqrt{x \cdot x + y} \]
      fma-def [=>] 99.5	\[ \sqrt{\color{blue}{\mathsf{fma}\left(x, x, y\right)}} \]

   if 2.0000000000000001e148 < x

   1. Initial program 3.0%

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

   2. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+156}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	88.4%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{y}{x} \cdot \left(-0.5 + \frac{y}{x} \cdot \frac{0.125}{x}\right) - x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 3
Accuracy	68.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-242}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 4
Accuracy	68.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-180}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	68.0%
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	34.7%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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)))