Linear.Quaternion:$clog from linear-1.19.1.3

Average Accuracy: 66.1% → 99.4%

Time: 2.9s

Precision: binary64

Cost: 13256

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+154) (- x) (if (<= x 2e+78) (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+154) {
		tmp = -x;
	} else if (x <= 2e+78) {
		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+154)
		tmp = Float64(-x);
	elseif (x <= 2e+78)
		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+154], (-x), If[LessEqual[x, 2e+78], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], x]]

Target

Original	66.1%
Target	99.1%
Herbie	99.4%

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

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000007e154

   1. Initial program 0.0%

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

   2. Taylor expanded in x around -inf 99.9%

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

   3. Simplified 99.9%

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

      [Start] 99.9	\[ -1 \cdot x \]
      mul-1-neg [=>] 99.9	\[ \color{blue}{-x} \]

   if -2.00000000000000007e154 < x < 2.00000000000000002e78

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \sqrt{x \cdot x + y} \]
      fma-def [=>] 99.9	\[ \sqrt{\color{blue}{\mathsf{fma}\left(x, x, y\right)}} \]

   if 2.00000000000000002e78 < x

   1. Initial program 30.4%

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

   2. Taylor expanded in x around inf 97.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+78}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	89.0%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-62}:\\ \;\;\;\;\frac{y}{x} \cdot \left(-0.5 + \frac{y}{x} \cdot \frac{0.125}{x}\right) - x\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 3
Accuracy	68.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.48 \cdot 10^{-201}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 4
Accuracy	68.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 5
Accuracy	68.1%
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	35.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023129 
(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)))