Linear.Quaternion:$clog from linear-1.19.1.3

Average Accuracy: 66.8% → 99.6%

Time: 7.0s

Precision: binary64

Cost: 13256

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+154)
   (- x)
   (if (<= x 3e+92) (sqrt (fma x x y)) (+ x (* 0.5 (/ y x))))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (x <= -5e+154) {
		tmp = -x;
	} else if (x <= 3e+92) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = x + (0.5 * (y / 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 <= -5e+154)
		tmp = Float64(-x);
	elseif (x <= 3e+92)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = Float64(x + Float64(0.5 * Float64(y / 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, -5e+154], (-x), If[LessEqual[x, 3e+92], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	66.8%
Target	99.1%
Herbie	99.6%

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

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000004e154

   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 -5.00000000000000004e154 < x < 3.00000000000000013e92

   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 3.00000000000000013e92 < x

   1. Initial program 27.3%

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

   2. Taylor expanded in x around inf 98.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{x} + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 2
Accuracy	89.1%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3
Accuracy	66.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{-230}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4
Accuracy	66.4%
Cost	260

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

Alternative 5
Accuracy	34.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023122 
(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)))