Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.8 → 0.1

Time: 3.6s

Precision: binary64

Cost: 13256

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+154)
   (- x)
   (if (<= x 2.2e+97) (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 <= -1e+154) {
		tmp = -x;
	} else if (x <= 2.2e+97) {
		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 <= -1e+154)
		tmp = Float64(-x);
	elseif (x <= 2.2e+97)
		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, -1e+154], (-x), If[LessEqual[x, 2.2e+97], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	21.8
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 < -1.00000000000000004e154

   1. Initial program 64.0

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

   2. Taylor expanded in x around -inf 0

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

   3. Simplified 0

      \[\leadsto \color{blue}{-x} \]
      Proof
      (neg.f64 x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)): 0 points increase in error, 0 points decrease in error

   if -1.00000000000000004e154 < x < 2.2000000000000001e97

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}} \]
      Proof
      (sqrt.f64 (fma.f64 x x y)): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x x) y))): 2 points increase in error, 0 points decrease in error

   if 2.2000000000000001e97 < x

   1. Initial program 47.9

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

   2. Taylor expanded in x around inf 0.6

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 2
Error	6.9
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-81}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3
Error	20.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-253}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4
Error	20.3
Cost	580

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

Alternative 5
Error	20.4
Cost	260

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

Alternative 6
Error	41.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(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)))