Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.5 → 0.1

Time: 4.7s

Precision: binary64

Cost: 13256

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-0.5}{x} \cdot y - x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -1e+154)
   (- (* (/ -0.5 x) y) x)
   (if (<= x 9.5e+112) (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 <= -1e+154) {
		tmp = ((-0.5 / x) * y) - x;
	} else if (x <= 9.5e+112) {
		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 <= -1e+154)
		tmp = Float64(Float64(Float64(-0.5 / x) * y) - x);
	elseif (x <= 9.5e+112)
		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, -1e+154], N[(N[(N[(-0.5 / x), $MachinePrecision] * y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 9.5e+112], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], x]]

Target

Original	21.5
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

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x} + -1 \cdot x} \]

   3. Simplified 0

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{x}{y}} - x} \]
      Proof

      [Start] 0	\[ -0.5 \cdot \frac{y}{x} + -1 \cdot x \]
      mul-1-neg [=>] 0	\[ -0.5 \cdot \frac{y}{x} + \color{blue}{\left(-x\right)} \]
      unsub-neg [=>] 0	\[ \color{blue}{-0.5 \cdot \frac{y}{x} - x} \]
      associate-*r/ [=>] 0	\[ \color{blue}{\frac{-0.5 \cdot y}{x}} - x \]
      associate-/l* [=>] 0	\[ \color{blue}{\frac{-0.5}{\frac{x}{y}}} - x \]

   4. Applied egg-rr 0

      \[\leadsto \color{blue}{\frac{-0.5}{x} \cdot y} - x \]

   if -1.00000000000000004e154 < x < 9.5000000000000008e112

   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

      [Start] 0.0	\[ \sqrt{x \cdot x + y} \]
      fma-def [=>] 0.0	\[ \sqrt{\color{blue}{\mathsf{fma}\left(x, x, y\right)}} \]

   if 9.5000000000000008e112 < x

   1. Initial program 51.2

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

   2. Taylor expanded in x around inf 0.4

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-0.5}{x} \cdot y - x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-0.5}{x} \cdot y - x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	7.2
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-31}:\\ \;\;\;\;\left(\left(1 + -0.5 \cdot \frac{y}{x}\right) + -1\right) - x\\ \mathbf{elif}\;x \leq 2.22 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 3
Error	21.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(1 + -0.5 \cdot \frac{y}{x}\right) + -1\right) - x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	20.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-232}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{x} \cdot 0.5\\ \end{array} \]

Alternative 5
Error	20.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-154}:\\ \;\;\;\;\frac{-0.5}{x} \cdot y - x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	20.9
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-311}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	42.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023027 
(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)))