Linear.Quaternion:$clog from linear-1.19.1.3

Average Error: 21.2 → 0.7

Time: 6.0s

Precision: binary64

Cost: 6984

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5720483880855185 \cdot 10^{+155}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5720483880855185e+155)
   (- x)
   (if (<= x 3.4e+41) (sqrt (+ (* 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 <= -1.5720483880855185e+155) {
		tmp = -x;
	} else if (x <= 3.4e+41) {
		tmp = sqrt(((x * x) + y));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(((x * x) + y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.5720483880855185d+155)) then
        tmp = -x
    else if (x <= 3.4d+41) then
        tmp = sqrt(((x * x) + y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(((x * x) + y));
}

↓

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.5720483880855185e+155) {
		tmp = -x;
	} else if (x <= 3.4e+41) {
		tmp = Math.sqrt(((x * x) + y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	return math.sqrt(((x * x) + y))

↓

def code(x, y):
	tmp = 0
	if x <= -1.5720483880855185e+155:
		tmp = -x
	elif x <= 3.4e+41:
		tmp = math.sqrt(((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 <= -1.5720483880855185e+155)
		tmp = Float64(-x);
	elseif (x <= 3.4e+41)
		tmp = sqrt(Float64(Float64(x * x) + y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(((x * x) + y));
end

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.5720483880855185e+155)
		tmp = -x;
	elseif (x <= 3.4e+41)
		tmp = sqrt(((x * x) + y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[Sqrt[N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[x, -1.5720483880855185e+155], (-x), If[LessEqual[x, 3.4e+41], N[Sqrt[N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], x]]

Target

Original	21.2
Target	0.5
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if x < -1.5720483880855185e155

   1. Initial program 64.0

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

   2. Taylor expanded in x around -inf 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{-x} \]
      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 -1.5720483880855185e155 < x < 3.39999999999999998e41

   1. Initial program 0.1

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

   if 3.39999999999999998e41 < x

   1. Initial program 38.4

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

   2. Simplified 38.4

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}} \]
      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))): 1 points increase in error, 1 points decrease in error

   3. Taylor expanded in x around inf 2.3

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

   4. Applied egg-rr 2.3

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

   5. Taylor expanded in x around inf 2.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5720483880855185 \cdot 10^{+155}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-40}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 70:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 2
Error	20.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-224}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 3
Error	20.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-303}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 4
Error	20.3
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	42.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022297 
(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)))