Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 69.7% → 99.2%

Time: 3.1s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{x \cdot x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return sqrt(Float64(Float64(x * x) + y))
end

MATLAB

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

Wolfram

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

Initial Program: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot x + y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return sqrt(Float64(Float64(x * x) + y))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e+198) (sqrt (+ (* x x) y)) x))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e+198) {
		tmp = sqrt(((x * x) + y));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 5d+198) then
        tmp = sqrt(((x * x) + y))
    else
        tmp = x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e+198) {
		tmp = Math.sqrt(((x * x) + y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	tmp = 0
	if (x * x) <= 5e+198:
		tmp = math.sqrt(((x * x) + y))
	else:
		tmp = x
	return tmp

Julia

x = abs(x)
function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e+198)
		tmp = sqrt(Float64(Float64(x * x) + y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 5e+198)
		tmp = sqrt(((x * x) + y));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+198], N[Sqrt[N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000049e198

   1. Initial program 100.0%

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

   if 5.00000000000000049e198 < (*.f64 x x)

   1. Initial program 22.8%

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

   2. Taylor expanded in x around inf 50.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+198}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= x 1.22e-79)
   (sqrt y)
   (if (<= x 2.5e-64) x (if (<= x 1.2e-29) (sqrt y) (+ x (* 0.5 (/ y x)))))))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if (x <= 1.22e-79) {
		tmp = sqrt(y);
	} else if (x <= 2.5e-64) {
		tmp = x;
	} else if (x <= 1.2e-29) {
		tmp = sqrt(y);
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.22d-79) then
        tmp = sqrt(y)
    else if (x <= 2.5d-64) then
        tmp = x
    else if (x <= 1.2d-29) then
        tmp = sqrt(y)
    else
        tmp = x + (0.5d0 * (y / x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	double tmp;
	if (x <= 1.22e-79) {
		tmp = Math.sqrt(y);
	} else if (x <= 2.5e-64) {
		tmp = x;
	} else if (x <= 1.2e-29) {
		tmp = Math.sqrt(y);
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	tmp = 0
	if x <= 1.22e-79:
		tmp = math.sqrt(y)
	elif x <= 2.5e-64:
		tmp = x
	elif x <= 1.2e-29:
		tmp = math.sqrt(y)
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

x = abs(x)
function code(x, y)
	tmp = 0.0
	if (x <= 1.22e-79)
		tmp = sqrt(y);
	elseif (x <= 2.5e-64)
		tmp = x;
	elseif (x <= 1.2e-29)
		tmp = sqrt(y);
	else
		tmp = Float64(x + Float64(0.5 * Float64(y / x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.22e-79)
		tmp = sqrt(y);
	elseif (x <= 2.5e-64)
		tmp = x;
	elseif (x <= 1.2e-29)
		tmp = sqrt(y);
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := If[LessEqual[x, 1.22e-79], N[Sqrt[y], $MachinePrecision], If[LessEqual[x, 2.5e-64], x, If[LessEqual[x, 1.2e-29], N[Sqrt[y], $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.22e-79 or 2.50000000000000017e-64 < x < 1.19999999999999996e-29

   1. Initial program 76.3%

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

   2. Taylor expanded in x around 0 49.8%

      \[\leadsto \color{blue}{\sqrt{y}} \]

   if 1.22e-79 < x < 2.50000000000000017e-64

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   if 1.19999999999999996e-29 < x

   1. Initial program 48.2%

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

   2. Taylor expanded in x around inf 98.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 68.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x + 0.5 \cdot \frac{y}{x} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y) :precision binary64 (+ x (* 0.5 (/ y x))))

C

x = abs(x);
double code(double x, double y) {
	return x + (0.5 * (y / x));
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (0.5d0 * (y / x))
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	return x + (0.5 * (y / x));
}

Python

x = abs(x)
def code(x, y):
	return x + (0.5 * (y / x))

Julia

x = abs(x)
function code(x, y)
	return Float64(x + Float64(0.5 * Float64(y / x)))
end

MATLAB

x = abs(x)
function tmp = code(x, y)
	tmp = x + (0.5 * (y / x));
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

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

2. Taylor expanded in x around inf 31.6%

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

3. Final simplification 31.6%

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

Alternative 4: 67.7% accurate, 105.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y) :precision binary64 x)

C

x = abs(x);
double code(double x, double y) {
	return x;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	return x;
}

Python

x = abs(x)
def code(x, y):
	return x

Julia

x = abs(x)
function code(x, y)
	return x
end

MATLAB

x = abs(x)
function tmp = code(x, y)
	tmp = x;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := x

Derivation

1. Initial program 68.3%

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

2. Taylor expanded in x around inf 31.9%

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

3. Final simplification 31.9%

   \[\leadsto x \]

Developer target: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{y}{x} + x\\ \mathbf{if}\;x < -1.5097698010472593 \cdot 10^{+153}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;x < 5.582399551122541 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* 0.5 (/ y x)) x)))
   (if (< x -1.5097698010472593e+153)
     (- t_0)
     (if (< x 5.582399551122541e+57) (sqrt (+ (* x x) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = (0.5 * (y / x)) + x;
	double tmp;
	if (x < -1.5097698010472593e+153) {
		tmp = -t_0;
	} else if (x < 5.582399551122541e+57) {
		tmp = sqrt(((x * x) + y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (y / x)) + x
    if (x < (-1.5097698010472593d+153)) then
        tmp = -t_0
    else if (x < 5.582399551122541d+57) then
        tmp = sqrt(((x * x) + y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (0.5 * (y / x)) + x;
	double tmp;
	if (x < -1.5097698010472593e+153) {
		tmp = -t_0;
	} else if (x < 5.582399551122541e+57) {
		tmp = Math.sqrt(((x * x) + y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (0.5 * (y / x)) + x
	tmp = 0
	if x < -1.5097698010472593e+153:
		tmp = -t_0
	elif x < 5.582399551122541e+57:
		tmp = math.sqrt(((x * x) + y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(0.5 * Float64(y / x)) + x)
	tmp = 0.0
	if (x < -1.5097698010472593e+153)
		tmp = Float64(-t_0);
	elseif (x < 5.582399551122541e+57)
		tmp = sqrt(Float64(Float64(x * x) + y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (0.5 * (y / x)) + x;
	tmp = 0.0;
	if (x < -1.5097698010472593e+153)
		tmp = -t_0;
	elseif (x < 5.582399551122541e+57)
		tmp = sqrt(((x * x) + y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[Less[x, -1.5097698010472593e+153], (-t$95$0), If[Less[x, 5.582399551122541e+57], N[Sqrt[N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024024 
(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)))