Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 68.2% → 99.9%

Time: 2.2s

Alternatives: 5

Speedup: 105.0×

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: 68.2% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e+257) (sqrt (fma x x y)) x))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e+257) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 4e+257)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e+257], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000012e257

   1. Initial program 100.0%

      \[\sqrt{x \cdot x + y} \]
   2. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, x, y\right)}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(x, x, y\right)}} \]

   if 4.00000000000000012e257 < (*.f64 x x)

   1. Initial program 16.2%

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

   2. Taylor expanded in x around inf 47.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e+257) (sqrt (+ (* x x) y)) x))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e+257) {
		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) <= 4d+257) 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) <= 4e+257) {
		tmp = Math.sqrt(((x * x) + y));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	tmp = 0
	if (x * x) <= 4e+257:
		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) <= 4e+257)
		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) <= 4e+257)
		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], 4e+257], N[Sqrt[N[(N[(x * x), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000012e257

   1. Initial program 100.0%

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

   if 4.00000000000000012e257 < (*.f64 x x)

   1. Initial program 16.2%

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

   2. Taylor expanded in x around inf 47.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{+257}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-54} \lor \neg \left(x \leq 7 \cdot 10^{-37}\right) \land x \leq 1.4 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= x 2e-54) (and (not (<= x 7e-37)) (<= x 1.4e-17)))
   (sqrt y)
   (+ x (* 0.5 (/ y x)))))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if ((x <= 2e-54) || (!(x <= 7e-37) && (x <= 1.4e-17))) {
		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 <= 2d-54) .or. (.not. (x <= 7d-37)) .and. (x <= 1.4d-17)) 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 <= 2e-54) || (!(x <= 7e-37) && (x <= 1.4e-17))) {
		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 <= 2e-54) or (not (x <= 7e-37) and (x <= 1.4e-17)):
		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 <= 2e-54) || (!(x <= 7e-37) && (x <= 1.4e-17)))
		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 <= 2e-54) || (~((x <= 7e-37)) && (x <= 1.4e-17)))
		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[Or[LessEqual[x, 2e-54], And[N[Not[LessEqual[x, 7e-37]], $MachinePrecision], LessEqual[x, 1.4e-17]]], N[Sqrt[y], $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-54 or 7.0000000000000003e-37 < x < 1.3999999999999999e-17

   1. Initial program 73.1%

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

   2. Taylor expanded in x around 0 47.3%

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

   if 2.0000000000000001e-54 < x < 7.0000000000000003e-37 or 1.3999999999999999e-17 < x

   1. Initial program 53.9%

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

   2. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{x} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-54} \lor \neg \left(x \leq 7 \cdot 10^{-37}\right) \land x \leq 1.4 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4: 68.9% 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 66.3%

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

2. Taylor expanded in x around inf 36.7%

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

3. Final simplification 36.7%

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

Alternative 5: 68.5% 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 66.3%

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

2. Taylor expanded in x around inf 36.7%

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

3. Final simplification 36.7%

   \[\leadsto x \]

Developer target: 98.7% 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 2023240 
(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)))