Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 69.4% → 99.7%

Time: 3.4s

Alternatives: 7

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: 69.4% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+154) (- x) (if (<= x 1.95e+111) (sqrt (fma x x y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+154) {
		tmp = -x;
	} else if (x <= 1.95e+111) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+154)
		tmp = Float64(-x);
	elseif (x <= 1.95e+111)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+154], (-x), If[LessEqual[x, 1.95e+111], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000007e154

   1. Initial program 6.6%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -2.00000000000000007e154 < x < 1.9499999999999999e111

   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 1.9499999999999999e111 < x

   1. Initial program 26.7%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 10^{+111}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+154) (- x) (if (<= x 1e+111) (sqrt (+ y (* x x))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+154) {
		tmp = -x;
	} else if (x <= 1e+111) {
		tmp = sqrt((y + (x * x)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+154)) then
        tmp = -x
    else if (x <= 1d+111) then
        tmp = sqrt((y + (x * x)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+154) {
		tmp = -x;
	} else if (x <= 1e+111) {
		tmp = Math.sqrt((y + (x * x)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e+154:
		tmp = -x
	elif x <= 1e+111:
		tmp = math.sqrt((y + (x * x)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+154)
		tmp = Float64(-x);
	elseif (x <= 1e+111)
		tmp = sqrt(Float64(y + Float64(x * x)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+154)
		tmp = -x;
	elseif (x <= 1e+111)
		tmp = sqrt((y + (x * x)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+154], (-x), If[LessEqual[x, 1e+111], N[Sqrt[N[(y + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000007e154

   1. Initial program 6.6%

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

   2. Taylor expanded in x around -inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 100.0%

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

   4. Simplified 100.0%

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

   if -2.00000000000000007e154 < x < 9.99999999999999957e110

   1. Initial program 100.0%

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

   if 9.99999999999999957e110 < x

   1. Initial program 26.7%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 10^{+111}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.4e-49)
   (- (* y (/ -0.5 x)) x)
   (if (<= x 1.9e-10) (sqrt y) (+ x (* 0.5 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-49) {
		tmp = (y * (-0.5 / x)) - x;
	} else if (x <= 1.9e-10) {
		tmp = sqrt(y);
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.4d-49)) then
        tmp = (y * ((-0.5d0) / x)) - x
    else if (x <= 1.9d-10) then
        tmp = sqrt(y)
    else
        tmp = x + (0.5d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-49) {
		tmp = (y * (-0.5 / x)) - x;
	} else if (x <= 1.9e-10) {
		tmp = Math.sqrt(y);
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.4e-49:
		tmp = (y * (-0.5 / x)) - x
	elif x <= 1.9e-10:
		tmp = math.sqrt(y)
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.4e-49)
		tmp = Float64(Float64(y * Float64(-0.5 / x)) - x);
	elseif (x <= 1.9e-10)
		tmp = sqrt(y);
	else
		tmp = Float64(x + Float64(0.5 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.4e-49)
		tmp = (y * (-0.5 / x)) - x;
	elseif (x <= 1.9e-10)
		tmp = sqrt(y);
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.4e-49], N[(N[(y * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.9e-10], 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 < -8.3999999999999995e-49

   1. Initial program 52.2%

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

   2. Taylor expanded in x around -inf 74.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x} + \left(-1 \cdot x + 0.125 \cdot \frac{{y}^{2}}{{x}^{3}}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 74.5%

         \[\leadsto -0.5 \cdot \frac{y}{x} + \color{blue}{\left(0.125 \cdot \frac{{y}^{2}}{{x}^{3}} + -1 \cdot x\right)} \]

      2. mul-1-neg 74.5%

         \[\leadsto -0.5 \cdot \frac{y}{x} + \left(0.125 \cdot \frac{{y}^{2}}{{x}^{3}} + \color{blue}{\left(-x\right)}\right) \]

      3. unsub-neg 74.5%

         \[\leadsto -0.5 \cdot \frac{y}{x} + \color{blue}{\left(0.125 \cdot \frac{{y}^{2}}{{x}^{3}} - x\right)} \]

      4. associate-+r- 74.5%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{y}{x} + 0.125 \cdot \frac{{y}^{2}}{{x}^{3}}\right) - x} \]

      5. unpow2 74.5%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + 0.125 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{3}}\right) - x \]

      6. associate-*r/ 74.5%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \color{blue}{\frac{0.125 \cdot \left(y \cdot y\right)}{{x}^{3}}}\right) - x \]

      7. unpow3 74.5%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \frac{0.125 \cdot \left(y \cdot y\right)}{\color{blue}{\left(x \cdot x\right) \cdot x}}\right) - x \]

      8. associate-*r* 74.5%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \frac{\color{blue}{\left(0.125 \cdot y\right) \cdot y}}{\left(x \cdot x\right) \cdot x}\right) - x \]

      9. times-frac 94.3%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \color{blue}{\frac{0.125 \cdot y}{x \cdot x} \cdot \frac{y}{x}}\right) - x \]

      10. distribute-rgt-out 94.3%

          \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(-0.5 + \frac{0.125 \cdot y}{x \cdot x}\right)} - x \]

      11. *-commutative 94.3%

          \[\leadsto \frac{y}{x} \cdot \left(-0.5 + \frac{\color{blue}{y \cdot 0.125}}{x \cdot x}\right) - x \]

   4. Simplified 94.3%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(-0.5 + \frac{y \cdot 0.125}{x \cdot x}\right) - x} \]

   5. Taylor expanded in y around 0 94.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x}} - x \]
   6. Step-by-step derivation

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.4%

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

      3. associate-*r/ 94.4%

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

   7. Simplified 94.4%

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

   if -8.3999999999999995e-49 < x < 1.8999999999999999e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 88.6%

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

   if 1.8999999999999999e-10 < x

   1. Initial program 47.8%

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

   2. Taylor expanded in x around inf 97.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4: 67.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-309) (- x) (+ x (* 0.5 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = -x;
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = -x
    else
        tmp = x + (0.5d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = -x;
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-309:
		tmp = -x
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(-x);
	else
		tmp = Float64(x + Float64(0.5 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = -x;
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-309], (-x), N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 65.2%

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

   2. Taylor expanded in x around -inf 71.4%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 71.4%

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

   4. Simplified 71.4%

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

   if -1.000000000000002e-309 < x

   1. Initial program 67.1%

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

   2. Taylor expanded in x around inf 68.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 5: 67.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-309) (- (* y (/ -0.5 x)) x) (+ x (* 0.5 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = (y * (-0.5 / x)) - x;
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = (y * ((-0.5d0) / x)) - x
    else
        tmp = x + (0.5d0 * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = (y * (-0.5 / x)) - x;
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-309:
		tmp = (y * (-0.5 / x)) - x
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(Float64(y * Float64(-0.5 / x)) - x);
	else
		tmp = Float64(x + Float64(0.5 * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = (y * (-0.5 / x)) - x;
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-309], N[(N[(y * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 65.2%

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

   2. Taylor expanded in x around -inf 55.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x} + \left(-1 \cdot x + 0.125 \cdot \frac{{y}^{2}}{{x}^{3}}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 55.3%

         \[\leadsto -0.5 \cdot \frac{y}{x} + \color{blue}{\left(0.125 \cdot \frac{{y}^{2}}{{x}^{3}} + -1 \cdot x\right)} \]

      2. mul-1-neg 55.3%

         \[\leadsto -0.5 \cdot \frac{y}{x} + \left(0.125 \cdot \frac{{y}^{2}}{{x}^{3}} + \color{blue}{\left(-x\right)}\right) \]

      3. unsub-neg 55.3%

         \[\leadsto -0.5 \cdot \frac{y}{x} + \color{blue}{\left(0.125 \cdot \frac{{y}^{2}}{{x}^{3}} - x\right)} \]

      4. associate-+r- 55.3%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{y}{x} + 0.125 \cdot \frac{{y}^{2}}{{x}^{3}}\right) - x} \]

      5. unpow2 55.3%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + 0.125 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{3}}\right) - x \]

      6. associate-*r/ 55.3%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \color{blue}{\frac{0.125 \cdot \left(y \cdot y\right)}{{x}^{3}}}\right) - x \]

      7. unpow3 55.3%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \frac{0.125 \cdot \left(y \cdot y\right)}{\color{blue}{\left(x \cdot x\right) \cdot x}}\right) - x \]

      8. associate-*r* 55.3%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \frac{\color{blue}{\left(0.125 \cdot y\right) \cdot y}}{\left(x \cdot x\right) \cdot x}\right) - x \]

      9. times-frac 70.6%

         \[\leadsto \left(-0.5 \cdot \frac{y}{x} + \color{blue}{\frac{0.125 \cdot y}{x \cdot x} \cdot \frac{y}{x}}\right) - x \]

      10. distribute-rgt-out 70.6%

          \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(-0.5 + \frac{0.125 \cdot y}{x \cdot x}\right)} - x \]

      11. *-commutative 70.6%

          \[\leadsto \frac{y}{x} \cdot \left(-0.5 + \frac{\color{blue}{y \cdot 0.125}}{x \cdot x}\right) - x \]

   4. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(-0.5 + \frac{y \cdot 0.125}{x \cdot x}\right) - x} \]

   5. Taylor expanded in y around 0 71.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x}} - x \]
   6. Step-by-step derivation

      1. *-commutative 71.7%

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

      2. associate-*l/ 71.7%

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

      3. associate-*r/ 71.7%

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

   7. Simplified 71.7%

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

   if -1.000000000000002e-309 < x

   1. Initial program 67.1%

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

   2. Taylor expanded in x around inf 68.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 6: 67.1% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1e-309) (- x) x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1d-309)) then
        tmp = -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-309) {
		tmp = -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-309:
		tmp = -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(-x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-309], (-x), x]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 65.2%

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

   2. Taylor expanded in x around -inf 71.4%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 71.4%

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

   4. Simplified 71.4%

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

   if -1.000000000000002e-309 < x

   1. Initial program 67.1%

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

   2. Taylor expanded in x around inf 68.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 34.0% accurate, 105.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 66.2%

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

2. Taylor expanded in x around inf 37.4%

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

3. Final simplification 37.4%

   \[\leadsto x \]

Developer target: 99.2% 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 2023185 
(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)))