Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 68.4% → 99.8%

Time: 3.5s

Alternatives: 8

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: 68.4% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+154)
   (- x)
   (if (<= x 2.8e+102) (sqrt (fma x x y)) (+ x (* 0.5 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+154) {
		tmp = -x;
	} else if (x <= 2.8e+102) {
		tmp = sqrt(fma(x, x, y));
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+154)
		tmp = Float64(-x);
	elseif (x <= 2.8e+102)
		tmp = sqrt(fma(x, x, y));
	else
		tmp = Float64(x + Float64(0.5 * Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+154], (-x), If[LessEqual[x, 2.8e+102], N[Sqrt[N[(x * x + y), $MachinePrecision]], $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 < 2.80000000000000018e102

   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 2.80000000000000018e102 < x

   1. Initial program 29.1%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e+154)
   (- x)
   (if (<= x 2e+102) (sqrt (+ y (* x x))) (+ x (* 0.5 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e+154) {
		tmp = -x;
	} else if (x <= 2e+102) {
		tmp = sqrt((y + (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 <= (-2d+154)) then
        tmp = -x
    else if (x <= 2d+102) then
        tmp = sqrt((y + (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 <= -2e+154) {
		tmp = -x;
	} else if (x <= 2e+102) {
		tmp = Math.sqrt((y + (x * x)));
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e+154:
		tmp = -x
	elif x <= 2e+102:
		tmp = math.sqrt((y + (x * x)))
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e+154)
		tmp = Float64(-x);
	elseif (x <= 2e+102)
		tmp = sqrt(Float64(y + Float64(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 <= -2e+154)
		tmp = -x;
	elseif (x <= 2e+102)
		tmp = sqrt((y + (x * x)));
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e+154], (-x), If[LessEqual[x, 2e+102], N[Sqrt[N[(y + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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 < 1.99999999999999995e102

   1. Initial program 100.0%

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

   if 1.99999999999999995e102 < x

   1. Initial program 29.1%

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

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 3: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e-58)
   (- (* y (/ -0.5 x)) x)
   (if (<= x 3.2e-43) (sqrt y) (+ x (* 0.5 (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e-58) {
		tmp = (y * (-0.5 / x)) - x;
	} else if (x <= 3.2e-43) {
		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 <= (-1.5d-58)) then
        tmp = (y * ((-0.5d0) / x)) - x
    else if (x <= 3.2d-43) 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 <= -1.5e-58) {
		tmp = (y * (-0.5 / x)) - x;
	} else if (x <= 3.2e-43) {
		tmp = Math.sqrt(y);
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e-58:
		tmp = (y * (-0.5 / x)) - x
	elif x <= 3.2e-43:
		tmp = math.sqrt(y)
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.5e-58)
		tmp = Float64(Float64(y * Float64(-0.5 / x)) - x);
	elseif (x <= 3.2e-43)
		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 <= -1.5e-58)
		tmp = (y * (-0.5 / x)) - x;
	elseif (x <= 3.2e-43)
		tmp = sqrt(y);
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.5e-58], N[(N[(y * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 3.2e-43], 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 < -1.50000000000000004e-58

   1. Initial program 61.1%

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

   2. Taylor expanded in x around -inf 90.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x} + -1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

      3. *-commutative 90.2%

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

   4. Simplified 90.2%

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

   5. Taylor expanded in y around 0 90.2%

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

      1. associate-*r/ 90.2%

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

      2. *-commutative 90.2%

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

      3. associate-*r/ 90.2%

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

   7. Simplified 90.2%

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

   if -1.50000000000000004e-58 < x < 3.19999999999999985e-43

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.6%

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

   if 3.19999999999999985e-43 < x

   1. Initial program 57.1%

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

   2. Taylor expanded in x around inf 88.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 4: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-310) (- x) (+ x (* 0.5 (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = -x;
	} else {
		tmp = x + (0.5 * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = -x
	else:
		tmp = x + (0.5 * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		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 <= -5e-310)
		tmp = -x;
	else
		tmp = x + (0.5 * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], (-x), N[(x + N[(0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in x around -inf 65.9%

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

      1. mul-1-neg 65.9%

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

   4. Simplified 65.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 66.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 0.5 \cdot \frac{y}{x}\\ \end{array} \]

Alternative 5: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3e-309) (- x) (+ x (* y (/ 0.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3e-309) {
		tmp = -x;
	} else {
		tmp = x + (y * (0.5 / 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 <= 3d-309) then
        tmp = -x
    else
        tmp = x + (y * (0.5d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3e-309) {
		tmp = -x;
	} else {
		tmp = x + (y * (0.5 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3e-309:
		tmp = -x
	else:
		tmp = x + (y * (0.5 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3e-309)
		tmp = Float64(-x);
	else
		tmp = Float64(x + Float64(y * Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3e-309)
		tmp = -x;
	else
		tmp = x + (y * (0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3e-309], (-x), N[(x + N[(y * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.000000000000001e-309

   1. Initial program 72.1%

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

   2. Taylor expanded in x around -inf 65.9%

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

      1. mul-1-neg 65.9%

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

   4. Simplified 65.9%

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

   if 3.000000000000001e-309 < x

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 66.4%

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

      1. add-sqr-sqrt 46.8%

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

      2. sqrt-unprod 66.0%

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

      3. *-commutative 66.0%

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

      4. *-commutative 66.0%

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

      5. swap-sqr 66.0%

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

      6. metadata-eval 66.0%

         \[\leadsto \sqrt{\left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot \color{blue}{0.25}} + x \]

      7. metadata-eval 66.0%

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

      8. swap-sqr 66.0%

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

      9. sqrt-unprod 38.9%

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

      10. add-sqr-sqrt 64.3%

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

      11. *-commutative 64.3%

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

      12. clear-num 64.3%

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

      13. un-div-inv 64.3%

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

   4. Applied egg-rr 64.3%

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

      1. associate-/r/ 64.3%

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

      2. *-commutative 64.3%

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

      3. add-sqr-sqrt 38.9%

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

      4. sqrt-unprod 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/r/ 66.0%

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

      7. *-commutative 66.0%

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

      8. associate-/r/ 66.0%

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

      9. frac-times 66.0%

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

      10. metadata-eval 66.0%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\frac{x}{y} \cdot \frac{x}{y}}} + x \]

      11. metadata-eval 66.0%

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

      12. frac-times 66.0%

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

      13. sqrt-unprod 48.3%

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

      14. add-sqr-sqrt 66.4%

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

      15. associate-/r/ 66.4%

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

   6. Applied egg-rr 66.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-309}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 6: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-310) (- (* y (/ -0.5 x)) x) (+ x (* y (/ 0.5 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (y * (-0.5 / x)) - x;
	} else {
		tmp = x + (y * (0.5 / 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 <= (-5d-310)) then
        tmp = (y * ((-0.5d0) / x)) - x
    else
        tmp = x + (y * (0.5d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (y * (-0.5 / x)) - x;
	} else {
		tmp = x + (y * (0.5 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = (y * (-0.5 / x)) - x
	else:
		tmp = x + (y * (0.5 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(y * Float64(-0.5 / x)) - x);
	else
		tmp = Float64(x + Float64(y * Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = (y * (-0.5 / x)) - x;
	else
		tmp = x + (y * (0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], N[(N[(y * N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], N[(x + N[(y * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in x around -inf 66.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{x} + -1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 66.8%

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

      2. unsub-neg 66.8%

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

      3. *-commutative 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in y around 0 66.8%

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

      1. associate-*r/ 66.8%

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

      2. *-commutative 66.8%

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

      3. associate-*r/ 66.8%

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

   7. Simplified 66.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 66.4%

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

      1. add-sqr-sqrt 46.8%

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

      2. sqrt-unprod 66.0%

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

      3. *-commutative 66.0%

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

      4. *-commutative 66.0%

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

      5. swap-sqr 66.0%

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

      6. metadata-eval 66.0%

         \[\leadsto \sqrt{\left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot \color{blue}{0.25}} + x \]

      7. metadata-eval 66.0%

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

      8. swap-sqr 66.0%

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

      9. sqrt-unprod 38.9%

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

      10. add-sqr-sqrt 64.3%

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

      11. *-commutative 64.3%

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

      12. clear-num 64.3%

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

      13. un-div-inv 64.3%

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

   4. Applied egg-rr 64.3%

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

      1. associate-/r/ 64.3%

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

      2. *-commutative 64.3%

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

      3. add-sqr-sqrt 38.9%

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

      4. sqrt-unprod 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/r/ 66.0%

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

      7. *-commutative 66.0%

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

      8. associate-/r/ 66.0%

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

      9. frac-times 66.0%

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

      10. metadata-eval 66.0%

          \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\frac{x}{y} \cdot \frac{x}{y}}} + x \]

      11. metadata-eval 66.0%

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

      12. frac-times 66.0%

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

      13. sqrt-unprod 48.3%

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

      14. add-sqr-sqrt 66.4%

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

      15. associate-/r/ 66.4%

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

   6. Applied egg-rr 66.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{-0.5}{x} - x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 7: 67.8% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5e-310) (- x) x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-310:
		tmp = -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(-x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-310], (-x), x]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 72.1%

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

   2. Taylor expanded in x around -inf 65.9%

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

      1. mul-1-neg 65.9%

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

   4. Simplified 65.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 66.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 34.4% 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 71.2%

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

2. Taylor expanded in x around inf 34.7%

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

3. Final simplification 34.7%

   \[\leadsto x \]

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 2023196 
(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)))