Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 68.8% → 99.0%

Time: 3.6s

Alternatives: 4

Speedup: 0.9×

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: 68.8% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x\_m, x\_m, y\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* x_m x_m) 4e+164) (sqrt (fma x_m x_m y)) (* 1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((x_m * x_m) <= 4e+164) {
		tmp = sqrt(fma(x_m, x_m, y));
	} else {
		tmp = 1.0 * x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 4e+164)
		tmp = sqrt(fma(x_m, x_m, y));
	else
		tmp = Float64(1.0 * x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 4e+164], N[Sqrt[N[(x$95$m * x$95$m + y), $MachinePrecision]], $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4e164

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 4e164 < (*.f64 x x)

   1. Initial program 34.0%

      \[\sqrt{x \cdot x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{{x}^{2}}, \frac{1}{2}, 1\right)} \cdot x \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{{x}^{2}}}, \frac{1}{2}, 1\right) \cdot x \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, \frac{1}{2}, 1\right) \cdot x \]

      8. lower-*.f64 46.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, 0.5, 1\right) \cdot x \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x \cdot x}, 0.5, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 46.3%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 10^{-63}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* x_m x_m) 1e-63) (sqrt y) (* 1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((x_m * x_m) <= 1e-63) {
		tmp = sqrt(y);
	} else {
		tmp = 1.0 * x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x_m * x_m) <= 1d-63) then
        tmp = sqrt(y)
    else
        tmp = 1.0d0 * x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if ((x_m * x_m) <= 1e-63) {
		tmp = Math.sqrt(y);
	} else {
		tmp = 1.0 * x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if (x_m * x_m) <= 1e-63:
		tmp = math.sqrt(y)
	else:
		tmp = 1.0 * x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1e-63)
		tmp = sqrt(y);
	else
		tmp = Float64(1.0 * x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((x_m * x_m) <= 1e-63)
		tmp = sqrt(y);
	else
		tmp = 1.0 * x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[(x$95$m * x$95$m), $MachinePrecision], 1e-63], N[Sqrt[y], $MachinePrecision], N[(1.0 * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000007e-63

   1. Initial program 100.0%

      \[\sqrt{x \cdot x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sqrt{y}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f64 90.2

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

   5. Applied rewrites 90.2%

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

   if 1.00000000000000007e-63 < (*.f64 x x)

   1. Initial program 52.5%

      \[\sqrt{x \cdot x + y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{{x}^{2}}, \frac{1}{2}, 1\right)} \cdot x \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{{x}^{2}}}, \frac{1}{2}, 1\right) \cdot x \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, \frac{1}{2}, 1\right) \cdot x \]

      8. lower-*.f64 41.8

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, 0.5, 1\right) \cdot x \]

   5. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x \cdot x}, 0.5, 1\right) \cdot x} \]

   6. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (* 1.0 x_m))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return 1.0 * x_m;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = 1.0d0 * x_m
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return 1.0 * x_m;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return 1.0 * x_m

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(1.0 * x_m)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y)
	tmp = 1.0 * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(1.0 * x$95$m), $MachinePrecision]

Derivation

1. Initial program 67.0%

   \[\sqrt{x \cdot x + y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{{x}^{2}}, \frac{1}{2}, 1\right)} \cdot x \]

   6. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{{x}^{2}}}, \frac{1}{2}, 1\right) \cdot x \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, \frac{1}{2}, 1\right) \cdot x \]

   8. lower-*.f64 30.5

      \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{x \cdot x}}, 0.5, 1\right) \cdot x \]

5. Applied rewrites 30.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x \cdot x}, 0.5, 1\right) \cdot x} \]

6. Taylor expanded in x around inf

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 30.9%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Alternative 4: 1.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ -x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (- x_m))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return -x_m;
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y):
	return -x_m

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(-x_m)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := (-x$95$m)

Derivation

1. Initial program 67.0%

   \[\sqrt{x \cdot x + y} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 38.7

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

5. Applied rewrites 38.7%

   \[\leadsto \color{blue}{-x} \]
6. Add Preprocessing

Developer Target 1: 98.3% accurate, 0.6× 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 2024329 
(FPCore (x y)
  :name "Linear.Quaternion:$clog from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1509769801047259300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ (* 1/2 (/ y x)) x)) (if (< x 5582399551122541000000000000000000000000000000000000000000) (sqrt (+ (* x x) y)) (+ (* 1/2 (/ y x)) x))))

  (sqrt (+ (* x x) y)))