Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 68.7% → 99.7%

Time: 4.1s

Alternatives: 5

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: 68.7% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \cdot x_m \leq 10^{+228}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x_m, x_m, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* x_m x_m) 1e+228) (sqrt (fma x_m x_m y)) x_m))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((x_m * x_m) <= 1e+228) {
		tmp = sqrt(fma(x_m, x_m, y));
	} else {
		tmp = x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1e+228)
		tmp = sqrt(fma(x_m, x_m, y));
	else
		tmp = 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], 1e+228], N[Sqrt[N[(x$95$m * x$95$m + y), $MachinePrecision]], $MachinePrecision], x$95$m]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999992e227

   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 9.9999999999999992e227 < (*.f64 x x)

   1. Initial program 20.3%

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

   2. Taylor expanded in x around inf 48.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+228}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, x, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \cdot x_m \leq 10^{+228}:\\ \;\;\;\;\sqrt{x_m \cdot x_m + y}\\ \mathbf{else}:\\ \;\;\;\;x_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* x_m x_m) 1e+228) (sqrt (+ (* x_m x_m) y)) x_m))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((x_m * x_m) <= 1e+228) {
		tmp = sqrt(((x_m * x_m) + y));
	} else {
		tmp = 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+228) then
        tmp = sqrt(((x_m * x_m) + y))
    else
        tmp = 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+228) {
		tmp = Math.sqrt(((x_m * x_m) + y));
	} else {
		tmp = x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if (x_m * x_m) <= 1e+228:
		tmp = math.sqrt(((x_m * x_m) + y))
	else:
		tmp = x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(x_m * x_m) <= 1e+228)
		tmp = sqrt(Float64(Float64(x_m * x_m) + y));
	else
		tmp = 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+228)
		tmp = sqrt(((x_m * x_m) + y));
	else
		tmp = 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+228], N[Sqrt[N[(N[(x$95$m * x$95$m), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision], x$95$m]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999992e227

   1. Initial program 100.0%

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

   if 9.9999999999999992e227 < (*.f64 x x)

   1. Initial program 20.3%

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

   2. Taylor expanded in x around inf 48.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+228}:\\ \;\;\;\;\sqrt{x \cdot x + y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x_m + y \cdot \frac{0.5}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 2e-35) (sqrt y) (+ x_m (* y (/ 0.5 x_m)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2e-35) {
		tmp = sqrt(y);
	} else {
		tmp = x_m + (y * (0.5 / 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 <= 2d-35) then
        tmp = sqrt(y)
    else
        tmp = x_m + (y * (0.5d0 / 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 <= 2e-35) {
		tmp = Math.sqrt(y);
	} else {
		tmp = x_m + (y * (0.5 / x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 2e-35:
		tmp = math.sqrt(y)
	else:
		tmp = x_m + (y * (0.5 / x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 2e-35)
		tmp = sqrt(y);
	else
		tmp = Float64(x_m + Float64(y * Float64(0.5 / x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 2e-35)
		tmp = sqrt(y);
	else
		tmp = x_m + (y * (0.5 / x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 2e-35], N[Sqrt[y], $MachinePrecision], N[(x$95$m + N[(y * N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000002e-35

   1. Initial program 77.4%

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

   2. Taylor expanded in x around 0 48.4%

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

   if 2.00000000000000002e-35 < x

   1. Initial program 52.4%

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

   2. Taylor expanded in x around inf 77.5%

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

   3. Taylor expanded in y around 0 89.0%

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

      1. associate-*r/ 89.0%

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

      2. associate-/l* 89.0%

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

   5. Simplified 89.0%

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

      1. associate-/r/ 89.0%

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

   7. Applied egg-rr 89.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{0.5}{x}\\ \end{array} \]

Alternative 4: 68.3% accurate, 15.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (+ x_m (* y (/ 0.5 x_m))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return x_m + (y * (0.5 / 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 + (y * (0.5d0 / x_m))
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y):
	return x_m + (y * (0.5 / x_m))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(x$95$m + N[(y * N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.2%

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

2. Taylor expanded in x around inf 26.4%

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

3. Taylor expanded in y around 0 30.8%

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

   1. associate-*r/ 30.8%

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

   2. associate-/l* 30.8%

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

5. Simplified 30.8%

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

   1. associate-/r/ 30.8%

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

7. Applied egg-rr 30.8%

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

8. Final simplification 30.8%

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

Alternative 5: 67.8% accurate, 105.0× 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 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 69.2%

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

2. Taylor expanded in x around inf 31.0%

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

3. Final simplification 31.0%

   \[\leadsto x \]

Developer target: 98.5% 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 2023334 
(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)))