Linear.Quaternion:$clog from linear-1.19.1.3

Percentage Accurate: 68.7% → 99.6%

Time: 6.1s

Alternatives: 3

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: 68.7% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 10^{+226}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x\_m, x\_m, y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* x_m x_m) 1e+226) (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+226) {
		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+226)
		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+226], 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) < 9.99999999999999961e225

   1. Initial program 100.0%

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

      1. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if 9.99999999999999961e225 < (*.f64 x x)

   1. Initial program 24.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \sqrt{\color{blue}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 24.0

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

   5. Simplified 24.0%

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

      1. sqrt-prod N/A

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

      2. rem-square-sqrt 57.2

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

   7. Applied egg-rr 57.2%

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

Alternative 2: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \cdot x\_m \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (if (<= (* x_m x_m) 5e-91) (sqrt y) x_m))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((x_m * x_m) <= 5e-91) {
		tmp = sqrt(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) <= 5d-91) then
        tmp = sqrt(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) <= 5e-91) {
		tmp = Math.sqrt(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) <= 5e-91:
		tmp = math.sqrt(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) <= 5e-91)
		tmp = sqrt(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) <= 5e-91)
		tmp = sqrt(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], 5e-91], N[Sqrt[y], $MachinePrecision], x$95$m]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999997e-91

   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 87.8

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

   5. Simplified 87.8%

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

   if 4.99999999999999997e-91 < (*.f64 x x)

   1. Initial program 57.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \sqrt{\color{blue}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 48.6

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

   5. Simplified 48.6%

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

      1. sqrt-prod N/A

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

      2. rem-square-sqrt 52.0

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

   7. Applied egg-rr 52.0%

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

Alternative 3: 68.1% accurate, 19.0× 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 72.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \sqrt{\color{blue}{{x}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 36.2

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

5. Simplified 36.2%

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

   1. sqrt-prod N/A

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

   2. rem-square-sqrt 36.2

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

7. Applied egg-rr 36.2%

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

Developer Target 1: 98.5% 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 2024208 
(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)))