Kahan p9 Example

Percentage Accurate: 68.6% → 100.0%

Time: 8.9s

Alternatives: 9

Speedup: 2.1×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{\frac{x + y_m}{\mathsf{hypot}\left(x, y_m\right)}}{\frac{\mathsf{hypot}\left(x, y_m\right)}{x - y_m}} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (/ (+ x y_m) (hypot x y_m)) (/ (hypot x y_m) (- x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return ((x + y_m) / hypot(x, y_m)) / (hypot(x, y_m) / (x - y_m));
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return ((x + y_m) / Math.hypot(x, y_m)) / (Math.hypot(x, y_m) / (x - y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return ((x + y_m) / math.hypot(x, y_m)) / (math.hypot(x, y_m) / (x - y_m))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) / Float64(hypot(x, y_m) / Float64(x - y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = ((x + y_m) / hypot(x, y_m)) / (hypot(x, y_m) / (x - y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \frac{x + y_m}{\mathsf{hypot}\left(x, y_m\right)} \cdot \frac{x - y_m}{\mathsf{hypot}\left(x, y_m\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (/ (+ x y_m) (hypot x y_m)) (/ (- x y_m) (hypot x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / hypot(x, y_m));
}

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return ((x + y_m) / math.hypot(x, y_m)) * ((x - y_m) / math.hypot(x, y_m))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) * Float64(Float64(x - y_m) / hypot(x, y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / hypot(x, y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.32 \cdot 10^{-170}:\\ \;\;\;\;1 - 2 \cdot {\left(\frac{y_m}{x}\right)}^{2}\\ \mathbf{elif}\;y_m \leq 10^{-13}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.32e-170)
   (- 1.0 (* 2.0 (pow (/ y_m x) 2.0)))
   (if (<= y_m 1e-13)
     (/ (* (+ x y_m) (- x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.32e-170) {
		tmp = 1.0 - (2.0 * pow((y_m / x), 2.0));
	} else if (y_m <= 1e-13) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.32d-170) then
        tmp = 1.0d0 - (2.0d0 * ((y_m / x) ** 2.0d0))
    else if (y_m <= 1d-13) then
        tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.32e-170) {
		tmp = 1.0 - (2.0 * Math.pow((y_m / x), 2.0));
	} else if (y_m <= 1e-13) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.32e-170:
		tmp = 1.0 - (2.0 * math.pow((y_m / x), 2.0))
	elif y_m <= 1e-13:
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.32e-170)
		tmp = Float64(1.0 - Float64(2.0 * (Float64(y_m / x) ^ 2.0)));
	elseif (y_m <= 1e-13)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.32e-170)
		tmp = 1.0 - (2.0 * ((y_m / x) ^ 2.0));
	elseif (y_m <= 1e-13)
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.32e-170], N[(1.0 - N[(2.0 * N[Power[N[(y$95$m / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1e-13], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(\frac{x}{y_m} \cdot \frac{x}{y_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ x y_m) (- x y_m)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 2.0) t_0 (+ -1.0 (* 2.0 (* (/ x y_m) (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (-1.0d0) + (2.0d0 * ((x / y_m) * (x / y_m)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(x * x) + Float64(y_m * y_m)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(-1.0 + Float64(2.0 * Float64(Float64(x / y_m) * Float64(x / y_m))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(-1.0 + N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.35 \cdot 10^{-155} \lor \neg \left(y_m \leq 8.5 \cdot 10^{-79}\right) \land y_m \leq 1.3 \cdot 10^{-65}:\\ \;\;\;\;\left(1 - \frac{y_m}{x}\right) \cdot \left(1 + \frac{y_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 1.35e-155) (and (not (<= y_m 8.5e-79)) (<= y_m 1.3e-65)))
   (* (- 1.0 (/ y_m x)) (+ 1.0 (/ y_m x)))
   -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.35e-155) || (!(y_m <= 8.5e-79) && (y_m <= 1.3e-65))) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 1.35d-155) .or. (.not. (y_m <= 8.5d-79)) .and. (y_m <= 1.3d-65)) then
        tmp = (1.0d0 - (y_m / x)) * (1.0d0 + (y_m / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.35e-155) || (!(y_m <= 8.5e-79) && (y_m <= 1.3e-65))) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 1.35e-155) or (not (y_m <= 8.5e-79) and (y_m <= 1.3e-65)):
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 1.35e-155) || (!(y_m <= 8.5e-79) && (y_m <= 1.3e-65)))
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(1.0 + Float64(y_m / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 1.35e-155) || (~((y_m <= 8.5e-79)) && (y_m <= 1.3e-65)))
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 1.35e-155], And[N[Not[LessEqual[y$95$m, 8.5e-79]], $MachinePrecision], LessEqual[y$95$m, 1.3e-65]]], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(1 - \frac{y_m}{x}\right) \cdot \left(1 + \frac{y_m}{x}\right)\\ \mathbf{if}\;y_m \leq 1.36 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 8.5 \cdot 10^{-79}:\\ \;\;\;\;-1 + 2 \cdot \left(\frac{x}{y_m} \cdot \frac{x}{y_m}\right)\\ \mathbf{elif}\;y_m \leq 9 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (- 1.0 (/ y_m x)) (+ 1.0 (/ y_m x)))))
   (if (<= y_m 1.36e-155)
     t_0
     (if (<= y_m 8.5e-79)
       (+ -1.0 (* 2.0 (* (/ x y_m) (/ x y_m))))
       (if (<= y_m 9e-66) t_0 -1.0)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	double tmp;
	if (y_m <= 1.36e-155) {
		tmp = t_0;
	} else if (y_m <= 8.5e-79) {
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)));
	} else if (y_m <= 9e-66) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (y_m / x)) * (1.0d0 + (y_m / x))
    if (y_m <= 1.36d-155) then
        tmp = t_0
    else if (y_m <= 8.5d-79) then
        tmp = (-1.0d0) + (2.0d0 * ((x / y_m) * (x / y_m)))
    else if (y_m <= 9d-66) then
        tmp = t_0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	double tmp;
	if (y_m <= 1.36e-155) {
		tmp = t_0;
	} else if (y_m <= 8.5e-79) {
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)));
	} else if (y_m <= 9e-66) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (1.0 - (y_m / x)) * (1.0 + (y_m / x))
	tmp = 0
	if y_m <= 1.36e-155:
		tmp = t_0
	elif y_m <= 8.5e-79:
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)))
	elif y_m <= 9e-66:
		tmp = t_0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(1.0 + Float64(y_m / x)))
	tmp = 0.0
	if (y_m <= 1.36e-155)
		tmp = t_0;
	elseif (y_m <= 8.5e-79)
		tmp = Float64(-1.0 + Float64(2.0 * Float64(Float64(x / y_m) * Float64(x / y_m))));
	elseif (y_m <= 9e-66)
		tmp = t_0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	tmp = 0.0;
	if (y_m <= 1.36e-155)
		tmp = t_0;
	elseif (y_m <= 8.5e-79)
		tmp = -1.0 + (2.0 * ((x / y_m) * (x / y_m)));
	elseif (y_m <= 9e-66)
		tmp = t_0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 1.36e-155], t$95$0, If[LessEqual[y$95$m, 8.5e-79], N[(-1.0 + N[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 9e-66], t$95$0, -1.0]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 81.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y_m \leq 9 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-155)
   1.0
   (if (<= y_m 7.5e-79) -1.0 (if (<= y_m 9e-66) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-155) {
		tmp = 1.0;
	} else if (y_m <= 7.5e-79) {
		tmp = -1.0;
	} else if (y_m <= 9e-66) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.3d-155) then
        tmp = 1.0d0
    else if (y_m <= 7.5d-79) then
        tmp = -1.0d0
    else if (y_m <= 9d-66) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-155) {
		tmp = 1.0;
	} else if (y_m <= 7.5e-79) {
		tmp = -1.0;
	} else if (y_m <= 9e-66) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.3e-155:
		tmp = 1.0
	elif y_m <= 7.5e-79:
		tmp = -1.0
	elif y_m <= 9e-66:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.3e-155)
		tmp = 1.0;
	elseif (y_m <= 7.5e-79)
		tmp = -1.0;
	elseif (y_m <= 9e-66)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.3e-155)
		tmp = 1.0;
	elseif (y_m <= 7.5e-79)
		tmp = -1.0;
	elseif (y_m <= 9e-66)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.3e-155], 1.0, If[LessEqual[y$95$m, 7.5e-79], -1.0, If[LessEqual[y$95$m, 9e-66], 1.0, -1.0]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 80.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.3 \cdot 10^{-155}:\\ \;\;\;\;\frac{x - y_m}{x}\\ \mathbf{elif}\;y_m \leq 1.35 \cdot 10^{-80}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y_m \leq 9 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-155)
   (/ (- x y_m) x)
   (if (<= y_m 1.35e-80) -1.0 (if (<= y_m 9e-66) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-155) {
		tmp = (x - y_m) / x;
	} else if (y_m <= 1.35e-80) {
		tmp = -1.0;
	} else if (y_m <= 9e-66) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.3d-155) then
        tmp = (x - y_m) / x
    else if (y_m <= 1.35d-80) then
        tmp = -1.0d0
    else if (y_m <= 9d-66) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-155) {
		tmp = (x - y_m) / x;
	} else if (y_m <= 1.35e-80) {
		tmp = -1.0;
	} else if (y_m <= 9e-66) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.3e-155:
		tmp = (x - y_m) / x
	elif y_m <= 1.35e-80:
		tmp = -1.0
	elif y_m <= 9e-66:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.3e-155)
		tmp = Float64(Float64(x - y_m) / x);
	elseif (y_m <= 1.35e-80)
		tmp = -1.0;
	elseif (y_m <= 9e-66)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.3e-155)
		tmp = (x - y_m) / x;
	elseif (y_m <= 1.35e-80)
		tmp = -1.0;
	elseif (y_m <= 9e-66)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.3e-155], N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y$95$m, 1.35e-80], -1.0, If[LessEqual[y$95$m, 9e-66], 1.0, -1.0]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 66.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return -1.0;
}

Fortran

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = -1.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t_0 \land t_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (and (< 0.5 t_0) (< t_0 2.0))
     (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
     (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = fabs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	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 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(2.0 / Float64(1.0 + Float64(Float64(x / y) * Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023341 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))