Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.8%

Time: 6.2s

Alternatives: 6

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p_m 2.0) x)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0:
		tmp = -p_m / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p_m * 2.0), x)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0)
		tmp = -p_m / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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

Alternative 2: 69.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 3.55 \cdot 10^{-271}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 3.6 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.55 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 6.2 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.2 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.25}{p_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 3.55e-271)
     1.0
     (if (<= p_m 1e-256)
       t_0
       (if (<= p_m 3.6e-167)
         1.0
         (if (<= p_m 1.55e-152)
           t_0
           (if (<= p_m 6.2e-127)
             1.0
             (if (<= p_m 1.2e-115)
               t_0
               (if (<= p_m 1e-27)
                 1.0
                 (sqrt (+ 0.5 (* x (/ 0.25 p_m)))))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 3.55e-271) {
		tmp = 1.0;
	} else if (p_m <= 1e-256) {
		tmp = t_0;
	} else if (p_m <= 3.6e-167) {
		tmp = 1.0;
	} else if (p_m <= 1.55e-152) {
		tmp = t_0;
	} else if (p_m <= 6.2e-127) {
		tmp = 1.0;
	} else if (p_m <= 1.2e-115) {
		tmp = t_0;
	} else if (p_m <= 1e-27) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (x * (0.25 / p_m))));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 3.55d-271) then
        tmp = 1.0d0
    else if (p_m <= 1d-256) then
        tmp = t_0
    else if (p_m <= 3.6d-167) then
        tmp = 1.0d0
    else if (p_m <= 1.55d-152) then
        tmp = t_0
    else if (p_m <= 6.2d-127) then
        tmp = 1.0d0
    else if (p_m <= 1.2d-115) then
        tmp = t_0
    else if (p_m <= 1d-27) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (x * (0.25d0 / p_m))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 3.55e-271) {
		tmp = 1.0;
	} else if (p_m <= 1e-256) {
		tmp = t_0;
	} else if (p_m <= 3.6e-167) {
		tmp = 1.0;
	} else if (p_m <= 1.55e-152) {
		tmp = t_0;
	} else if (p_m <= 6.2e-127) {
		tmp = 1.0;
	} else if (p_m <= 1.2e-115) {
		tmp = t_0;
	} else if (p_m <= 1e-27) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (x * (0.25 / p_m))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 3.55e-271:
		tmp = 1.0
	elif p_m <= 1e-256:
		tmp = t_0
	elif p_m <= 3.6e-167:
		tmp = 1.0
	elif p_m <= 1.55e-152:
		tmp = t_0
	elif p_m <= 6.2e-127:
		tmp = 1.0
	elif p_m <= 1.2e-115:
		tmp = t_0
	elif p_m <= 1e-27:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (x * (0.25 / p_m))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 3.55e-271)
		tmp = 1.0;
	elseif (p_m <= 1e-256)
		tmp = t_0;
	elseif (p_m <= 3.6e-167)
		tmp = 1.0;
	elseif (p_m <= 1.55e-152)
		tmp = t_0;
	elseif (p_m <= 6.2e-127)
		tmp = 1.0;
	elseif (p_m <= 1.2e-115)
		tmp = t_0;
	elseif (p_m <= 1e-27)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(x * Float64(0.25 / p_m))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 3.55e-271)
		tmp = 1.0;
	elseif (p_m <= 1e-256)
		tmp = t_0;
	elseif (p_m <= 3.6e-167)
		tmp = 1.0;
	elseif (p_m <= 1.55e-152)
		tmp = t_0;
	elseif (p_m <= 6.2e-127)
		tmp = 1.0;
	elseif (p_m <= 1.2e-115)
		tmp = t_0;
	elseif (p_m <= 1e-27)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (x * (0.25 / p_m))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 3.55e-271], 1.0, If[LessEqual[p$95$m, 1e-256], t$95$0, If[LessEqual[p$95$m, 3.6e-167], 1.0, If[LessEqual[p$95$m, 1.55e-152], t$95$0, If[LessEqual[p$95$m, 6.2e-127], 1.0, If[LessEqual[p$95$m, 1.2e-115], t$95$0, If[LessEqual[p$95$m, 1e-27], 1.0, N[Sqrt[N[(0.5 + N[(x * N[(0.25 / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 3: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 1.25 \cdot 10^{-270}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 3.7 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 3.9 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 2.65 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.9 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 3 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 1.25e-270)
     1.0
     (if (<= p_m 3.7e-256)
       t_0
       (if (<= p_m 3.9e-167)
         1.0
         (if (<= p_m 2.65e-152)
           t_0
           (if (<= p_m 5.2e-127)
             1.0
             (if (<= p_m 1.9e-115)
               t_0
               (if (<= p_m 3e-28) 1.0 (sqrt 0.5))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 1.25e-270) {
		tmp = 1.0;
	} else if (p_m <= 3.7e-256) {
		tmp = t_0;
	} else if (p_m <= 3.9e-167) {
		tmp = 1.0;
	} else if (p_m <= 2.65e-152) {
		tmp = t_0;
	} else if (p_m <= 5.2e-127) {
		tmp = 1.0;
	} else if (p_m <= 1.9e-115) {
		tmp = t_0;
	} else if (p_m <= 3e-28) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 1.25d-270) then
        tmp = 1.0d0
    else if (p_m <= 3.7d-256) then
        tmp = t_0
    else if (p_m <= 3.9d-167) then
        tmp = 1.0d0
    else if (p_m <= 2.65d-152) then
        tmp = t_0
    else if (p_m <= 5.2d-127) then
        tmp = 1.0d0
    else if (p_m <= 1.9d-115) then
        tmp = t_0
    else if (p_m <= 3d-28) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 1.25e-270) {
		tmp = 1.0;
	} else if (p_m <= 3.7e-256) {
		tmp = t_0;
	} else if (p_m <= 3.9e-167) {
		tmp = 1.0;
	} else if (p_m <= 2.65e-152) {
		tmp = t_0;
	} else if (p_m <= 5.2e-127) {
		tmp = 1.0;
	} else if (p_m <= 1.9e-115) {
		tmp = t_0;
	} else if (p_m <= 3e-28) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 1.25e-270:
		tmp = 1.0
	elif p_m <= 3.7e-256:
		tmp = t_0
	elif p_m <= 3.9e-167:
		tmp = 1.0
	elif p_m <= 2.65e-152:
		tmp = t_0
	elif p_m <= 5.2e-127:
		tmp = 1.0
	elif p_m <= 1.9e-115:
		tmp = t_0
	elif p_m <= 3e-28:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 1.25e-270)
		tmp = 1.0;
	elseif (p_m <= 3.7e-256)
		tmp = t_0;
	elseif (p_m <= 3.9e-167)
		tmp = 1.0;
	elseif (p_m <= 2.65e-152)
		tmp = t_0;
	elseif (p_m <= 5.2e-127)
		tmp = 1.0;
	elseif (p_m <= 1.9e-115)
		tmp = t_0;
	elseif (p_m <= 3e-28)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 1.25e-270)
		tmp = 1.0;
	elseif (p_m <= 3.7e-256)
		tmp = t_0;
	elseif (p_m <= 3.9e-167)
		tmp = 1.0;
	elseif (p_m <= 2.65e-152)
		tmp = t_0;
	elseif (p_m <= 5.2e-127)
		tmp = 1.0;
	elseif (p_m <= 1.9e-115)
		tmp = t_0;
	elseif (p_m <= 3e-28)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 1.25e-270], 1.0, If[LessEqual[p$95$m, 3.7e-256], t$95$0, If[LessEqual[p$95$m, 3.9e-167], 1.0, If[LessEqual[p$95$m, 2.65e-152], t$95$0, If[LessEqual[p$95$m, 5.2e-127], 1.0, If[LessEqual[p$95$m, 1.9e-115], t$95$0, If[LessEqual[p$95$m, 3e-28], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 4: 68.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 4.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 4.2e-61) (/ (- p_m) x) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.2e-61) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 4.2d-61) then
        tmp = -p_m / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.2e-61) {
		tmp = -p_m / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 4.2e-61:
		tmp = -p_m / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 4.2e-61)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 4.2e-61)
		tmp = -p_m / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 4.2e-61], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

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

Alternative 5: 28.5% accurate, 35.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5e-310) (/ (- p_m) x) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p_m / x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -p_m / x
    else
        tmp = p_m / x
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p_m / x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e-310:
		tmp = -p_m / x
	else:
		tmp = p_m / x
	return tmp

Julia

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

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -p_m / x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e-310], N[((-p$95$m) / x), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

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

Alternative 6: 6.3% accurate, 71.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \frac{p_m}{x} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (/ p_m x))

C

p_m = fabs(p);
double code(double p_m, double x) {
	return p_m / x;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = p_m / x
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return p_m / x;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return p_m / x

Julia

p_m = abs(p)
function code(p_m, x)
	return Float64(p_m / x)
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = p_m / x;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[(p$95$m / x), $MachinePrecision]

Derivation

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

Developer target: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2023343 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))