Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.7%

Time: 8.2s

Alternatives: 7

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: 79.7% 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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.5)
   (/ (- p) x)
   (sqrt (+ 0.5 (* x (/ 0.5 (hypot x (* p 2.0))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.5

   1. Initial program 14.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 14.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 14.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 14.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 14.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 14.1%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 13.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 13.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around -inf 54.4%

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

      1. mul-1-neg 54.4%

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

   8. Simplified 54.4%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 99.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 99.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

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

Alternative 2: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := p \cdot \sqrt{{x}^{-2}}\\ t_1 := \frac{-p}{x}\\ t_2 := 1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{if}\;p \leq 3.8 \cdot 10^{-246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p \leq 2.55 \cdot 10^{-228}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;p \leq 2.3 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 5.4 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 920000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.55 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.25}{p}}\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (* p (sqrt (pow x -2.0))))
        (t_1 (/ (- p) x))
        (t_2 (+ 1.0 (* -0.5 (* (/ p x) (/ p x))))))
   (if (<= p 3.8e-246)
     t_1
     (if (<= p 2.55e-228)
       t_2
       (if (<= p 2.3e-165)
         t_0
         (if (<= p 5.4e-132)
           t_2
           (if (<= p 7.5e-66)
             t_0
             (if (<= p 920000000000.0)
               1.0
               (if (<= p 2.55e+64)
                 (sqrt (+ 0.5 (* x (/ 0.25 p))))
                 (if (<= p 2.6e+67) t_1 (sqrt 0.5)))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p * sqrt(pow(x, -2.0));
	double t_1 = -p / x;
	double t_2 = 1.0 + (-0.5 * ((p / x) * (p / x)));
	double tmp;
	if (p <= 3.8e-246) {
		tmp = t_1;
	} else if (p <= 2.55e-228) {
		tmp = t_2;
	} else if (p <= 2.3e-165) {
		tmp = t_0;
	} else if (p <= 5.4e-132) {
		tmp = t_2;
	} else if (p <= 7.5e-66) {
		tmp = t_0;
	} else if (p <= 920000000000.0) {
		tmp = 1.0;
	} else if (p <= 2.55e+64) {
		tmp = sqrt((0.5 + (x * (0.25 / p))));
	} else if (p <= 2.6e+67) {
		tmp = t_1;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = p * sqrt((x ** (-2.0d0)))
    t_1 = -p / x
    t_2 = 1.0d0 + ((-0.5d0) * ((p / x) * (p / x)))
    if (p <= 3.8d-246) then
        tmp = t_1
    else if (p <= 2.55d-228) then
        tmp = t_2
    else if (p <= 2.3d-165) then
        tmp = t_0
    else if (p <= 5.4d-132) then
        tmp = t_2
    else if (p <= 7.5d-66) then
        tmp = t_0
    else if (p <= 920000000000.0d0) then
        tmp = 1.0d0
    else if (p <= 2.55d+64) then
        tmp = sqrt((0.5d0 + (x * (0.25d0 / p))))
    else if (p <= 2.6d+67) then
        tmp = t_1
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p * Math.sqrt(Math.pow(x, -2.0));
	double t_1 = -p / x;
	double t_2 = 1.0 + (-0.5 * ((p / x) * (p / x)));
	double tmp;
	if (p <= 3.8e-246) {
		tmp = t_1;
	} else if (p <= 2.55e-228) {
		tmp = t_2;
	} else if (p <= 2.3e-165) {
		tmp = t_0;
	} else if (p <= 5.4e-132) {
		tmp = t_2;
	} else if (p <= 7.5e-66) {
		tmp = t_0;
	} else if (p <= 920000000000.0) {
		tmp = 1.0;
	} else if (p <= 2.55e+64) {
		tmp = Math.sqrt((0.5 + (x * (0.25 / p))));
	} else if (p <= 2.6e+67) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p * math.sqrt(math.pow(x, -2.0))
	t_1 = -p / x
	t_2 = 1.0 + (-0.5 * ((p / x) * (p / x)))
	tmp = 0
	if p <= 3.8e-246:
		tmp = t_1
	elif p <= 2.55e-228:
		tmp = t_2
	elif p <= 2.3e-165:
		tmp = t_0
	elif p <= 5.4e-132:
		tmp = t_2
	elif p <= 7.5e-66:
		tmp = t_0
	elif p <= 920000000000.0:
		tmp = 1.0
	elif p <= 2.55e+64:
		tmp = math.sqrt((0.5 + (x * (0.25 / p))))
	elif p <= 2.6e+67:
		tmp = t_1
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p * sqrt((x ^ -2.0)))
	t_1 = Float64(Float64(-p) / x)
	t_2 = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) * Float64(p / x))))
	tmp = 0.0
	if (p <= 3.8e-246)
		tmp = t_1;
	elseif (p <= 2.55e-228)
		tmp = t_2;
	elseif (p <= 2.3e-165)
		tmp = t_0;
	elseif (p <= 5.4e-132)
		tmp = t_2;
	elseif (p <= 7.5e-66)
		tmp = t_0;
	elseif (p <= 920000000000.0)
		tmp = 1.0;
	elseif (p <= 2.55e+64)
		tmp = sqrt(Float64(0.5 + Float64(x * Float64(0.25 / p))));
	elseif (p <= 2.6e+67)
		tmp = t_1;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p * sqrt((x ^ -2.0));
	t_1 = -p / x;
	t_2 = 1.0 + (-0.5 * ((p / x) * (p / x)));
	tmp = 0.0;
	if (p <= 3.8e-246)
		tmp = t_1;
	elseif (p <= 2.55e-228)
		tmp = t_2;
	elseif (p <= 2.3e-165)
		tmp = t_0;
	elseif (p <= 5.4e-132)
		tmp = t_2;
	elseif (p <= 7.5e-66)
		tmp = t_0;
	elseif (p <= 920000000000.0)
		tmp = 1.0;
	elseif (p <= 2.55e+64)
		tmp = sqrt((0.5 + (x * (0.25 / p))));
	elseif (p <= 2.6e+67)
		tmp = t_1;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p * N[Sqrt[N[Power[x, -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-p) / x), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, 3.8e-246], t$95$1, If[LessEqual[p, 2.55e-228], t$95$2, If[LessEqual[p, 2.3e-165], t$95$0, If[LessEqual[p, 5.4e-132], t$95$2, If[LessEqual[p, 7.5e-66], t$95$0, If[LessEqual[p, 920000000000.0], 1.0, If[LessEqual[p, 2.55e+64], N[Sqrt[N[(0.5 + N[(x * N[(0.25 / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[p, 2.6e+67], t$95$1, N[Sqrt[0.5], $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if p < 3.79999999999999976e-246 or 2.55000000000000012e64 < p < 2.6e67

   1. Initial program 73.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 72.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 73.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 72.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 73.7%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 73.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 73.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around -inf 11.9%

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

      1. mul-1-neg 11.9%

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

   8. Simplified 11.9%

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

   if 3.79999999999999976e-246 < p < 2.5500000000000001e-228 or 2.3e-165 < p < 5.3999999999999998e-132

   1. Initial program 78.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 78.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 78.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 78.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 78.4%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 78.4%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 78.4%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around inf 77.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 77.0%

         \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{p \cdot p}}{{x}^{2}} \]

      2. unpow2 77.0%

         \[\leadsto 1 + -0.5 \cdot \frac{p \cdot p}{\color{blue}{x \cdot x}} \]

      3. times-frac 77.0%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   8. Applied egg-rr 77.0%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   if 2.5500000000000001e-228 < p < 2.3e-165 or 5.3999999999999998e-132 < p < 7.49999999999999995e-66

   1. Initial program 47.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 47.5%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}\right)} \]

      2. log1p-def 47.5%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      3. expm1-udef 47.5%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\left(e^{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}\right)} \]

      4. add-exp-log 47.5%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \left(\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)\right)} \]

      5. associate-+r- 47.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(1 + \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) - 1\right)}} \]

   3. Applied egg-rr 47.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right) - 1\right)}} \]

   4. Taylor expanded in x around -inf 16.9%

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

      1. div-inv 16.9%

         \[\leadsto \sqrt{\color{blue}{{p}^{2} \cdot \frac{1}{{x}^{2}}}} \]

      2. sqrt-prod 41.5%

         \[\leadsto \color{blue}{\sqrt{{p}^{2}} \cdot \sqrt{\frac{1}{{x}^{2}}}} \]

      3. unpow2 41.5%

         \[\leadsto \sqrt{\color{blue}{p \cdot p}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      4. sqrt-prod 59.2%

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

      5. add-sqr-sqrt 59.2%

         \[\leadsto \color{blue}{p} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      6. *-commutative 59.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{2}}} \cdot p} \]

      7. pow-flip 59.2%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(-2\right)}}} \cdot p \]

      8. metadata-eval 59.2%

         \[\leadsto \sqrt{{x}^{\color{blue}{-2}}} \cdot p \]

   6. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\sqrt{{x}^{-2}} \cdot p} \]

   if 7.49999999999999995e-66 < p < 9.2e11

   1. Initial program 83.9%

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

   2. Taylor expanded in x around inf 62.3%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 9.2e11 < p < 2.55000000000000012e64

   1. Initial program 71.1%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 70.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 70.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 70.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 71.1%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 71.4%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around 0 49.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.25}{p}} \cdot x} \]

   if 2.6e67 < p

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0 96.3%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.8 \cdot 10^{-246}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.55 \cdot 10^{-228}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{elif}\;p \leq 2.3 \cdot 10^{-165}:\\ \;\;\;\;p \cdot \sqrt{{x}^{-2}}\\ \mathbf{elif}\;p \leq 5.4 \cdot 10^{-132}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-66}:\\ \;\;\;\;p \cdot \sqrt{{x}^{-2}}\\ \mathbf{elif}\;p \leq 920000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.55 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.25}{p}}\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 5.2 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 7.8 \cdot 10^{-227}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{elif}\;p \leq 4.7 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.02 \cdot 10^{-80}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 6800000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 5.2e-243)
     t_0
     (if (<= p 7.8e-227)
       (+ 1.0 (* -0.5 (* (/ p x) (/ p x))))
       (if (<= p 4.7e-163)
         t_0
         (if (<= p 1.02e-80)
           1.0
           (if (<= p 7e-67) t_0 (if (<= p 6800000000.0) 1.0 (sqrt 0.5)))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 5.2e-243) {
		tmp = t_0;
	} else if (p <= 7.8e-227) {
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)));
	} else if (p <= 4.7e-163) {
		tmp = t_0;
	} else if (p <= 1.02e-80) {
		tmp = 1.0;
	} else if (p <= 7e-67) {
		tmp = t_0;
	} else if (p <= 6800000000.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (p <= 5.2d-243) then
        tmp = t_0
    else if (p <= 7.8d-227) then
        tmp = 1.0d0 + ((-0.5d0) * ((p / x) * (p / x)))
    else if (p <= 4.7d-163) then
        tmp = t_0
    else if (p <= 1.02d-80) then
        tmp = 1.0d0
    else if (p <= 7d-67) then
        tmp = t_0
    else if (p <= 6800000000.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 5.2e-243) {
		tmp = t_0;
	} else if (p <= 7.8e-227) {
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)));
	} else if (p <= 4.7e-163) {
		tmp = t_0;
	} else if (p <= 1.02e-80) {
		tmp = 1.0;
	} else if (p <= 7e-67) {
		tmp = t_0;
	} else if (p <= 6800000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= 5.2e-243:
		tmp = t_0
	elif p <= 7.8e-227:
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)))
	elif p <= 4.7e-163:
		tmp = t_0
	elif p <= 1.02e-80:
		tmp = 1.0
	elif p <= 7e-67:
		tmp = t_0
	elif p <= 6800000000.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 5.2e-243)
		tmp = t_0;
	elseif (p <= 7.8e-227)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) * Float64(p / x))));
	elseif (p <= 4.7e-163)
		tmp = t_0;
	elseif (p <= 1.02e-80)
		tmp = 1.0;
	elseif (p <= 7e-67)
		tmp = t_0;
	elseif (p <= 6800000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= 5.2e-243)
		tmp = t_0;
	elseif (p <= 7.8e-227)
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)));
	elseif (p <= 4.7e-163)
		tmp = t_0;
	elseif (p <= 1.02e-80)
		tmp = 1.0;
	elseif (p <= 7e-67)
		tmp = t_0;
	elseif (p <= 6800000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[((-p) / x), $MachinePrecision]}, If[LessEqual[p, 5.2e-243], t$95$0, If[LessEqual[p, 7.8e-227], N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 4.7e-163], t$95$0, If[LessEqual[p, 1.02e-80], 1.0, If[LessEqual[p, 7e-67], t$95$0, If[LessEqual[p, 6800000000.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 5.1999999999999995e-243 or 7.7999999999999999e-227 < p < 4.7e-163 or 1.02000000000000005e-80 < p < 7.0000000000000001e-67

   1. Initial program 70.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 69.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 69.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 69.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 70.6%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 70.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around -inf 17.1%

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

      1. mul-1-neg 17.1%

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

   8. Simplified 17.1%

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

   if 5.1999999999999995e-243 < p < 7.7999999999999999e-227

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{p \cdot p}}{{x}^{2}} \]

      2. unpow2 100.0%

         \[\leadsto 1 + -0.5 \cdot \frac{p \cdot p}{\color{blue}{x \cdot x}} \]

      3. times-frac 100.0%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   8. Applied egg-rr 100.0%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   if 4.7e-163 < p < 1.02000000000000005e-80 or 7.0000000000000001e-67 < p < 6.8e9

   1. Initial program 72.3%

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

   2. Taylor expanded in x around inf 58.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 6.8e9 < p

   1. Initial program 91.7%

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

   2. Taylor expanded in x around 0 86.7%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5.2 \cdot 10^{-243}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 7.8 \cdot 10^{-227}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{elif}\;p \leq 4.7 \cdot 10^{-163}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.02 \cdot 10^{-80}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 6800000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ t_1 := 1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{if}\;p \leq 3.6 \cdot 10^{-244}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p \leq 7.6 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)) (t_1 (+ 1.0 (* -0.5 (* (/ p x) (/ p x))))))
   (if (<= p 3.6e-244)
     t_0
     (if (<= p 3.8e-228)
       t_1
       (if (<= p 7.6e-166)
         t_0
         (if (<= p 8.5e-85) t_1 (if (<= p 1.4e-64) t_0 (sqrt 0.5))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double t_1 = 1.0 + (-0.5 * ((p / x) * (p / x)));
	double tmp;
	if (p <= 3.6e-244) {
		tmp = t_0;
	} else if (p <= 3.8e-228) {
		tmp = t_1;
	} else if (p <= 7.6e-166) {
		tmp = t_0;
	} else if (p <= 8.5e-85) {
		tmp = t_1;
	} else if (p <= 1.4e-64) {
		tmp = t_0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -p / x
    t_1 = 1.0d0 + ((-0.5d0) * ((p / x) * (p / x)))
    if (p <= 3.6d-244) then
        tmp = t_0
    else if (p <= 3.8d-228) then
        tmp = t_1
    else if (p <= 7.6d-166) then
        tmp = t_0
    else if (p <= 8.5d-85) then
        tmp = t_1
    else if (p <= 1.4d-64) then
        tmp = t_0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -p / x;
	double t_1 = 1.0 + (-0.5 * ((p / x) * (p / x)));
	double tmp;
	if (p <= 3.6e-244) {
		tmp = t_0;
	} else if (p <= 3.8e-228) {
		tmp = t_1;
	} else if (p <= 7.6e-166) {
		tmp = t_0;
	} else if (p <= 8.5e-85) {
		tmp = t_1;
	} else if (p <= 1.4e-64) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	t_1 = 1.0 + (-0.5 * ((p / x) * (p / x)))
	tmp = 0
	if p <= 3.6e-244:
		tmp = t_0
	elif p <= 3.8e-228:
		tmp = t_1
	elif p <= 7.6e-166:
		tmp = t_0
	elif p <= 8.5e-85:
		tmp = t_1
	elif p <= 1.4e-64:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	t_1 = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) * Float64(p / x))))
	tmp = 0.0
	if (p <= 3.6e-244)
		tmp = t_0;
	elseif (p <= 3.8e-228)
		tmp = t_1;
	elseif (p <= 7.6e-166)
		tmp = t_0;
	elseif (p <= 8.5e-85)
		tmp = t_1;
	elseif (p <= 1.4e-64)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	t_1 = 1.0 + (-0.5 * ((p / x) * (p / x)));
	tmp = 0.0;
	if (p <= 3.6e-244)
		tmp = t_0;
	elseif (p <= 3.8e-228)
		tmp = t_1;
	elseif (p <= 7.6e-166)
		tmp = t_0;
	elseif (p <= 8.5e-85)
		tmp = t_1;
	elseif (p <= 1.4e-64)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[((-p) / x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, 3.6e-244], t$95$0, If[LessEqual[p, 3.8e-228], t$95$1, If[LessEqual[p, 7.6e-166], t$95$0, If[LessEqual[p, 8.5e-85], t$95$1, If[LessEqual[p, 1.4e-64], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.59999999999999975e-244 or 3.7999999999999999e-228 < p < 7.59999999999999964e-166 or 8.50000000000000052e-85 < p < 1.40000000000000002e-64

   1. Initial program 70.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 69.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 69.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 69.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 70.6%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 70.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around -inf 17.1%

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

      1. mul-1-neg 17.1%

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

   8. Simplified 17.1%

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

   if 3.59999999999999975e-244 < p < 3.7999999999999999e-228 or 7.59999999999999964e-166 < p < 8.50000000000000052e-85

   1. Initial program 64.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 64.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 64.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 64.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 64.8%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 64.6%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 64.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around inf 59.1%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 59.1%

         \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{p \cdot p}}{{x}^{2}} \]

      2. unpow2 59.1%

         \[\leadsto 1 + -0.5 \cdot \frac{p \cdot p}{\color{blue}{x \cdot x}} \]

      3. times-frac 59.1%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   8. Applied egg-rr 59.1%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   if 1.40000000000000002e-64 < p

   1. Initial program 89.8%

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

   2. Taylor expanded in x around 0 75.1%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.6 \cdot 10^{-244}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-228}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{elif}\;p \leq 7.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-85}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 50.6% accurate, 16.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= x 4.3e-140) (/ (- p) x) (+ 1.0 (* -0.5 (* (/ p x) (/ p x))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= 4.3e-140) {
		tmp = -p / x;
	} else {
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.3d-140) then
        tmp = -p / x
    else
        tmp = 1.0d0 + ((-0.5d0) * ((p / x) * (p / x)))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= 4.3e-140) {
		tmp = -p / x;
	} else {
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= 4.3e-140:
		tmp = -p / x
	else:
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= 4.3e-140)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) * Float64(p / x))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= 4.3e-140)
		tmp = -p / x;
	else
		tmp = 1.0 + (-0.5 * ((p / x) * (p / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, 4.3e-140], N[((-p) / x), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.29999999999999962e-140

   1. Initial program 58.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 57.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 57.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 57.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 58.5%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 57.9%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around -inf 27.8%

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

      1. mul-1-neg 27.8%

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

   8. Simplified 27.8%

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

   if 4.29999999999999962e-140 < x

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 99.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around inf 55.5%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 55.5%

         \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{p \cdot p}}{{x}^{2}} \]

      2. unpow2 55.5%

         \[\leadsto 1 + -0.5 \cdot \frac{p \cdot p}{\color{blue}{x \cdot x}} \]

      3. times-frac 55.5%

         \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]

   8. Applied egg-rr 55.5%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{p}{x} \cdot \frac{p}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\\ \end{array} \]

Alternative 6: 28.2% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -2e-310) (/ (- p) x) (/ p x)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -p / x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = -p / x
    else
        tmp = p / x
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = -p / x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -2e-310:
		tmp = -p / x
	else:
		tmp = p / x
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = Float64(p / x);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = -p / x;
	else
		tmp = p / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -2e-310], N[((-p) / x), $MachinePrecision], N[(p / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 57.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 56.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. expm1-udef 56.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)} - 1} \]

   3. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 56.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 57.6%

         \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}} \]

      3. associate-/r/ 57.1%

         \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot x}} \]

   6. Taylor expanded in x around -inf 28.4%

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

      1. mul-1-neg 28.4%

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

   8. Simplified 28.4%

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

   if -1.999999999999994e-310 < x

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}\right)} \]

      2. log1p-def 100.0%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      3. expm1-udef 100.0%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\left(e^{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}\right)} \]

      4. add-exp-log 100.0%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \left(\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)\right)} \]

      5. associate-+r- 100.0%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(1 + \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) - 1\right)}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right) - 1\right)}} \]

   4. Taylor expanded in x around -inf 5.3%

      \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

   5. Taylor expanded in p around 0 3.6%

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

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]

Alternative 7: 6.4% accurate, 71.7× speedup

Math

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

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (/ p x))

C

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

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = p / x
end function

Java

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

Python

p = abs(p)
def code(p, x):
	return p / x

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 75.8%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
2. Step-by-step derivation

   1. expm1-log1p-u 75.8%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}\right)} \]

   2. log1p-def 75.8%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

   3. expm1-udef 75.8%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\left(e^{\log \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)}\right)} \]

   4. add-exp-log 75.8%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \left(\color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} - 1\right)\right)} \]

   5. associate-+r- 75.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(1 + \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) - 1\right)}} \]

3. Applied egg-rr 75.8%

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(\left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right) - 1\right)}} \]

4. Taylor expanded in x around -inf 15.9%

   \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

5. Taylor expanded in p around 0 18.6%

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

6. Final simplification 18.6%

   \[\leadsto \frac{p}{x} \]

Developer target: 79.7% 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 2023305 
(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))))))))