Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.9%

Time: 8.1s

Alternatives: 5

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: 79.3% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{p\_m}{-x} - -1.5 \cdot {\left(\frac{p\_m}{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(t\_0 + 1\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.5)
     (- (/ p_m (- x)) (* -1.5 (pow (/ p_m x) 3.0)))
     (sqrt (* 0.5 (+ t_0 1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (p_m / -x) - (-1.5 * pow((p_m / x), 3.0));
	} else {
		tmp = sqrt((0.5 * (t_0 + 1.0)));
	}
	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 = x / sqrt(((p_m * (4.0d0 * p_m)) + (x * x)))
    if (t_0 <= (-0.5d0)) then
        tmp = (p_m / -x) - ((-1.5d0) * ((p_m / x) ** 3.0d0))
    else
        tmp = sqrt((0.5d0 * (t_0 + 1.0d0)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (p_m / -x) - (-1.5 * Math.pow((p_m / x), 3.0));
	} else {
		tmp = Math.sqrt((0.5 * (t_0 + 1.0)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))
	tmp = 0
	if t_0 <= -0.5:
		tmp = (p_m / -x) - (-1.5 * math.pow((p_m / x), 3.0))
	else:
		tmp = math.sqrt((0.5 * (t_0 + 1.0)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(p_m / Float64(-x)) - Float64(-1.5 * (Float64(p_m / x) ^ 3.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(t_0 + 1.0)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = (p_m / -x) - (-1.5 * ((p_m / x) ^ 3.0));
	else
		tmp = sqrt((0.5 * (t_0 + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(p$95$m / (-x)), $MachinePrecision] - N[(-1.5 * N[Power[N[(p$95$m / x), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 18.5%

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

      1. +-commutative 18.5%

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

      2. sqr-neg 18.5%

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

      3. associate-*l* 18.5%

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

      4. sqr-neg 18.5%

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

      5. fma-define 18.5%

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

      6. sqr-neg 18.5%

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

      7. fma-define 18.5%

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

      8. associate-*l* 18.5%

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

      9. +-commutative 18.5%

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

   3. Simplified 18.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 43.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 43.4%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-rgt-out 43.4%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}}{x} \]

      3. metadata-eval 43.4%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}}{x} \]

   7. Simplified 43.4%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]

   8. Taylor expanded in p around 0 49.4%

      \[\leadsto -\color{blue}{p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 49.4%

         \[\leadsto -p \cdot \color{blue}{\left(\frac{1}{x} + -1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)} \]

      2. distribute-lft-in 49.4%

         \[\leadsto -\color{blue}{\left(p \cdot \frac{1}{x} + p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)\right)} \]

      3. associate-*r/ 49.5%

         \[\leadsto -\left(\color{blue}{\frac{p \cdot 1}{x}} + p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)\right) \]

      4. *-rgt-identity 49.5%

         \[\leadsto -\left(\frac{\color{blue}{p}}{x} + p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)\right) \]

      5. *-commutative 49.5%

         \[\leadsto -\left(\frac{p}{x} + p \cdot \color{blue}{\left(\frac{{p}^{2}}{{x}^{3}} \cdot -1.5\right)}\right) \]

      6. associate-*r* 49.5%

         \[\leadsto -\left(\frac{p}{x} + \color{blue}{\left(p \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot -1.5}\right) \]

      7. associate-/l* 49.4%

         \[\leadsto -\left(\frac{p}{x} + \color{blue}{\frac{p \cdot {p}^{2}}{{x}^{3}}} \cdot -1.5\right) \]

      8. unpow2 49.4%

         \[\leadsto -\left(\frac{p}{x} + \frac{p \cdot \color{blue}{\left(p \cdot p\right)}}{{x}^{3}} \cdot -1.5\right) \]

      9. cube-mult 49.4%

         \[\leadsto -\left(\frac{p}{x} + \frac{\color{blue}{{p}^{3}}}{{x}^{3}} \cdot -1.5\right) \]

      10. *-commutative 49.4%

          \[\leadsto -\left(\frac{p}{x} + \color{blue}{-1.5 \cdot \frac{{p}^{3}}{{x}^{3}}}\right) \]

      11. cube-div 53.8%

          \[\leadsto -\left(\frac{p}{x} + -1.5 \cdot \color{blue}{{\left(\frac{p}{x}\right)}^{3}}\right) \]

   10. Simplified 53.8%

       \[\leadsto -\color{blue}{\left(\frac{p}{x} + -1.5 \cdot {\left(\frac{p}{x}\right)}^{3}\right)} \]

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) 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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{p}{-x} - -1.5 \cdot {\left(\frac{p}{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 6.3 \cdot 10^{-257}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 4.5 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;t\_0 - -1.5 \cdot {\left(\frac{p\_m}{x}\right)}^{3}\\ \mathbf{elif}\;p\_m \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 6.3e-257)
     t_0
     (if (<= p_m 4.5e-128)
       1.0
       (if (<= p_m 1.05e-80)
         (- t_0 (* -1.5 (pow (/ p_m x) 3.0)))
         (if (<= p_m 4.6e-57) 1.0 (if (<= p_m 4.5e-38) t_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 <= 6.3e-257) {
		tmp = t_0;
	} else if (p_m <= 4.5e-128) {
		tmp = 1.0;
	} else if (p_m <= 1.05e-80) {
		tmp = t_0 - (-1.5 * pow((p_m / x), 3.0));
	} else if (p_m <= 4.6e-57) {
		tmp = 1.0;
	} else if (p_m <= 4.5e-38) {
		tmp = t_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 <= 6.3d-257) then
        tmp = t_0
    else if (p_m <= 4.5d-128) then
        tmp = 1.0d0
    else if (p_m <= 1.05d-80) then
        tmp = t_0 - ((-1.5d0) * ((p_m / x) ** 3.0d0))
    else if (p_m <= 4.6d-57) then
        tmp = 1.0d0
    else if (p_m <= 4.5d-38) then
        tmp = t_0
    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 <= 6.3e-257) {
		tmp = t_0;
	} else if (p_m <= 4.5e-128) {
		tmp = 1.0;
	} else if (p_m <= 1.05e-80) {
		tmp = t_0 - (-1.5 * Math.pow((p_m / x), 3.0));
	} else if (p_m <= 4.6e-57) {
		tmp = 1.0;
	} else if (p_m <= 4.5e-38) {
		tmp = t_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 <= 6.3e-257:
		tmp = t_0
	elif p_m <= 4.5e-128:
		tmp = 1.0
	elif p_m <= 1.05e-80:
		tmp = t_0 - (-1.5 * math.pow((p_m / x), 3.0))
	elif p_m <= 4.6e-57:
		tmp = 1.0
	elif p_m <= 4.5e-38:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 6.3e-257)
		tmp = t_0;
	elseif (p_m <= 4.5e-128)
		tmp = 1.0;
	elseif (p_m <= 1.05e-80)
		tmp = Float64(t_0 - Float64(-1.5 * (Float64(p_m / x) ^ 3.0)));
	elseif (p_m <= 4.6e-57)
		tmp = 1.0;
	elseif (p_m <= 4.5e-38)
		tmp = t_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 <= 6.3e-257)
		tmp = t_0;
	elseif (p_m <= 4.5e-128)
		tmp = 1.0;
	elseif (p_m <= 1.05e-80)
		tmp = t_0 - (-1.5 * ((p_m / x) ^ 3.0));
	elseif (p_m <= 4.6e-57)
		tmp = 1.0;
	elseif (p_m <= 4.5e-38)
		tmp = t_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, 6.3e-257], t$95$0, If[LessEqual[p$95$m, 4.5e-128], 1.0, If[LessEqual[p$95$m, 1.05e-80], N[(t$95$0 - N[(-1.5 * N[Power[N[(p$95$m / x), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 4.6e-57], 1.0, If[LessEqual[p$95$m, 4.5e-38], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 6.29999999999999993e-257 or 4.6e-57 < p < 4.50000000000000009e-38

   1. Initial program 73.9%

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

      1. +-commutative 73.9%

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

      2. sqr-neg 73.9%

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

      3. associate-*l* 73.9%

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

      4. sqr-neg 73.9%

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

      5. fma-define 73.9%

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

      6. sqr-neg 73.9%

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

      7. fma-define 73.9%

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

      8. associate-*l* 73.9%

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

      9. +-commutative 73.9%

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

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 14.3%

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

      1. mul-1-neg 14.3%

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

      2. distribute-neg-frac2 14.3%

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

   7. Simplified 14.3%

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

   if 6.29999999999999993e-257 < p < 4.4999999999999999e-128 or 1.05000000000000001e-80 < p < 4.6e-57

   1. Initial program 72.0%

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

   3. Taylor expanded in x around inf 62.3%

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

   if 4.4999999999999999e-128 < p < 1.05000000000000001e-80

   1. Initial program 36.1%

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

      1. +-commutative 36.1%

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

      2. sqr-neg 36.1%

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

      3. associate-*l* 36.1%

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

      4. sqr-neg 36.1%

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

      5. fma-define 36.1%

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

      6. sqr-neg 36.1%

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

      7. fma-define 36.1%

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

      8. associate-*l* 36.1%

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

      9. +-commutative 36.1%

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

   3. Simplified 36.1%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 67.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 67.1%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-rgt-out 67.1%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}}{x} \]

      3. metadata-eval 67.1%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}}{x} \]

   7. Simplified 67.1%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]

   8. Taylor expanded in p around 0 67.9%

      \[\leadsto -\color{blue}{p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 67.9%

         \[\leadsto -p \cdot \color{blue}{\left(\frac{1}{x} + -1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)} \]

      2. distribute-lft-in 67.9%

         \[\leadsto -\color{blue}{\left(p \cdot \frac{1}{x} + p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)\right)} \]

      3. associate-*r/ 68.3%

         \[\leadsto -\left(\color{blue}{\frac{p \cdot 1}{x}} + p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)\right) \]

      4. *-rgt-identity 68.3%

         \[\leadsto -\left(\frac{\color{blue}{p}}{x} + p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)\right) \]

      5. *-commutative 68.3%

         \[\leadsto -\left(\frac{p}{x} + p \cdot \color{blue}{\left(\frac{{p}^{2}}{{x}^{3}} \cdot -1.5\right)}\right) \]

      6. associate-*r* 68.3%

         \[\leadsto -\left(\frac{p}{x} + \color{blue}{\left(p \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot -1.5}\right) \]

      7. associate-/l* 68.3%

         \[\leadsto -\left(\frac{p}{x} + \color{blue}{\frac{p \cdot {p}^{2}}{{x}^{3}}} \cdot -1.5\right) \]

      8. unpow2 68.3%

         \[\leadsto -\left(\frac{p}{x} + \frac{p \cdot \color{blue}{\left(p \cdot p\right)}}{{x}^{3}} \cdot -1.5\right) \]

      9. cube-mult 68.3%

         \[\leadsto -\left(\frac{p}{x} + \frac{\color{blue}{{p}^{3}}}{{x}^{3}} \cdot -1.5\right) \]

      10. *-commutative 68.3%

          \[\leadsto -\left(\frac{p}{x} + \color{blue}{-1.5 \cdot \frac{{p}^{3}}{{x}^{3}}}\right) \]

      11. cube-div 68.5%

          \[\leadsto -\left(\frac{p}{x} + -1.5 \cdot \color{blue}{{\left(\frac{p}{x}\right)}^{3}}\right) \]

   10. Simplified 68.5%

       \[\leadsto -\color{blue}{\left(\frac{p}{x} + -1.5 \cdot {\left(\frac{p}{x}\right)}^{3}\right)} \]

   if 4.50000000000000009e-38 < p

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0 88.5%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6.3 \cdot 10^{-257}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 4.5 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{p}{-x} - -1.5 \cdot {\left(\frac{p}{x}\right)}^{3}\\ \mathbf{elif}\;p \leq 4.6 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 1.25 \cdot 10^{-256}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 7.3 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.65 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 1.25e-256)
     t_0
     (if (<= p_m 7.3e-128)
       1.0
       (if (<= p_m 2.1e-81)
         t_0
         (if (<= p_m 1.65e-59) 1.0 (if (<= p_m 4.5e-38) t_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-256) {
		tmp = t_0;
	} else if (p_m <= 7.3e-128) {
		tmp = 1.0;
	} else if (p_m <= 2.1e-81) {
		tmp = t_0;
	} else if (p_m <= 1.65e-59) {
		tmp = 1.0;
	} else if (p_m <= 4.5e-38) {
		tmp = t_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-256) then
        tmp = t_0
    else if (p_m <= 7.3d-128) then
        tmp = 1.0d0
    else if (p_m <= 2.1d-81) then
        tmp = t_0
    else if (p_m <= 1.65d-59) then
        tmp = 1.0d0
    else if (p_m <= 4.5d-38) then
        tmp = t_0
    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-256) {
		tmp = t_0;
	} else if (p_m <= 7.3e-128) {
		tmp = 1.0;
	} else if (p_m <= 2.1e-81) {
		tmp = t_0;
	} else if (p_m <= 1.65e-59) {
		tmp = 1.0;
	} else if (p_m <= 4.5e-38) {
		tmp = t_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-256:
		tmp = t_0
	elif p_m <= 7.3e-128:
		tmp = 1.0
	elif p_m <= 2.1e-81:
		tmp = t_0
	elif p_m <= 1.65e-59:
		tmp = 1.0
	elif p_m <= 4.5e-38:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 1.25e-256)
		tmp = t_0;
	elseif (p_m <= 7.3e-128)
		tmp = 1.0;
	elseif (p_m <= 2.1e-81)
		tmp = t_0;
	elseif (p_m <= 1.65e-59)
		tmp = 1.0;
	elseif (p_m <= 4.5e-38)
		tmp = t_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-256)
		tmp = t_0;
	elseif (p_m <= 7.3e-128)
		tmp = 1.0;
	elseif (p_m <= 2.1e-81)
		tmp = t_0;
	elseif (p_m <= 1.65e-59)
		tmp = 1.0;
	elseif (p_m <= 4.5e-38)
		tmp = t_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-256], t$95$0, If[LessEqual[p$95$m, 7.3e-128], 1.0, If[LessEqual[p$95$m, 2.1e-81], t$95$0, If[LessEqual[p$95$m, 1.65e-59], 1.0, If[LessEqual[p$95$m, 4.5e-38], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.25e-256 or 7.29999999999999972e-128 < p < 2.0999999999999999e-81 or 1.64999999999999991e-59 < p < 4.50000000000000009e-38

   1. Initial program 71.7%

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

      1. +-commutative 71.7%

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

      2. sqr-neg 71.7%

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

      3. associate-*l* 71.7%

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

      4. sqr-neg 71.7%

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

      5. fma-define 71.7%

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

      6. sqr-neg 71.7%

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

      7. fma-define 71.7%

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

      8. associate-*l* 71.7%

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

      9. +-commutative 71.7%

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

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   7. Simplified 17.3%

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

   if 1.25e-256 < p < 7.29999999999999972e-128 or 2.0999999999999999e-81 < p < 1.64999999999999991e-59

   1. Initial program 72.0%

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

   3. Taylor expanded in x around inf 62.3%

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

   if 4.50000000000000009e-38 < p

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0 88.5%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.25 \cdot 10^{-256}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 7.3 \cdot 10^{-128}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.65 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 6 \cdot 10^{-81} \lor \neg \left(p\_m \leq 6.5 \cdot 10^{-68}\right) \land p\_m \leq 8 \cdot 10^{-38}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (or (<= p_m 6e-81) (and (not (<= p_m 6.5e-68)) (<= p_m 8e-38)))
   (/ p_m (- x))
   (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((p_m <= 6e-81) || (!(p_m <= 6.5e-68) && (p_m <= 8e-38))) {
		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 <= 6d-81) .or. (.not. (p_m <= 6.5d-68)) .and. (p_m <= 8d-38)) 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 <= 6e-81) || (!(p_m <= 6.5e-68) && (p_m <= 8e-38))) {
		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 <= 6e-81) or (not (p_m <= 6.5e-68) and (p_m <= 8e-38)):
		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 <= 6e-81) || (!(p_m <= 6.5e-68) && (p_m <= 8e-38)))
		tmp = Float64(p_m / Float64(-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 <= 6e-81) || (~((p_m <= 6.5e-68)) && (p_m <= 8e-38)))
		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[Or[LessEqual[p$95$m, 6e-81], And[N[Not[LessEqual[p$95$m, 6.5e-68]], $MachinePrecision], LessEqual[p$95$m, 8e-38]]], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 5.9999999999999998e-81 or 6.4999999999999997e-68 < p < 7.9999999999999997e-38

   1. Initial program 71.0%

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

      1. +-commutative 71.0%

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

      2. sqr-neg 71.0%

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

      3. associate-*l* 71.0%

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

      4. sqr-neg 71.0%

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

      5. fma-define 71.0%

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

      6. sqr-neg 71.0%

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

      7. fma-define 71.0%

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

      8. associate-*l* 71.0%

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

      9. +-commutative 71.0%

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

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 21.0%

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

      1. mul-1-neg 21.0%

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

      2. distribute-neg-frac2 21.0%

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

   7. Simplified 21.0%

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

   if 5.9999999999999998e-81 < p < 6.4999999999999997e-68 or 7.9999999999999997e-38 < p

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0 86.0%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6 \cdot 10^{-81} \lor \neg \left(p \leq 6.5 \cdot 10^{-68}\right) \land p \leq 8 \cdot 10^{-38}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.0% accurate, 53.8× 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 / Float64(-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

1. Initial program 78.0%

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

   1. +-commutative 78.0%

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

   2. sqr-neg 78.0%

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

   3. associate-*l* 78.0%

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

   4. sqr-neg 78.0%

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

   5. fma-define 78.0%

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

   6. sqr-neg 78.0%

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

   7. fma-define 78.0%

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

   8. associate-*l* 78.0%

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

   9. +-commutative 78.0%

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

3. Simplified 78.0%

   \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
4. Add Preprocessing

5. Taylor expanded in x around -inf 17.0%

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

   1. mul-1-neg 17.0%

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

   2. distribute-neg-frac2 17.0%

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

7. Simplified 17.0%

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

Developer target: 79.3% 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 2024102 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (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))))))))