Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.8%

Time: 8.7s

Alternatives: 6

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;1.5 \cdot {\left(\frac{p\_m}{x}\right)}^{3} - \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.5)
   (- (* 1.5 (pow (/ p_m x) 3.0)) (/ p_m x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p_m 2.0) x)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5) {
		tmp = (1.5 * pow((p_m / x), 3.0)) - (p_m / x);
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5) {
		tmp = (1.5 * Math.pow((p_m / x), 3.0)) - (p_m / x);
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p_m * 2.0), x)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5:
		tmp = (1.5 * math.pow((p_m / x), 3.0)) - (p_m / x)
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p_m * 2.0), x)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.5)
		tmp = Float64(Float64(1.5 * (Float64(p_m / x) ^ 3.0)) - Float64(p_m / x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5)
		tmp = (1.5 * ((p_m / x) ^ 3.0)) - (p_m / x);
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p_m * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(1.5 * N[Power[N[(p$95$m / x), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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 17.4%

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

      1. add-sqr-sqrt 17.4%

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

      2. hypot-def 17.4%

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

      3. associate-*l* 17.4%

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

      4. sqrt-prod 17.4%

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

      5. metadata-eval 17.4%

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

      6. sqrt-unprod 10.1%

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

      7. add-sqr-sqrt 17.4%

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

   4. Applied egg-rr 17.4%

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

   5. Taylor expanded in x around -inf 46.8%

      \[\leadsto \sqrt{\color{blue}{0.5 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{{x}^{4}} + \left(0.5 \cdot \frac{-2 \cdot \left({p}^{2} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right) + 8 \cdot {p}^{6}}{{x}^{6}} + \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. fma-def 46.8%

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{{x}^{4}}, 0.5 \cdot \frac{-2 \cdot \left({p}^{2} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right) + 8 \cdot {p}^{6}}{{x}^{6}} + \frac{{p}^{2}}{{x}^{2}}\right)}} \]

      2. distribute-rgt-out 46.8%

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, \frac{\color{blue}{{p}^{4} \cdot \left(-4 + -2\right)}}{{x}^{4}}, 0.5 \cdot \frac{-2 \cdot \left({p}^{2} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right) + 8 \cdot {p}^{6}}{{x}^{6}} + \frac{{p}^{2}}{{x}^{2}}\right)} \]

      3. metadata-eval 46.8%

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, \frac{{p}^{4} \cdot \color{blue}{-6}}{{x}^{4}}, 0.5 \cdot \frac{-2 \cdot \left({p}^{2} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right) + 8 \cdot {p}^{6}}{{x}^{6}} + \frac{{p}^{2}}{{x}^{2}}\right)} \]

      4. fma-def 46.8%

         \[\leadsto \sqrt{\mathsf{fma}\left(0.5, \frac{{p}^{4} \cdot -6}{{x}^{4}}, \color{blue}{\mathsf{fma}\left(0.5, \frac{-2 \cdot \left({p}^{2} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right) + 8 \cdot {p}^{6}}{{x}^{6}}, \frac{{p}^{2}}{{x}^{2}}\right)}\right)} \]

   7. Simplified 46.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, \frac{{p}^{4} \cdot -6}{{x}^{4}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, {p}^{6} \cdot -6, 8 \cdot {p}^{6}\right)}{{x}^{6}}, {\left(\frac{p}{x}\right)}^{2}\right)\right)}} \]

   8. Taylor expanded in x around -inf 51.5%

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

      1. neg-mul-1 51.5%

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

      2. +-commutative 51.5%

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

      3. unsub-neg 51.5%

         \[\leadsto \color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{3}} - \frac{p}{x}} \]

      4. cube-div 58.7%

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

   10. Simplified 58.7%

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-def 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 59.4%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;1.5 \cdot {\left(\frac{p}{x}\right)}^{3} - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := p\_m \cdot \left|\frac{1}{x}\right|\\ \mathbf{if}\;p\_m \leq 4.5 \cdot 10^{-265}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.52 \cdot 10^{-203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 9.5 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* p_m (fabs (/ 1.0 x)))))
   (if (<= p_m 4.5e-265)
     1.0
     (if (<= p_m 1.52e-203)
       t_0
       (if (<= p_m 1.5e-182) 1.0 (if (<= p_m 9.5e-42) t_0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m * fabs((1.0 / x));
	double tmp;
	if (p_m <= 4.5e-265) {
		tmp = 1.0;
	} else if (p_m <= 1.52e-203) {
		tmp = t_0;
	} else if (p_m <= 1.5e-182) {
		tmp = 1.0;
	} else if (p_m <= 9.5e-42) {
		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 * abs((1.0d0 / x))
    if (p_m <= 4.5d-265) then
        tmp = 1.0d0
    else if (p_m <= 1.52d-203) then
        tmp = t_0
    else if (p_m <= 1.5d-182) then
        tmp = 1.0d0
    else if (p_m <= 9.5d-42) 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 * Math.abs((1.0 / x));
	double tmp;
	if (p_m <= 4.5e-265) {
		tmp = 1.0;
	} else if (p_m <= 1.52e-203) {
		tmp = t_0;
	} else if (p_m <= 1.5e-182) {
		tmp = 1.0;
	} else if (p_m <= 9.5e-42) {
		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 * math.fabs((1.0 / x))
	tmp = 0
	if p_m <= 4.5e-265:
		tmp = 1.0
	elif p_m <= 1.52e-203:
		tmp = t_0
	elif p_m <= 1.5e-182:
		tmp = 1.0
	elif p_m <= 9.5e-42:
		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 * abs(Float64(1.0 / x)))
	tmp = 0.0
	if (p_m <= 4.5e-265)
		tmp = 1.0;
	elseif (p_m <= 1.52e-203)
		tmp = t_0;
	elseif (p_m <= 1.5e-182)
		tmp = 1.0;
	elseif (p_m <= 9.5e-42)
		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 * abs((1.0 / x));
	tmp = 0.0;
	if (p_m <= 4.5e-265)
		tmp = 1.0;
	elseif (p_m <= 1.52e-203)
		tmp = t_0;
	elseif (p_m <= 1.5e-182)
		tmp = 1.0;
	elseif (p_m <= 9.5e-42)
		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 * N[Abs[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 4.5e-265], 1.0, If[LessEqual[p$95$m, 1.52e-203], t$95$0, If[LessEqual[p$95$m, 1.5e-182], 1.0, If[LessEqual[p$95$m, 9.5e-42], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.5000000000000003e-265 or 1.51999999999999993e-203 < p < 1.5000000000000001e-182

   1. Initial program 75.3%

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

   3. Taylor expanded in x around inf 41.3%

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

   if 4.5000000000000003e-265 < p < 1.51999999999999993e-203 or 1.5000000000000001e-182 < p < 9.49999999999999948e-42

   1. Initial program 52.1%

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

   3. Taylor expanded in x around -inf 35.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]
   4. Step-by-step derivation

      1. pow1/2 35.0%

         \[\leadsto \color{blue}{{\left(0.5 \cdot \left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)\right)}^{0.5}} \]

      2. associate-*r* 35.0%

         \[\leadsto {\color{blue}{\left(\left(0.5 \cdot 2\right) \cdot \frac{{p}^{2}}{{x}^{2}}\right)}}^{0.5} \]

      3. metadata-eval 35.0%

         \[\leadsto {\left(\color{blue}{1} \cdot \frac{{p}^{2}}{{x}^{2}}\right)}^{0.5} \]

      4. *-un-lft-identity 35.0%

         \[\leadsto {\color{blue}{\left(\frac{{p}^{2}}{{x}^{2}}\right)}}^{0.5} \]

      5. div-inv 35.0%

         \[\leadsto {\color{blue}{\left({p}^{2} \cdot \frac{1}{{x}^{2}}\right)}}^{0.5} \]

      6. unpow-prod-down 39.5%

         \[\leadsto \color{blue}{{\left({p}^{2}\right)}^{0.5} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.5}} \]

      7. pow1/2 39.5%

         \[\leadsto \color{blue}{\sqrt{{p}^{2}}} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.5} \]

      8. unpow2 39.5%

         \[\leadsto \sqrt{\color{blue}{p \cdot p}} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.5} \]

      9. sqrt-prod 59.7%

         \[\leadsto \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.5} \]

      10. add-sqr-sqrt 60.0%

          \[\leadsto \color{blue}{p} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.5} \]

      11. pow-flip 60.0%

          \[\leadsto p \cdot {\color{blue}{\left({x}^{\left(-2\right)}\right)}}^{0.5} \]

      12. metadata-eval 60.0%

          \[\leadsto p \cdot {\left({x}^{\color{blue}{-2}}\right)}^{0.5} \]

   5. Applied egg-rr 60.0%

      \[\leadsto \color{blue}{p \cdot {\left({x}^{-2}\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/2 60.0%

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

      2. metadata-eval 60.0%

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

      3. pow-sqr 59.9%

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

      4. unpow-1 59.9%

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

      5. unpow-1 59.9%

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

      6. rem-sqrt-square 59.9%

         \[\leadsto p \cdot \color{blue}{\left|\frac{1}{x}\right|} \]

   7. Simplified 59.9%

      \[\leadsto \color{blue}{p \cdot \left|\frac{1}{x}\right|} \]

   if 9.49999999999999948e-42 < p

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0 87.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.5 \cdot 10^{-265}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.52 \cdot 10^{-203}:\\ \;\;\;\;p \cdot \left|\frac{1}{x}\right|\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-42}:\\ \;\;\;\;p \cdot \left|\frac{1}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p\_m}{x}\\ \mathbf{if}\;p\_m \leq 4.2 \cdot 10^{-252}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.52 \cdot 10^{-203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.9 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 9.2 \cdot 10^{-42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 4.2e-252)
     1.0
     (if (<= p_m 1.52e-203)
       t_0
       (if (<= p_m 1.9e-182) 1.0 (if (<= p_m 9.2e-42) 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 <= 4.2e-252) {
		tmp = 1.0;
	} else if (p_m <= 1.52e-203) {
		tmp = t_0;
	} else if (p_m <= 1.9e-182) {
		tmp = 1.0;
	} else if (p_m <= 9.2e-42) {
		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 <= 4.2d-252) then
        tmp = 1.0d0
    else if (p_m <= 1.52d-203) then
        tmp = t_0
    else if (p_m <= 1.9d-182) then
        tmp = 1.0d0
    else if (p_m <= 9.2d-42) 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 <= 4.2e-252) {
		tmp = 1.0;
	} else if (p_m <= 1.52e-203) {
		tmp = t_0;
	} else if (p_m <= 1.9e-182) {
		tmp = 1.0;
	} else if (p_m <= 9.2e-42) {
		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 <= 4.2e-252:
		tmp = 1.0
	elif p_m <= 1.52e-203:
		tmp = t_0
	elif p_m <= 1.9e-182:
		tmp = 1.0
	elif p_m <= 9.2e-42:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 4.2e-252)
		tmp = 1.0;
	elseif (p_m <= 1.52e-203)
		tmp = t_0;
	elseif (p_m <= 1.9e-182)
		tmp = 1.0;
	elseif (p_m <= 9.2e-42)
		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 <= 4.2e-252)
		tmp = 1.0;
	elseif (p_m <= 1.52e-203)
		tmp = t_0;
	elseif (p_m <= 1.9e-182)
		tmp = 1.0;
	elseif (p_m <= 9.2e-42)
		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, 4.2e-252], 1.0, If[LessEqual[p$95$m, 1.52e-203], t$95$0, If[LessEqual[p$95$m, 1.9e-182], 1.0, If[LessEqual[p$95$m, 9.2e-42], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.2e-252 or 1.51999999999999993e-203 < p < 1.9000000000000002e-182

   1. Initial program 75.4%

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

   3. Taylor expanded in x around inf 41.8%

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

   if 4.2e-252 < p < 1.51999999999999993e-203 or 1.9000000000000002e-182 < p < 9.20000000000000015e-42

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf 35.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

   4. Taylor expanded in p around -inf 60.1%

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

      1. associate-*r/ 60.1%

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

      2. mul-1-neg 60.1%

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

   6. Simplified 60.1%

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

   if 9.20000000000000015e-42 < p

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0 87.5%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.2 \cdot 10^{-252}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.52 \cdot 10^{-203}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-182}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.15e-41) (/ (- p_m) x) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.15e-41) {
		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 <= 1.15d-41) 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 <= 1.15e-41) {
		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 <= 1.15e-41:
		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 <= 1.15e-41)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.15e-41)
		tmp = -p_m / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.15e-41], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.15000000000000005e-41

   1. Initial program 69.7%

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

   3. Taylor expanded in x around -inf 21.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

   4. Taylor expanded in p around -inf 23.0%

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

      1. associate-*r/ 23.0%

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

      2. mul-1-neg 23.0%

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

   6. Simplified 23.0%

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

   if 1.15000000000000005e-41 < p

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0 87.5%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.15 \cdot 10^{-41}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.9% accurate, 23.8× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -2e-310) (/ (- p_m) x) (/ p_m x)))

C

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

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = -p_m / x
    else
        tmp = p_m / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2e-310], N[((-p$95$m) / x), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 56.5%

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

   3. Taylor expanded in x around -inf 30.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

   4. Taylor expanded in p around -inf 32.6%

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

      1. associate-*r/ 32.6%

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

      2. mul-1-neg 32.6%

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

   6. Simplified 32.6%

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

   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. Add Preprocessing

   3. Taylor expanded in x around -inf 4.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

   4. Taylor expanded in p around 0 3.6%

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

4. Final simplification 18.4%

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

Alternative 6: 6.7% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around -inf 18.1%

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

4. Taylor expanded in p around 0 17.6%

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

5. Final simplification 17.6%

   \[\leadsto \frac{p}{x} \]
6. Add Preprocessing

Developer target: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(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))))))))