Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.7%

Time: 7.3s

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: 79.7% 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

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

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)
   (/ (- 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 = -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 = -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 = -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(-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 = -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[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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 16.5%

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

   3. Taylor expanded in x around -inf 45.9%

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

   4. Taylor expanded in p around -inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. mul-1-neg 52.5%

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

   6. Simplified 52.5%

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

   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. 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 48.0%

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

      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%

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

Alternative 2: 69.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p\_m}{x}\\ \mathbf{if}\;p\_m \leq 7.8 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 2.8 \cdot 10^{-262}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.3 \cdot 10^{-184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.56 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.6 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.46 \cdot 10^{-58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 7.8e-305)
     t_0
     (if (<= p_m 2.8e-262)
       1.0
       (if (<= p_m 3.3e-184)
         t_0
         (if (<= p_m 1.56e-132)
           1.0
           (if (<= p_m 1.6e-115)
             t_0
             (if (<= p_m 1.46e-58) 1.0 (sqrt 0.5)))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 7.8e-305) {
		tmp = t_0;
	} else if (p_m <= 2.8e-262) {
		tmp = 1.0;
	} else if (p_m <= 3.3e-184) {
		tmp = t_0;
	} else if (p_m <= 1.56e-132) {
		tmp = 1.0;
	} else if (p_m <= 1.6e-115) {
		tmp = t_0;
	} else if (p_m <= 1.46e-58) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 7.8d-305) then
        tmp = t_0
    else if (p_m <= 2.8d-262) then
        tmp = 1.0d0
    else if (p_m <= 3.3d-184) then
        tmp = t_0
    else if (p_m <= 1.56d-132) then
        tmp = 1.0d0
    else if (p_m <= 1.6d-115) then
        tmp = t_0
    else if (p_m <= 1.46d-58) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 7.8e-305) {
		tmp = t_0;
	} else if (p_m <= 2.8e-262) {
		tmp = 1.0;
	} else if (p_m <= 3.3e-184) {
		tmp = t_0;
	} else if (p_m <= 1.56e-132) {
		tmp = 1.0;
	} else if (p_m <= 1.6e-115) {
		tmp = t_0;
	} else if (p_m <= 1.46e-58) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 7.8e-305:
		tmp = t_0
	elif p_m <= 2.8e-262:
		tmp = 1.0
	elif p_m <= 3.3e-184:
		tmp = t_0
	elif p_m <= 1.56e-132:
		tmp = 1.0
	elif p_m <= 1.6e-115:
		tmp = t_0
	elif p_m <= 1.46e-58:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 7.8e-305)
		tmp = t_0;
	elseif (p_m <= 2.8e-262)
		tmp = 1.0;
	elseif (p_m <= 3.3e-184)
		tmp = t_0;
	elseif (p_m <= 1.56e-132)
		tmp = 1.0;
	elseif (p_m <= 1.6e-115)
		tmp = t_0;
	elseif (p_m <= 1.46e-58)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 7.8e-305)
		tmp = t_0;
	elseif (p_m <= 2.8e-262)
		tmp = 1.0;
	elseif (p_m <= 3.3e-184)
		tmp = t_0;
	elseif (p_m <= 1.56e-132)
		tmp = 1.0;
	elseif (p_m <= 1.6e-115)
		tmp = t_0;
	elseif (p_m <= 1.46e-58)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 7.8e-305], t$95$0, If[LessEqual[p$95$m, 2.8e-262], 1.0, If[LessEqual[p$95$m, 3.3e-184], t$95$0, If[LessEqual[p$95$m, 1.56e-132], 1.0, If[LessEqual[p$95$m, 1.6e-115], t$95$0, If[LessEqual[p$95$m, 1.46e-58], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.8000000000000005e-305 or 2.79999999999999975e-262 < p < 3.2999999999999997e-184 or 1.56000000000000012e-132 < p < 1.6e-115

   1. Initial program 73.3%

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

   3. Taylor expanded in x around -inf 15.2%

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

   4. Taylor expanded in p around -inf 15.2%

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

      1. associate-*r/ 15.2%

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

      2. mul-1-neg 15.2%

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

   6. Simplified 15.2%

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

   if 7.8000000000000005e-305 < p < 2.79999999999999975e-262 or 3.2999999999999997e-184 < p < 1.56000000000000012e-132 or 1.6e-115 < p < 1.4600000000000001e-58

   1. Initial program 84.7%

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

   3. Taylor expanded in x around inf 66.2%

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

   if 1.4600000000000001e-58 < p

   1. Initial program 94.5%

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

   3. Taylor expanded in x around 0 89.0%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-262}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.56 \cdot 10^{-132}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.46 \cdot 10^{-58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.05 \cdot 10^{-63}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.05e-63) (/ (- p_m) x) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.05e-63) {
		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 <= 2.05d-63) 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 <= 2.05e-63) {
		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 <= 2.05e-63:
		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 <= 2.05e-63)
		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 <= 2.05e-63)
		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, 2.05e-63], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 2.0499999999999999e-63

   1. Initial program 74.9%

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

   3. Taylor expanded in x around -inf 16.3%

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

   4. Taylor expanded in p around -inf 17.5%

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

      1. associate-*r/ 17.5%

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

      2. mul-1-neg 17.5%

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

   6. Simplified 17.5%

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

   if 2.0499999999999999e-63 < p

   1. Initial program 94.7%

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

   3. Taylor expanded in x around 0 87.0%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.05 \cdot 10^{-63}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 28.3% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 61.8%

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

   3. Taylor expanded in x around -inf 23.5%

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

   4. Taylor expanded in p around -inf 25.5%

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

      1. associate-*r/ 25.5%

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

      2. mul-1-neg 25.5%

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

   6. Simplified 25.5%

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

   if -4.999999999999985e-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 5.0%

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

   4. Taylor expanded in p around 0 3.5%

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

4. Final simplification 14.8%

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

Alternative 5: 6.4% 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 80.4%

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

3. Taylor expanded in x around -inf 14.4%

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

4. Taylor expanded in p around 0 17.1%

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

5. Final simplification 17.1%

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

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 2024031 
(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))))))))