Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.8%

Time: 10.6s

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: 79.3% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 24.4%

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

   3. Taylor expanded in x around -inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. associate-/l* 60.9%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 60.9%

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

      4. associate-/l* 61.2%

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

   5. Simplified 61.2%

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

      1. distribute-rgt-neg-out 61.2%

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

      2. neg-sub0 61.2%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 60.9%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 61.4%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 61.4%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 61.4%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 61.7%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 61.7%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 61.7%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 61.7%

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

      2. distribute-neg-frac 61.7%

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

   9. Simplified 61.7%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. hypot-define 99.8%

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

      3. associate-*l* 99.8%

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

      4. sqrt-prod 99.8%

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

      5. metadata-eval 99.8%

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

      6. sqrt-unprod 47.6%

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

      7. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 92.4%

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

Alternative 2: 68.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 1.3 \cdot 10^{-248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 9.5 \cdot 10^{-104}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 8.2 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 1.3e-248)
     t_0
     (if (<= p_m 9.5e-104)
       1.0
       (if (<= p_m 8.2e-84)
         t_0
         (if (<= p_m 2.75e-30) 1.0 (sqrt (+ 0.5 (/ (* x 0.25) p_m)))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 1.3e-248) {
		tmp = t_0;
	} else if (p_m <= 9.5e-104) {
		tmp = 1.0;
	} else if (p_m <= 8.2e-84) {
		tmp = t_0;
	} else if (p_m <= 2.75e-30) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 1.3d-248) then
        tmp = t_0
    else if (p_m <= 9.5d-104) then
        tmp = 1.0d0
    else if (p_m <= 8.2d-84) then
        tmp = t_0
    else if (p_m <= 2.75d-30) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 1.3e-248) {
		tmp = t_0;
	} else if (p_m <= 9.5e-104) {
		tmp = 1.0;
	} else if (p_m <= 8.2e-84) {
		tmp = t_0;
	} else if (p_m <= 2.75e-30) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 1.3e-248:
		tmp = t_0
	elif p_m <= 9.5e-104:
		tmp = 1.0
	elif p_m <= 8.2e-84:
		tmp = t_0
	elif p_m <= 2.75e-30:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 1.3e-248)
		tmp = t_0;
	elseif (p_m <= 9.5e-104)
		tmp = 1.0;
	elseif (p_m <= 8.2e-84)
		tmp = t_0;
	elseif (p_m <= 2.75e-30)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 1.3e-248)
		tmp = t_0;
	elseif (p_m <= 9.5e-104)
		tmp = 1.0;
	elseif (p_m <= 8.2e-84)
		tmp = t_0;
	elseif (p_m <= 2.75e-30)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 1.3e-248], t$95$0, If[LessEqual[p$95$m, 9.5e-104], 1.0, If[LessEqual[p$95$m, 8.2e-84], t$95$0, If[LessEqual[p$95$m, 2.75e-30], 1.0, N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.30000000000000003e-248 or 9.5000000000000002e-104 < p < 8.2000000000000001e-84

   1. Initial program 81.6%

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

   3. Taylor expanded in x around -inf 15.8%

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

      1. mul-1-neg 15.8%

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

      2. associate-/l* 15.8%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 15.8%

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

      4. associate-/l* 15.8%

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

   5. Simplified 15.8%

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

      1. distribute-rgt-neg-out 15.8%

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

      2. neg-sub0 15.8%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 15.8%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 15.9%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 15.9%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 15.9%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 15.9%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 15.9%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 15.9%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 15.9%

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

      2. distribute-neg-frac 15.9%

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

   9. Simplified 15.9%

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

   if 1.30000000000000003e-248 < p < 9.5000000000000002e-104 or 8.2000000000000001e-84 < p < 2.74999999999999988e-30

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf 65.8%

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

   if 2.74999999999999988e-30 < p

   1. Initial program 97.2%

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

      1. add-sqr-sqrt 97.2%

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

      2. hypot-define 97.2%

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

      3. associate-*l* 97.2%

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

      4. sqrt-prod 97.2%

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

      5. metadata-eval 97.2%

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

      6. sqrt-unprod 97.2%

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

      7. add-sqr-sqrt 97.2%

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

   4. Applied egg-rr 97.2%

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

   5. Taylor expanded in x around 0 90.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(1 + 0.5 \cdot \frac{x}{p}\right)}} \]
   6. Step-by-step derivation

      1. associate-*r/ 90.6%

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

   7. Simplified 90.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(1 + \frac{0.5 \cdot x}{p}\right)}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 90.6%

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

      2. distribute-lft-in 90.6%

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

      3. metadata-eval 90.6%

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

      4. associate-/l* 90.6%

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

      5. associate-*r* 90.6%

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

      6. metadata-eval 90.6%

         \[\leadsto 1 \cdot \sqrt{0.5 + \color{blue}{0.25} \cdot \frac{x}{p}} \]

   9. Applied egg-rr 90.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{0.5 + 0.25 \cdot \frac{x}{p}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 90.6%

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

       2. associate-*r/ 90.6%

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

   11. Simplified 90.6%

       \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.25 \cdot x}{p}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-104}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 9.2 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.6 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 2 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 9.2e-250)
     t_0
     (if (<= p_m 3.3e-105)
       1.0
       (if (<= p_m 3.6e-84) t_0 (if (<= p_m 2e-30) 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 <= 9.2e-250) {
		tmp = t_0;
	} else if (p_m <= 3.3e-105) {
		tmp = 1.0;
	} else if (p_m <= 3.6e-84) {
		tmp = t_0;
	} else if (p_m <= 2e-30) {
		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 <= 9.2d-250) then
        tmp = t_0
    else if (p_m <= 3.3d-105) then
        tmp = 1.0d0
    else if (p_m <= 3.6d-84) then
        tmp = t_0
    else if (p_m <= 2d-30) 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 <= 9.2e-250) {
		tmp = t_0;
	} else if (p_m <= 3.3e-105) {
		tmp = 1.0;
	} else if (p_m <= 3.6e-84) {
		tmp = t_0;
	} else if (p_m <= 2e-30) {
		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 <= 9.2e-250:
		tmp = t_0
	elif p_m <= 3.3e-105:
		tmp = 1.0
	elif p_m <= 3.6e-84:
		tmp = t_0
	elif p_m <= 2e-30:
		tmp = 1.0
	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 <= 9.2e-250)
		tmp = t_0;
	elseif (p_m <= 3.3e-105)
		tmp = 1.0;
	elseif (p_m <= 3.6e-84)
		tmp = t_0;
	elseif (p_m <= 2e-30)
		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 <= 9.2e-250)
		tmp = t_0;
	elseif (p_m <= 3.3e-105)
		tmp = 1.0;
	elseif (p_m <= 3.6e-84)
		tmp = t_0;
	elseif (p_m <= 2e-30)
		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, 9.2e-250], t$95$0, If[LessEqual[p$95$m, 3.3e-105], 1.0, If[LessEqual[p$95$m, 3.6e-84], t$95$0, If[LessEqual[p$95$m, 2e-30], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 9.1999999999999998e-250 or 3.2999999999999999e-105 < p < 3.60000000000000003e-84

   1. Initial program 81.6%

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

   3. Taylor expanded in x around -inf 15.8%

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

      1. mul-1-neg 15.8%

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

      2. associate-/l* 15.8%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 15.8%

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

      4. associate-/l* 15.8%

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

   5. Simplified 15.8%

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

      1. distribute-rgt-neg-out 15.8%

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

      2. neg-sub0 15.8%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 15.8%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 15.9%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 15.9%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 15.9%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 15.9%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 15.9%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 15.9%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 15.9%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 15.9%

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

      2. distribute-neg-frac 15.9%

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

   9. Simplified 15.9%

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

   if 9.1999999999999998e-250 < p < 3.2999999999999999e-105 or 3.60000000000000003e-84 < p < 2e-30

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf 65.8%

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

   if 2e-30 < p

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0 89.6%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 9.2 \cdot 10^{-250}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.3 \cdot 10^{-105}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.15e-61) (/ 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-61) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.15d-61) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.15e-61) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.15e-61:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.15e-61)
		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 <= 1.15e-61)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.15e-61], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.14999999999999996e-61

   1. Initial program 81.4%

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

   3. Taylor expanded in x around -inf 17.2%

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

      1. mul-1-neg 17.2%

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

      2. associate-/l* 17.2%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 17.2%

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

      4. associate-/l* 17.3%

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

   5. Simplified 17.3%

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

      1. distribute-rgt-neg-out 17.3%

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

      2. neg-sub0 17.3%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 17.2%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 17.3%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 17.3%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 17.3%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 17.4%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 17.4%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 17.4%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 17.4%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 17.4%

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

      2. distribute-neg-frac 17.4%

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

   9. Simplified 17.4%

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

   if 1.14999999999999996e-61 < p

   1. Initial program 93.8%

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

   3. Taylor expanded in x around 0 84.6%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.15 \cdot 10^{-61}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.0% 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(p_m / Float64(-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 69.8%

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

   3. Taylor expanded in x around -inf 26.2%

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

      1. mul-1-neg 26.2%

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

      2. associate-/l* 26.2%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 26.2%

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

      4. associate-/l* 26.3%

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

   5. Simplified 26.3%

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

      1. distribute-rgt-neg-out 26.3%

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

      2. neg-sub0 26.3%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 26.2%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 26.4%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 26.4%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 26.4%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 26.5%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 26.5%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 26.5%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 26.5%

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

      2. distribute-neg-frac 26.5%

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

   9. Simplified 26.5%

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

   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 4.0%

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

      1. mul-1-neg 4.0%

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

      2. associate-/l* 4.0%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 4.0%

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

      4. associate-/l* 4.0%

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

   5. Simplified 4.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 3.3%

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

      3. sqr-neg 3.3%

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

      4. sqrt-unprod 3.3%

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

      5. add-sqr-sqrt 3.3%

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

      6. associate-*r/ 3.3%

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

      7. sqrt-unprod 3.3%

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

      8. metadata-eval 3.3%

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

      9. metadata-eval 3.3%

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

      10. associate-*r/ 3.3%

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

      11. *-commutative 3.3%

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

      12. *-un-lft-identity 3.3%

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

   7. Applied egg-rr 3.3%

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

4. Final simplification 14.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]
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 85.1%

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

3. Taylor expanded in x around -inf 14.9%

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

   1. mul-1-neg 14.9%

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

   2. associate-/l* 14.9%

      \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

   3. distribute-rgt-neg-in 14.9%

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

   4. associate-/l* 15.0%

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

5. Simplified 15.0%

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

   1. add-sqr-sqrt 13.0%

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

   2. sqrt-unprod 14.7%

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

   3. sqr-neg 14.7%

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

   4. sqrt-unprod 1.7%

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

   5. add-sqr-sqrt 14.6%

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

   6. associate-*r/ 14.6%

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

   7. sqrt-unprod 14.7%

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

   8. metadata-eval 14.7%

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

   9. metadata-eval 14.7%

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

   10. associate-*r/ 14.7%

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

   11. *-commutative 14.7%

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

   12. *-un-lft-identity 14.7%

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

7. Applied egg-rr 14.7%

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

8. Final simplification 14.7%

   \[\leadsto \frac{p}{x} \]
9. 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 2024100 
(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))))))))