Given's Rotation SVD example

Percentage Accurate: 79.5% → 99.8%

Time: 7.4s

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: 79.5% 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

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

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 (* (/ x (hypot x (* p_m 2.0))) 0.5)))))

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 + ((x / hypot(x, (p_m * 2.0))) * 0.5)));
	}
	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 + ((x / Math.hypot(x, (p_m * 2.0))) * 0.5)));
	}
	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 + ((x / math.hypot(x, (p_m * 2.0))) * 0.5)))
	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(Float64(x / hypot(x, Float64(p_m * 2.0))) * 0.5)));
	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 + ((x / hypot(x, (p_m * 2.0))) * 0.5)));
	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[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $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 19.3%

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

      1. +-commutative 19.3%

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

      2. distribute-rgt-in 19.3%

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

      3. +-commutative 19.3%

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

      4. add-sqr-sqrt 19.3%

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

      5. hypot-define 19.3%

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

      6. associate-*l* 19.3%

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

      7. sqrt-prod 19.3%

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

      8. metadata-eval 19.3%

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

      9. sqrt-unprod 7.6%

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

      10. add-sqr-sqrt 19.3%

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

      11. metadata-eval 19.3%

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

   4. Applied egg-rr 19.3%

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

   5. Taylor expanded in x around -inf 59.0%

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

      1. associate-*r/ 59.0%

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

      2. neg-mul-1 59.0%

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

   7. Simplified 59.0%

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

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

   1. Initial program 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. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 53.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 88.3%

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

Alternative 2: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 5.5 \cdot 10^{-259}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 5 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 9.2 \cdot 10^{-91}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.85 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 5.5e-259)
     t_0
     (if (<= p_m 5e-172)
       1.0
       (if (<= p_m 9.2e-91) t_0 (if (<= p_m 1.85e-27) 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 <= 5.5e-259) {
		tmp = t_0;
	} else if (p_m <= 5e-172) {
		tmp = 1.0;
	} else if (p_m <= 9.2e-91) {
		tmp = t_0;
	} else if (p_m <= 1.85e-27) {
		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 <= 5.5d-259) then
        tmp = t_0
    else if (p_m <= 5d-172) then
        tmp = 1.0d0
    else if (p_m <= 9.2d-91) then
        tmp = t_0
    else if (p_m <= 1.85d-27) 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 <= 5.5e-259) {
		tmp = t_0;
	} else if (p_m <= 5e-172) {
		tmp = 1.0;
	} else if (p_m <= 9.2e-91) {
		tmp = t_0;
	} else if (p_m <= 1.85e-27) {
		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 <= 5.5e-259:
		tmp = t_0
	elif p_m <= 5e-172:
		tmp = 1.0
	elif p_m <= 9.2e-91:
		tmp = t_0
	elif p_m <= 1.85e-27:
		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 <= 5.5e-259)
		tmp = t_0;
	elseif (p_m <= 5e-172)
		tmp = 1.0;
	elseif (p_m <= 9.2e-91)
		tmp = t_0;
	elseif (p_m <= 1.85e-27)
		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 <= 5.5e-259)
		tmp = t_0;
	elseif (p_m <= 5e-172)
		tmp = 1.0;
	elseif (p_m <= 9.2e-91)
		tmp = t_0;
	elseif (p_m <= 1.85e-27)
		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, 5.5e-259], t$95$0, If[LessEqual[p$95$m, 5e-172], 1.0, If[LessEqual[p$95$m, 9.2e-91], t$95$0, If[LessEqual[p$95$m, 1.85e-27], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 5.50000000000000038e-259 or 4.9999999999999999e-172 < p < 9.19999999999999982e-91

   1. Initial program 70.9%

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

      1. +-commutative 70.9%

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

      2. distribute-rgt-in 70.9%

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

      3. +-commutative 70.9%

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

      4. add-sqr-sqrt 70.9%

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

      5. hypot-define 70.9%

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

      6. associate-*l* 70.9%

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

      7. sqrt-prod 70.9%

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

      8. metadata-eval 70.9%

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

      9. sqrt-unprod 8.7%

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

      10. add-sqr-sqrt 70.9%

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

      11. metadata-eval 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in x around -inf 20.1%

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

      1. associate-*r/ 20.1%

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

      2. neg-mul-1 20.1%

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

   7. Simplified 20.1%

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

   if 5.50000000000000038e-259 < p < 4.9999999999999999e-172 or 9.19999999999999982e-91 < p < 1.85000000000000014e-27

   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. Step-by-step derivation

      1. +-commutative 76.1%

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

      2. distribute-rgt-in 76.1%

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

      3. +-commutative 76.1%

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

      4. add-sqr-sqrt 76.1%

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

      5. hypot-define 76.1%

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

      6. associate-*l* 76.1%

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

      7. sqrt-prod 76.1%

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

      8. metadata-eval 76.1%

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

      9. sqrt-unprod 76.1%

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

      10. add-sqr-sqrt 76.1%

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

      11. metadata-eval 76.1%

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

   4. Applied egg-rr 76.1%

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

   5. Taylor expanded in x around inf 58.2%

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

      1. *-commutative 58.2%

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

   7. Simplified 58.2%

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

   8. Taylor expanded in p around 0 61.2%

      \[\leadsto \color{blue}{1} \]

   if 1.85000000000000014e-27 < p

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0 87.8%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-172}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.85 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -7.5e-244) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -7.5e-244) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7.5d-244)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    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 <= -7.5e-244) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -7.5e-244:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -7.5e-244)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -7.5e-244)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -7.5e-244], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.5000000000000003e-244

   1. Initial program 53.3%

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

      1. +-commutative 53.3%

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

      2. distribute-rgt-in 53.3%

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

      3. +-commutative 53.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. hypot-define 53.3%

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

      6. associate-*l* 53.3%

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

      7. sqrt-prod 53.3%

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

      8. metadata-eval 53.3%

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

      9. sqrt-unprod 21.9%

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

      10. add-sqr-sqrt 53.3%

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

      11. metadata-eval 53.3%

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

   4. Applied egg-rr 53.3%

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

   5. Taylor expanded in x around -inf 35.4%

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

      1. associate-*r/ 35.4%

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

      2. neg-mul-1 35.4%

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

   7. Simplified 35.4%

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

   if -7.5000000000000003e-244 < 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. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 57.7%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 46.6%

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

      1. *-commutative 46.6%

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

   7. Simplified 46.6%

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

   8. Taylor expanded in p around 0 56.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.92 \cdot 10^{+52}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.92e+52) 0.0 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.92e+52) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.92d+52)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0
    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 <= -1.92e+52) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.92e+52:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.92e+52)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.92e+52)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.92e+52], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.9199999999999999e52

   1. Initial program 48.9%

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

   3. Taylor expanded in x around -inf 25.2%

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

      1. neg-mul-1 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around 0 25.2%

      \[\leadsto \color{blue}{0} \]

   if -1.9199999999999999e52 < x

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. distribute-rgt-in 83.0%

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

      3. +-commutative 83.0%

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

      4. add-sqr-sqrt 83.0%

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

      5. hypot-define 83.0%

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

      6. associate-*l* 83.0%

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

      7. sqrt-prod 83.0%

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

      8. metadata-eval 83.0%

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

      9. sqrt-unprod 44.1%

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

      10. add-sqr-sqrt 83.0%

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

      11. metadata-eval 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in x around inf 29.9%

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

      1. *-commutative 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in p around 0 40.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 13.6% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.12 \cdot 10^{-220}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.001953125\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 1.12e-220) 0.0 0.001953125))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.12e-220) {
		tmp = 0.0;
	} else {
		tmp = 0.001953125;
	}
	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 <= 1.12d-220) then
        tmp = 0.0d0
    else
        tmp = 0.001953125d0
    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 <= 1.12e-220) {
		tmp = 0.0;
	} else {
		tmp = 0.001953125;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.12e-220:
		tmp = 0.0
	else:
		tmp = 0.001953125
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.12e-220)
		tmp = 0.0;
	else
		tmp = 0.001953125;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.12e-220)
		tmp = 0.0;
	else
		tmp = 0.001953125;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.12e-220], 0.0, 0.001953125]

Derivation

1. Split input into 2 regimes

2. if p < 1.12000000000000008e-220

   1. Initial program 74.4%

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

   3. Taylor expanded in x around -inf 11.2%

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

      1. neg-mul-1 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in x around 0 11.2%

      \[\leadsto \color{blue}{0} \]

   if 1.12000000000000008e-220 < p

   1. Initial program 80.1%

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

   3. Taylor expanded in x around 0 61.0%

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

   5. Applied egg-rr 12.2%

      \[\leadsto \color{blue}{0.001953125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 6.5% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 0 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 0.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 0.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 0.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 0.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 0.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 0.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 0.0

Derivation

1. Initial program 77.0%

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

3. Taylor expanded in x around -inf 7.7%

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

   1. neg-mul-1 7.7%

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

5. Simplified 7.7%

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

6. Taylor expanded in x around 0 7.7%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer Target 1: 79.6% 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 2024149 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))