Given's Rotation SVD example

Percentage Accurate: 78.8% → 99.7%

Time: 9.1s

Alternatives: 7

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: 78.8% 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

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

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 21.1%

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

      1. add-sqr-sqrt 21.1%

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

      2. hypot-define 21.1%

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

      3. associate-*l* 21.1%

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

      4. sqrt-prod 21.1%

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

      5. metadata-eval 21.1%

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

      6. sqrt-unprod 12.4%

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

      7. add-sqr-sqrt 21.1%

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

   4. Applied egg-rr 21.1%

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

   5. Taylor expanded in x around -inf 60.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]
   6. 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* 60.8%

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

      3. distribute-rgt-neg-in 60.8%

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

      4. *-commutative 60.8%

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

      5. associate-/l* 61.2%

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

      6. distribute-rgt-neg-in 61.2%

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

   7. Simplified 61.2%

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

      1. add-sqr-sqrt 61.1%

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

      2. sqrt-unprod 61.2%

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

      3. swap-sqr 61.0%

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

      4. rem-square-sqrt 61.4%

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

      5. sqr-neg 61.4%

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

      6. frac-times 61.2%

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

      7. rem-square-sqrt 61.5%

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

      8. pow2 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. associate-*r/ 61.5%

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

       2. metadata-eval 61.5%

          \[\leadsto p \cdot \sqrt{\frac{\color{blue}{1}}{{x}^{2}}} \]

   11. Simplified 61.5%

       \[\leadsto p \cdot \color{blue}{\sqrt{\frac{1}{{x}^{2}}}} \]

   12. Taylor expanded in x around -inf 61.7%

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

       1. associate-*r/ 61.7%

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

       2. neg-mul-1 61.7%

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

   14. 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 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-define 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 51.9%

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

      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 89.7%

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

Alternative 2: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.8 \cdot 10^{-269}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 3.2 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.8e-269) (/ p_m (- x)) (if (<= p_m 3.2e-21) 1.0 (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.8e-269) {
		tmp = p_m / -x;
	} else if (p_m <= 3.2e-21) {
		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) :: tmp
    if (p_m <= 1.8d-269) then
        tmp = p_m / -x
    else if (p_m <= 3.2d-21) 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 tmp;
	if (p_m <= 1.8e-269) {
		tmp = p_m / -x;
	} else if (p_m <= 3.2e-21) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.8e-269:
		tmp = p_m / -x
	elif p_m <= 3.2e-21:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.8e-269)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 3.2e-21)
		tmp = 1.0;
	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 <= 1.8e-269)
		tmp = p_m / -x;
	elseif (p_m <= 3.2e-21)
		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_] := If[LessEqual[p$95$m, 1.8e-269], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 3.2e-21], 1.0, N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.79999999999999999e-269

   1. Initial program 77.3%

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

      1. add-sqr-sqrt 77.3%

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

      2. hypot-define 77.3%

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

      3. associate-*l* 77.3%

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

      4. sqrt-prod 77.3%

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

      5. metadata-eval 77.3%

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

      6. sqrt-unprod 6.1%

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

      7. add-sqr-sqrt 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in x around -inf 12.6%

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

      1. mul-1-neg 12.6%

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

      2. associate-/l* 12.6%

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

      3. distribute-rgt-neg-in 12.6%

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

      4. *-commutative 12.6%

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

      5. associate-/l* 12.6%

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

      6. distribute-rgt-neg-in 12.6%

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

   7. Simplified 12.6%

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

      1. add-sqr-sqrt 10.2%

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

      2. sqrt-unprod 11.1%

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

      3. swap-sqr 11.2%

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

      4. rem-square-sqrt 11.2%

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

      5. sqr-neg 11.2%

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

      6. frac-times 11.2%

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

      7. rem-square-sqrt 11.2%

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

      8. pow2 11.2%

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

   9. Applied egg-rr 11.2%

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

       1. associate-*r/ 11.2%

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

       2. metadata-eval 11.2%

          \[\leadsto p \cdot \sqrt{\frac{\color{blue}{1}}{{x}^{2}}} \]

   11. Simplified 11.2%

       \[\leadsto p \cdot \color{blue}{\sqrt{\frac{1}{{x}^{2}}}} \]

   12. Taylor expanded in x around -inf 12.7%

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

       1. associate-*r/ 12.7%

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

       2. neg-mul-1 12.7%

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

   14. Simplified 12.7%

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

   if 1.79999999999999999e-269 < p < 3.2000000000000002e-21

   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 49.6%

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

   if 3.2000000000000002e-21 < p

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 87.0%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.8 \cdot 10^{-269}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.2 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2e-73) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2e-73) {
		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 <= 2d-73) 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 <= 2e-73) {
		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 <= 2e-73:
		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 <= 2e-73)
		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 <= 2e-73)
		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, 2e-73], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.99999999999999999e-73

   1. Initial program 72.6%

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

      1. add-sqr-sqrt 72.6%

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

      2. hypot-define 72.6%

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

      3. associate-*l* 72.6%

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

      4. sqrt-prod 72.6%

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

      5. metadata-eval 72.6%

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

      6. sqrt-unprod 19.5%

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

      7. add-sqr-sqrt 72.6%

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

   4. Applied egg-rr 72.6%

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

   5. Taylor expanded in x around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. associate-/l* 22.7%

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

      3. distribute-rgt-neg-in 22.7%

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

      4. *-commutative 22.7%

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

      5. associate-/l* 22.8%

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

      6. distribute-rgt-neg-in 22.8%

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

   7. Simplified 22.8%

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

      1. add-sqr-sqrt 20.8%

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

      2. sqrt-unprod 22.0%

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

      3. swap-sqr 22.0%

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

      4. rem-square-sqrt 22.1%

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

      5. sqr-neg 22.1%

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

      6. frac-times 22.0%

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

      7. rem-square-sqrt 22.1%

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

      8. pow2 22.1%

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

   9. Applied egg-rr 22.1%

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

       1. associate-*r/ 22.1%

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

       2. metadata-eval 22.1%

          \[\leadsto p \cdot \sqrt{\frac{\color{blue}{1}}{{x}^{2}}} \]

   11. Simplified 22.1%

       \[\leadsto p \cdot \color{blue}{\sqrt{\frac{1}{{x}^{2}}}} \]

   12. Taylor expanded in x around -inf 23.0%

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

       1. associate-*r/ 23.0%

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

       2. neg-mul-1 23.0%

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

   14. Simplified 23.0%

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

   if 1.99999999999999999e-73 < p

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 83.9%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2 \cdot 10^{-73}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.3% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.9e-150) (/ p_m (- x)) 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.9e-150) {
		tmp = p_m / -x;
	} else {
		tmp = 1.5;
	}
	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 (x <= (-3.9d-150)) then
        tmp = p_m / -x
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -3.9e-150) {
		tmp = p_m / -x;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.9e-150:
		tmp = p_m / -x
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.9e-150)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -3.9e-150)
		tmp = p_m / -x;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.9e-150], N[(p$95$m / (-x)), $MachinePrecision], 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000002e-150

   1. Initial program 58.8%

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

      1. add-sqr-sqrt 58.8%

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

      2. hypot-define 58.8%

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

      3. associate-*l* 58.8%

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

      4. sqrt-prod 58.8%

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

      5. metadata-eval 58.8%

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

      6. sqrt-unprod 30.7%

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

      7. add-sqr-sqrt 58.8%

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

   4. Applied egg-rr 58.8%

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

   5. Taylor expanded in x around -inf 33.4%

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

      1. mul-1-neg 33.4%

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

      2. associate-/l* 33.3%

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

      3. distribute-rgt-neg-in 33.3%

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

      4. *-commutative 33.3%

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

      5. associate-/l* 33.5%

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

      6. distribute-rgt-neg-in 33.5%

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

   7. Simplified 33.5%

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

      1. add-sqr-sqrt 33.5%

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

      2. sqrt-unprod 33.5%

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

      3. swap-sqr 33.4%

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

      4. rem-square-sqrt 33.6%

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

      5. sqr-neg 33.6%

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

      6. frac-times 33.5%

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

      7. rem-square-sqrt 33.7%

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

      8. pow2 33.7%

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

   9. Applied egg-rr 33.7%

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

       1. associate-*r/ 33.7%

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

       2. metadata-eval 33.7%

          \[\leadsto p \cdot \sqrt{\frac{\color{blue}{1}}{{x}^{2}}} \]

   11. Simplified 33.7%

       \[\leadsto p \cdot \color{blue}{\sqrt{\frac{1}{{x}^{2}}}} \]

   12. Taylor expanded in x around -inf 33.8%

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

       1. associate-*r/ 33.8%

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

       2. neg-mul-1 33.8%

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

   14. Simplified 33.8%

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

   if -3.9000000000000002e-150 < 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 0 54.8%

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

   5. Applied egg-rr 19.5%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-150}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 17.0% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2.4e+15) 0.0 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.4e+15) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	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 (x <= (-2.4d+15)) then
        tmp = 0.0d0
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -2.4e+15) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.4e+15:
		tmp = 0.0
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.4e+15)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -2.4e+15)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.4e+15], 0.0, 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e15

   1. Initial program 53.8%

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

   3. Taylor expanded in x around -inf 22.5%

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

      1. neg-mul-1 22.5%

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

   5. Simplified 22.5%

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

   6. Taylor expanded in x around 0 22.5%

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

   if -2.4e15 < x

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0 57.8%

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

   5. Applied egg-rr 17.3%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 15.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+15}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.7e+15) 0.0 0.125))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.7e+15) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	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 <= (-1.7d+15)) then
        tmp = 0.0d0
    else
        tmp = 0.125d0
    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 <= -1.7e+15) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.7e+15:
		tmp = 0.0
	else:
		tmp = 0.125
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.7e+15)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.7e+15)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.7e+15], 0.0, 0.125]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e15

   1. Initial program 53.8%

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

   3. Taylor expanded in x around -inf 22.5%

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

      1. neg-mul-1 22.5%

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

   5. Simplified 22.5%

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

   6. Taylor expanded in x around 0 22.5%

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

   if -1.7e15 < x

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0 57.8%

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

   5. Applied egg-rr 15.0%

      \[\leadsto \color{blue}{0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 6.2% 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 78.7%

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

3. Taylor expanded in x around -inf 8.0%

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

   1. neg-mul-1 8.0%

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

5. Simplified 8.0%

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

6. Taylor expanded in x around 0 8.0%

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

Developer Target 1: 78.8% 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 2024145 
(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))))))))