Given's Rotation SVD example

Percentage Accurate: 79.2% → 99.8%

Time: 8.0s

Alternatives: 7

Speedup: 1.0×

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: 79.2% 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, 0.3× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\sqrt[3]{{t_0}^{2}}, \sqrt[3]{t_0}, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (hypot x (* p 2.0)))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
     (/ p (- x))
     (sqrt (* 0.5 (fma (cbrt (pow t_0 2.0)) (cbrt t_0) 1.0))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = x / hypot(x, (p * 2.0));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = p / -x;
	} else {
		tmp = sqrt((0.5 * fma(cbrt(pow(t_0, 2.0)), cbrt(t_0), 1.0)));
	}
	return tmp;
}

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(x / hypot(x, Float64(p * 2.0)))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * fma(cbrt((t_0 ^ 2.0)), cbrt(t_0), 1.0)));
	end
	return tmp
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(p / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.5%

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

      1. clear-num 14.5%

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

      2. associate-/r/ 13.9%

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

      3. +-commutative 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. hypot-def 13.9%

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

      6. associate-*l* 13.9%

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

      7. sqrt-prod 13.9%

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

      8. metadata-eval 13.9%

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

      9. sqrt-unprod 9.1%

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

      10. add-sqr-sqrt 13.9%

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

   3. Applied egg-rr 13.9%

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

   5. Applied egg-rr 14.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 14.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 14.5%

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

      3. associate-*r/ 14.5%

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

      4. *-commutative 14.5%

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

   7. Simplified 14.5%

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

   8. Taylor expanded in x around -inf 61.4%

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

      1. mul-1-neg 61.4%

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

      2. *-lft-identity 61.4%

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

      3. distribute-rgt-neg-in 61.4%

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

      4. metadata-eval 61.4%

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

      5. distribute-frac-neg 61.4%

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

      6. times-frac 61.4%

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

      7. neg-mul-1 61.4%

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

      8. remove-double-neg 61.4%

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

      9. neg-mul-1 61.4%

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

   10. Simplified 61.4%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. add-cube-cbrt 99.3%

         \[\leadsto \sqrt{0.5 \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}} + 1\right)} \]

      3. fma-def 99.3%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, \sqrt[3]{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}, 1\right)}} \]

   3. Applied egg-rr 99.8%

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

4. Final simplification 89.5%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (/ p (- x))
   (sqrt (+ 0.5 (* x (/ 0.5 (hypot x (* p 2.0))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(p / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 14.5%

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

      1. clear-num 14.5%

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

      2. associate-/r/ 13.9%

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

      3. +-commutative 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. hypot-def 13.9%

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

      6. associate-*l* 13.9%

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

      7. sqrt-prod 13.9%

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

      8. metadata-eval 13.9%

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

      9. sqrt-unprod 9.1%

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

      10. add-sqr-sqrt 13.9%

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

   3. Applied egg-rr 13.9%

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

   5. Applied egg-rr 14.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 14.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 14.5%

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

      3. associate-*r/ 14.5%

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

      4. *-commutative 14.5%

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

   7. Simplified 14.5%

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

   8. Taylor expanded in x around -inf 61.4%

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

      1. mul-1-neg 61.4%

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

      2. *-lft-identity 61.4%

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

      3. distribute-rgt-neg-in 61.4%

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

      4. metadata-eval 61.4%

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

      5. distribute-frac-neg 61.4%

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

      6. times-frac 61.4%

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

      7. neg-mul-1 61.4%

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

      8. remove-double-neg 61.4%

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

      9. neg-mul-1 61.4%

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

   10. Simplified 61.4%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.4%

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

      3. +-commutative 99.4%

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

      4. add-sqr-sqrt 99.4%

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

      5. hypot-def 99.4%

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

      6. associate-*l* 99.4%

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

      7. sqrt-prod 99.4%

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

      8. metadata-eval 99.4%

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

      9. sqrt-unprod 44.8%

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

      10. add-sqr-sqrt 99.8%

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

   3. Applied egg-rr 99.8%

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

   5. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 98.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 89.4%

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

Alternative 3: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= x -7.5e-15)
   (/ p (- x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -7.5e-15) {
		tmp = p / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -7.5e-15) {
		tmp = p / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -7.5e-15:
		tmp = p / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -7.5e-15)
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -7.5e-15)
		tmp = p / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -7.5e-15], N[(p / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4999999999999996e-15

   1. Initial program 46.2%

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

      1. clear-num 46.2%

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

      2. associate-/r/ 45.6%

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

      3. +-commutative 45.6%

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

      4. add-sqr-sqrt 45.6%

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

      5. hypot-def 45.6%

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

      6. associate-*l* 45.6%

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

      7. sqrt-prod 45.6%

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

      8. metadata-eval 45.6%

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

      9. sqrt-unprod 25.6%

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

      10. add-sqr-sqrt 46.3%

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

   3. Applied egg-rr 46.3%

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

   5. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 46.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 46.9%

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

      3. associate-*r/ 46.9%

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

      4. *-commutative 46.9%

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

   7. Simplified 46.9%

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

   8. Taylor expanded in x around -inf 40.5%

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

      1. mul-1-neg 40.5%

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

      2. *-lft-identity 40.5%

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

      3. distribute-rgt-neg-in 40.5%

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

      4. metadata-eval 40.5%

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

      5. distribute-frac-neg 40.5%

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

      6. times-frac 40.5%

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

      7. neg-mul-1 40.5%

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

      8. remove-double-neg 40.5%

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

      9. neg-mul-1 40.5%

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

   10. Simplified 40.5%

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

   if -7.4999999999999996e-15 < x

   1. Initial program 89.5%

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

      1. add-sqr-sqrt 89.5%

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

      2. hypot-def 89.5%

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

      3. associate-*l* 89.5%

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

      4. sqrt-prod 89.5%

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

      5. metadata-eval 89.5%

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

      6. sqrt-unprod 39.2%

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

      7. add-sqr-sqrt 89.7%

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

   3. Applied egg-rr 89.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 4: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{p}{-x}\\ \mathbf{if}\;p \leq 2 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ p (- x))))
   (if (<= p 2e-158)
     t_0
     (if (<= p 3.4e-56)
       1.0
       (if (<= p 4.3e-44) t_0 (if (<= p 2.2e-29) 1.0 (sqrt 0.5)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (p <= 2e-158) {
		tmp = t_0;
	} else if (p <= 3.4e-56) {
		tmp = 1.0;
	} else if (p <= 4.3e-44) {
		tmp = t_0;
	} else if (p <= 2.2e-29) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p / -x
    if (p <= 2d-158) then
        tmp = t_0
    else if (p <= 3.4d-56) then
        tmp = 1.0d0
    else if (p <= 4.3d-44) then
        tmp = t_0
    else if (p <= 2.2d-29) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p / -x;
	double tmp;
	if (p <= 2e-158) {
		tmp = t_0;
	} else if (p <= 3.4e-56) {
		tmp = 1.0;
	} else if (p <= 4.3e-44) {
		tmp = t_0;
	} else if (p <= 2.2e-29) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p / -x
	tmp = 0
	if p <= 2e-158:
		tmp = t_0
	elif p <= 3.4e-56:
		tmp = 1.0
	elif p <= 4.3e-44:
		tmp = t_0
	elif p <= 2.2e-29:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p / Float64(-x))
	tmp = 0.0
	if (p <= 2e-158)
		tmp = t_0;
	elseif (p <= 3.4e-56)
		tmp = 1.0;
	elseif (p <= 4.3e-44)
		tmp = t_0;
	elseif (p <= 2.2e-29)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p / -x;
	tmp = 0.0;
	if (p <= 2e-158)
		tmp = t_0;
	elseif (p <= 3.4e-56)
		tmp = 1.0;
	elseif (p <= 4.3e-44)
		tmp = t_0;
	elseif (p <= 2.2e-29)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p / (-x)), $MachinePrecision]}, If[LessEqual[p, 2e-158], t$95$0, If[LessEqual[p, 3.4e-56], 1.0, If[LessEqual[p, 4.3e-44], t$95$0, If[LessEqual[p, 2.2e-29], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.00000000000000013e-158 or 3.39999999999999982e-56 < p < 4.30000000000000013e-44

   1. Initial program 73.0%

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

      1. clear-num 73.0%

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

      2. associate-/r/ 72.7%

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

      3. +-commutative 72.7%

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

      4. add-sqr-sqrt 72.7%

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

      5. hypot-def 72.7%

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

      6. associate-*l* 72.7%

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

      7. sqrt-prod 72.7%

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

      8. metadata-eval 72.7%

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

      9. sqrt-unprod 10.4%

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

      10. add-sqr-sqrt 72.9%

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

   3. Applied egg-rr 72.9%

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

   5. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 72.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 73.2%

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

      3. associate-*r/ 73.2%

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

      4. *-commutative 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in x around -inf 19.6%

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

      1. mul-1-neg 19.6%

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

      2. *-lft-identity 19.6%

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

      3. distribute-rgt-neg-in 19.6%

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

      4. metadata-eval 19.6%

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

      5. distribute-frac-neg 19.6%

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

      6. times-frac 19.6%

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

      7. neg-mul-1 19.6%

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

      8. remove-double-neg 19.6%

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

      9. neg-mul-1 19.6%

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

   10. Simplified 19.6%

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

   if 2.00000000000000013e-158 < p < 3.39999999999999982e-56 or 4.30000000000000013e-44 < p < 2.1999999999999999e-29

   1. Initial program 63.0%

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

      1. clear-num 63.0%

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

      2. associate-/r/ 63.5%

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

      3. +-commutative 63.5%

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

      4. add-sqr-sqrt 63.5%

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

      5. hypot-def 63.5%

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

      6. associate-*l* 63.5%

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

      7. sqrt-prod 63.5%

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

      8. metadata-eval 63.5%

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

      9. sqrt-unprod 63.5%

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

      10. add-sqr-sqrt 63.5%

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

   3. Applied egg-rr 63.5%

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

   5. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 62.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 63.0%

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

      3. associate-*r/ 63.0%

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

      4. *-commutative 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in x around inf 52.7%

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

   if 2.1999999999999999e-29 < p

   1. Initial program 89.1%

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

   2. Taylor expanded in x around 0 78.6%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2 \cdot 10^{-158}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 67.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= p 6e-35) (/ p (- x)) (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 6e-35) {
		tmp = p / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 6d-35) then
        tmp = p / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 6e-35) {
		tmp = p / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 6e-35:
		tmp = p / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 6e-35)
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 6e-35)
		tmp = p / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 6e-35], N[(p / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 5.99999999999999978e-35

   1. Initial program 71.8%

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

      1. clear-num 71.8%

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

      2. associate-/r/ 71.6%

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

      3. +-commutative 71.6%

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

      4. add-sqr-sqrt 71.6%

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

      5. hypot-def 71.6%

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

      6. associate-*l* 71.6%

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

      7. sqrt-prod 71.6%

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

      8. metadata-eval 71.6%

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

      9. sqrt-unprod 15.3%

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

      10. add-sqr-sqrt 71.8%

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

   3. Applied egg-rr 71.8%

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

   5. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 71.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 72.0%

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

      3. associate-*r/ 72.0%

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

      4. *-commutative 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in x around -inf 21.8%

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

      1. mul-1-neg 21.8%

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

      2. *-lft-identity 21.8%

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

      3. distribute-rgt-neg-in 21.8%

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

      4. metadata-eval 21.8%

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

      5. distribute-frac-neg 21.8%

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

      6. times-frac 21.8%

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

      7. neg-mul-1 21.8%

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

      8. remove-double-neg 21.8%

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

      9. neg-mul-1 21.8%

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

   10. Simplified 21.8%

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

   if 5.99999999999999978e-35 < p

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0 77.7%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 29.1% accurate, 35.5× speedup

Math

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

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -2e-310) (/ p (- x)) (/ p x)))

C

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

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = p / -x
    else
        tmp = p / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -2e-310], N[(p / (-x)), $MachinePrecision], N[(p / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 53.6%

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

      1. clear-num 53.5%

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

      2. associate-/r/ 53.3%

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

      3. +-commutative 53.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. hypot-def 53.3%

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

      6. associate-*l* 53.3%

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

      7. sqrt-prod 53.3%

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

      8. metadata-eval 53.3%

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

      9. sqrt-unprod 25.6%

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

      10. add-sqr-sqrt 53.7%

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

   3. Applied egg-rr 53.7%

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

   5. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 53.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)\right)} \]

      2. expm1-log1p 54.0%

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

      3. associate-*r/ 54.0%

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

      4. *-commutative 54.0%

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

   7. Simplified 54.0%

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

   8. Taylor expanded in x around -inf 34.6%

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

      1. mul-1-neg 34.6%

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

      2. *-lft-identity 34.6%

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

      3. distribute-rgt-neg-in 34.6%

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

      4. metadata-eval 34.6%

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

      5. distribute-frac-neg 34.6%

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

      6. times-frac 34.6%

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

      7. neg-mul-1 34.6%

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

      8. remove-double-neg 34.6%

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

      9. neg-mul-1 34.6%

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

   10. Simplified 34.6%

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

   if -1.999999999999994e-310 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around -inf 5.1%

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

      1. unpow2 5.1%

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

      2. unpow2 5.1%

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

      3. times-frac 5.4%

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

   4. Simplified 5.4%

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

   5. Taylor expanded in p around 0 3.2%

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

4. Final simplification 19.0%

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

Alternative 7: 6.8% accurate, 71.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \frac{p}{x} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (/ p x))

C

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

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = p / x
end function

Java

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

Python

p = abs(p)
def code(p, x):
	return p / x

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

2. Taylor expanded in x around -inf 19.7%

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

   1. unpow2 19.7%

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

   2. unpow2 19.7%

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

   3. times-frac 23.3%

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

4. Simplified 23.3%

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

5. Taylor expanded in p around 0 16.3%

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

6. Final simplification 16.3%

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

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 2023200 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

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