Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.8%

Time: 12.5s

Alternatives: 8

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 0.2× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (hypot x (* p_m 2.0)))))
   (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
     (/ (- p_m) x)
     (sqrt
      (*
       0.5
       (/
        (+ 1.0 (pow t_0 3.0))
        (fma t_0 (/ (+ -1.0 (pow t_0 2.0)) (- t_0 -1.0)) 1.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / hypot(x, (p_m * 2.0));
	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 + pow(t_0, 3.0)) / fma(t_0, ((-1.0 + pow(t_0, 2.0)) / (t_0 - -1.0)), 1.0))));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / hypot(x, Float64(p_m * 2.0)))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(Float64(1.0 + (t_0 ^ 3.0)) / fma(t_0, Float64(Float64(-1.0 + (t_0 ^ 2.0)) / Float64(t_0 - -1.0)), 1.0))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(1.0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(-1.0 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $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 15.0%

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

   3. Taylor expanded in x around -inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around -inf 60.4%

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

      1. associate-*r/ 60.4%

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

      2. mul-1-neg 60.4%

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

   8. Simplified 60.4%

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

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

   1. Initial program 99.9%

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

      1. flip3-+ 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}} \]
   5. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \color{blue}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + \left(-1\right)}, 1\right)}} \]

      2. flip-+ 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \color{blue}{\frac{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right) \cdot \left(-1\right)}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}}, 1\right)}} \]

      3. metadata-eval 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \color{blue}{-1} \cdot \left(-1\right)}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}, 1\right)}} \]

      4. metadata-eval 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - -1 \cdot \color{blue}{-1}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}, 1\right)}} \]

      5. metadata-eval 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \color{blue}{1}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}, 1\right)}} \]

      6. sub-neg 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\color{blue}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + \left(-1\right)}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}, 1\right)}} \]

      7. pow2 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{\color{blue}{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2}} + \left(-1\right)}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}, 1\right)}} \]

      8. metadata-eval 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2} + \color{blue}{-1}}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \left(-1\right)}, 1\right)}} \]

      9. metadata-eval 99.9%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{2} + -1}{\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - \color{blue}{-1}}, 1\right)}} \]

   6. Applied egg-rr 99.9%

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

4. Final simplification 90.2%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, p_m \cdot 2\right)}, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

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)
   (pow
    (pow (fma 0.5 (/ x (hypot x (* p_m 2.0))) 0.5) 1.5)
    0.3333333333333333)))

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 = pow(pow(fma(0.5, (x / hypot(x, (p_m * 2.0))), 0.5), 1.5), 0.3333333333333333);
	}
	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(Float64(-p_m) / x);
	else
		tmp = (fma(0.5, Float64(x / hypot(x, Float64(p_m * 2.0))), 0.5) ^ 1.5) ^ 0.3333333333333333;
	end
	return 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[Power[N[Power[N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $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 15.0%

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

   3. Taylor expanded in x around -inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around -inf 60.4%

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

      1. associate-*r/ 60.4%

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

      2. mul-1-neg 60.4%

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

   8. Simplified 60.4%

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

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

   1. Initial program 99.9%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 99.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p_m \cdot 2\right)}\right)}}\\ \end{array} \end{array} \]

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 (exp (log1p (/ x (hypot x (* p_m 2.0)))))))))

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 * exp(log1p((x / hypot(x, (p_m * 2.0)))))));
	}
	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 * Math.exp(Math.log1p((x / Math.hypot(x, (p_m * 2.0)))))));
	}
	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 * math.exp(math.log1p((x / math.hypot(x, (p_m * 2.0)))))))
	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(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * exp(log1p(Float64(x / hypot(x, Float64(p_m * 2.0)))))));
	end
	return 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[Exp[N[Log[1 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $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 15.0%

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

   3. Taylor expanded in x around -inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around -inf 60.4%

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

      1. associate-*r/ 60.4%

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

      2. mul-1-neg 60.4%

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

   8. Simplified 60.4%

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

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

   1. Initial program 99.9%

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

      1. add-exp-log 99.9%

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

      2. log1p-udef 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-def 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 99.9%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 99.9%

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

      11. sqrt-unprod 51.3%

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

      12. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -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 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 15.0%

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

   3. Taylor expanded in x around -inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-/l* 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around -inf 60.4%

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

      1. associate-*r/ 60.4%

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

      2. mul-1-neg 60.4%

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

   8. Simplified 60.4%

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

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

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. hypot-def 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. metadata-eval 99.9%

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

      6. sqrt-unprod 51.3%

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

      7. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 90.2%

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

Alternative 5: 65.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 1.28 \cdot 10^{-235}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 7 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 1.9 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p_m \cdot 2}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 1.28e-235)
     1.0
     (if (<= p_m 7e-207)
       t_0
       (if (<= p_m 1.9e-186)
         1.0
         (if (<= p_m 1.45e-40)
           t_0
           (if (<= p_m 3.4e+39)
             1.0
             (sqrt (* 0.5 (+ 1.0 (/ x (* p_m 2.0))))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 1.28e-235) {
		tmp = 1.0;
	} else if (p_m <= 7e-207) {
		tmp = t_0;
	} else if (p_m <= 1.9e-186) {
		tmp = 1.0;
	} else if (p_m <= 1.45e-40) {
		tmp = t_0;
	} else if (p_m <= 3.4e+39) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (p_m * 2.0)))));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 1.28d-235) then
        tmp = 1.0d0
    else if (p_m <= 7d-207) then
        tmp = t_0
    else if (p_m <= 1.9d-186) then
        tmp = 1.0d0
    else if (p_m <= 1.45d-40) then
        tmp = t_0
    else if (p_m <= 3.4d+39) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (p_m * 2.0d0)))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 1.28e-235) {
		tmp = 1.0;
	} else if (p_m <= 7e-207) {
		tmp = t_0;
	} else if (p_m <= 1.9e-186) {
		tmp = 1.0;
	} else if (p_m <= 1.45e-40) {
		tmp = t_0;
	} else if (p_m <= 3.4e+39) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 1.28e-235:
		tmp = 1.0
	elif p_m <= 7e-207:
		tmp = t_0
	elif p_m <= 1.9e-186:
		tmp = 1.0
	elif p_m <= 1.45e-40:
		tmp = t_0
	elif p_m <= 3.4e+39:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 1.28e-235)
		tmp = 1.0;
	elseif (p_m <= 7e-207)
		tmp = t_0;
	elseif (p_m <= 1.9e-186)
		tmp = 1.0;
	elseif (p_m <= 1.45e-40)
		tmp = t_0;
	elseif (p_m <= 3.4e+39)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(p_m * 2.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 1.28e-235)
		tmp = 1.0;
	elseif (p_m <= 7e-207)
		tmp = t_0;
	elseif (p_m <= 1.9e-186)
		tmp = 1.0;
	elseif (p_m <= 1.45e-40)
		tmp = t_0;
	elseif (p_m <= 3.4e+39)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 1.28e-235], 1.0, If[LessEqual[p$95$m, 7e-207], t$95$0, If[LessEqual[p$95$m, 1.9e-186], 1.0, If[LessEqual[p$95$m, 1.45e-40], t$95$0, If[LessEqual[p$95$m, 3.4e+39], 1.0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(p$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.28e-235 or 7.0000000000000003e-207 < p < 1.89999999999999987e-186 or 1.4499999999999999e-40 < p < 3.3999999999999999e39

   1. Initial program 78.2%

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

      1. flip3-+ 78.2%

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

   4. Applied egg-rr 78.2%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}} \]
   5. Step-by-step derivation

      1. add-cbrt-cube 78.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}}} \]

      2. pow1/3 78.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{{\left({\left(\frac{0.5 + 0.5 \cdot {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around inf 39.0%

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

   if 1.28e-235 < p < 7.0000000000000003e-207 or 1.89999999999999987e-186 < p < 1.4499999999999999e-40

   1. Initial program 44.5%

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

   3. Taylor expanded in x around -inf 39.5%

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

      1. associate-*r/ 39.5%

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

      2. associate-/l* 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in x around -inf 64.7%

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

      1. associate-*r/ 64.7%

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

      2. mul-1-neg 64.7%

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

   8. Simplified 64.7%

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

   if 3.3999999999999999e39 < p

   1. Initial program 97.0%

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

   3. Taylor expanded in p around inf 93.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.28 \cdot 10^{-235}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-207}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 5.2 \cdot 10^{-238}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 5.7 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 5.5 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 3.65 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 4 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 5.2e-238)
     1.0
     (if (<= p_m 5.7e-207)
       t_0
       (if (<= p_m 5.5e-197)
         1.0
         (if (<= p_m 3.65e-40) t_0 (if (<= p_m 4e+29) 1.0 (sqrt 0.5))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 5.2e-238) {
		tmp = 1.0;
	} else if (p_m <= 5.7e-207) {
		tmp = t_0;
	} else if (p_m <= 5.5e-197) {
		tmp = 1.0;
	} else if (p_m <= 3.65e-40) {
		tmp = t_0;
	} else if (p_m <= 4e+29) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 5.2d-238) then
        tmp = 1.0d0
    else if (p_m <= 5.7d-207) then
        tmp = t_0
    else if (p_m <= 5.5d-197) then
        tmp = 1.0d0
    else if (p_m <= 3.65d-40) then
        tmp = t_0
    else if (p_m <= 4d+29) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 5.2e-238) {
		tmp = 1.0;
	} else if (p_m <= 5.7e-207) {
		tmp = t_0;
	} else if (p_m <= 5.5e-197) {
		tmp = 1.0;
	} else if (p_m <= 3.65e-40) {
		tmp = t_0;
	} else if (p_m <= 4e+29) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 5.2e-238:
		tmp = 1.0
	elif p_m <= 5.7e-207:
		tmp = t_0
	elif p_m <= 5.5e-197:
		tmp = 1.0
	elif p_m <= 3.65e-40:
		tmp = t_0
	elif p_m <= 4e+29:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 5.2e-238)
		tmp = 1.0;
	elseif (p_m <= 5.7e-207)
		tmp = t_0;
	elseif (p_m <= 5.5e-197)
		tmp = 1.0;
	elseif (p_m <= 3.65e-40)
		tmp = t_0;
	elseif (p_m <= 4e+29)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 5.2e-238)
		tmp = 1.0;
	elseif (p_m <= 5.7e-207)
		tmp = t_0;
	elseif (p_m <= 5.5e-197)
		tmp = 1.0;
	elseif (p_m <= 3.65e-40)
		tmp = t_0;
	elseif (p_m <= 4e+29)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 5.2e-238], 1.0, If[LessEqual[p$95$m, 5.7e-207], t$95$0, If[LessEqual[p$95$m, 5.5e-197], 1.0, If[LessEqual[p$95$m, 3.65e-40], t$95$0, If[LessEqual[p$95$m, 4e+29], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 5.2000000000000002e-238 or 5.7e-207 < p < 5.50000000000000037e-197 or 3.65000000000000003e-40 < p < 3.99999999999999966e29

   1. Initial program 77.6%

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

      1. flip3-+ 77.6%

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

   4. Applied egg-rr 77.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}} \]
   5. Step-by-step derivation

      1. add-cbrt-cube 77.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}}} \]

      2. pow1/3 77.6%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{0.5 + 0.5 \cdot {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around inf 38.5%

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

   if 5.2000000000000002e-238 < p < 5.7e-207 or 5.50000000000000037e-197 < p < 3.65000000000000003e-40

   1. Initial program 44.5%

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

   3. Taylor expanded in x around -inf 39.5%

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

      1. associate-*r/ 39.5%

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

      2. associate-/l* 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in x around -inf 64.7%

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

      1. associate-*r/ 64.7%

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

      2. mul-1-neg 64.7%

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

   8. Simplified 64.7%

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

   if 3.99999999999999966e29 < p

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0 90.0%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5.2 \cdot 10^{-238}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.7 \cdot 10^{-207}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 5.5 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.65 \cdot 10^{-40}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.25e-121) (/ (- p_m) x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.25e-121) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.25d-121)) then
        tmp = -p_m / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -3.25e-121) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.25e-121:
		tmp = -p_m / x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.25e-121)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -3.25e-121)
		tmp = -p_m / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.25e-121], N[((-p$95$m) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.2500000000000001e-121

   1. Initial program 57.8%

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

   3. Taylor expanded in x around -inf 25.6%

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

      1. associate-*r/ 25.6%

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

      2. associate-/l* 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around -inf 31.4%

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

      1. associate-*r/ 31.4%

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

      2. mul-1-neg 31.4%

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

   8. Simplified 31.4%

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

   if -3.2500000000000001e-121 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ 99.2%

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

   4. Applied egg-rr 99.2%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}} \]
   5. Step-by-step derivation

      1. add-cbrt-cube 99.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}}} \]

      2. pow1/3 99.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left({\left(\frac{0.5 + 0.5 \cdot {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in x around inf 53.2%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.6% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 1 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 1.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 1.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 1.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 1.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 79.0%

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

   1. flip3-+ 79.0%

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

4. Applied egg-rr 79.0%

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}} \]
5. Step-by-step derivation

   1. add-cbrt-cube 79.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}}} \]

   2. pow1/3 79.0%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}} \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right) \cdot \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} - 1, 1\right)}}\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 79.0%

   \[\leadsto \color{blue}{{\left({\left(\frac{0.5 + 0.5 \cdot {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

7. Taylor expanded in x around inf 33.2%

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

8. Final simplification 33.2%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024017 
(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))))))))