Given's Rotation SVD example

Percentage Accurate: 80.1% → 99.9%

Time: 9.8s

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: 80.1% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\sqrt[3]{{t_0}^{2}}, \sqrt[3]{t_0}, 1\right)}\\ \end{array} \end{array} \]

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)))) -0.9999995)
     (/ (- 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)))) <= -0.9999995) {
		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)))) <= -0.9999995)
		tmp = Float64(Float64(-p) / 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], -0.9999995], 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)))) < -0.999999500000000041

   1. Initial program 18.3%

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

   2. Taylor expanded in x around -inf 54.0%

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

      1. unpow2 54.0%

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

      2. unpow2 54.0%

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

      3. times-frac 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in p around -inf 56.8%

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

      1. associate-*r/ 56.8%

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

      2. neg-mul-1 56.8%

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

   7. Simplified 56.8%

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

   if -0.999999500000000041 < (/.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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. add-cube-cbrt 99.8%

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

      3. fma-def 99.8%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\sqrt[3]{{\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}}, \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}, 1\right)}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9999995)
   (/ (- p) x)
   (sqrt (* 0.5 (fma (/ 1.0 (hypot x (* p 2.0))) x 1.0)))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999995) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * fma((1.0 / hypot(x, (p * 2.0))), x, 1.0)));
	}
	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)))) <= -0.9999995)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * fma(Float64(1.0 / hypot(x, Float64(p * 2.0))), x, 1.0)));
	end
	return 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], -0.9999995], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(1.0 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * x + 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)))) < -0.999999500000000041

   1. Initial program 18.3%

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

   2. Taylor expanded in x around -inf 54.0%

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

      1. unpow2 54.0%

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

      2. unpow2 54.0%

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

      3. times-frac 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in p around -inf 56.8%

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

      1. associate-*r/ 56.8%

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

      2. neg-mul-1 56.8%

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

   7. Simplified 56.8%

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

   if -0.999999500000000041 < (/.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. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. clear-num 99.8%

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

      3. associate-/r/ 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. hypot-def 99.9%

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

      8. associate-*l* 99.9%

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

      9. sqrt-prod 99.9%

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

      10. metadata-eval 99.9%

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

      11. sqrt-unprod 50.0%

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

      12. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 1\right)}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9999995)
   (/ (- 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 / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999995) {
		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 / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999995) {
		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 / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999995:
		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 (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.9999995)
		tmp = Float64(Float64(-p) / 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 / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999995)
		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[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999995], 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 (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999500000000041

   1. Initial program 18.3%

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

   2. Taylor expanded in x around -inf 54.0%

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

      1. unpow2 54.0%

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

      2. unpow2 54.0%

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

      3. times-frac 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in p around -inf 56.8%

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

      1. associate-*r/ 56.8%

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

      2. neg-mul-1 56.8%

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

   7. Simplified 56.8%

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

   if -0.999999500000000041 < (/.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. 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 50.0%

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

      7. add-sqr-sqrt 99.9%

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

   3. 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 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999995:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 4: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 3.6 \cdot 10^{-199}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 2.75 \cdot 10^{-101}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 3.6e-199)
     1.0
     (if (<= p 1.15e-147)
       t_0
       (if (<= p 2.75e-101)
         1.0
         (if (<= p 8.5e-94)
           t_0
           (if (<= p 9.5e-51)
             (sqrt (+ 0.5 (* 0.25 (/ x p))))
             (if (<= p 1.3e-37) t_0 (sqrt 0.5)))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 3.6e-199) {
		tmp = 1.0;
	} else if (p <= 1.15e-147) {
		tmp = t_0;
	} else if (p <= 2.75e-101) {
		tmp = 1.0;
	} else if (p <= 8.5e-94) {
		tmp = t_0;
	} else if (p <= 9.5e-51) {
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	} else if (p <= 1.3e-37) {
		tmp = t_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 <= 3.6d-199) then
        tmp = 1.0d0
    else if (p <= 1.15d-147) then
        tmp = t_0
    else if (p <= 2.75d-101) then
        tmp = 1.0d0
    else if (p <= 8.5d-94) then
        tmp = t_0
    else if (p <= 9.5d-51) then
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p))))
    else if (p <= 1.3d-37) then
        tmp = t_0
    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 <= 3.6e-199) {
		tmp = 1.0;
	} else if (p <= 1.15e-147) {
		tmp = t_0;
	} else if (p <= 2.75e-101) {
		tmp = 1.0;
	} else if (p <= 8.5e-94) {
		tmp = t_0;
	} else if (p <= 9.5e-51) {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p))));
	} else if (p <= 1.3e-37) {
		tmp = t_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 <= 3.6e-199:
		tmp = 1.0
	elif p <= 1.15e-147:
		tmp = t_0
	elif p <= 2.75e-101:
		tmp = 1.0
	elif p <= 8.5e-94:
		tmp = t_0
	elif p <= 9.5e-51:
		tmp = math.sqrt((0.5 + (0.25 * (x / p))))
	elif p <= 1.3e-37:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 3.6e-199)
		tmp = 1.0;
	elseif (p <= 1.15e-147)
		tmp = t_0;
	elseif (p <= 2.75e-101)
		tmp = 1.0;
	elseif (p <= 8.5e-94)
		tmp = t_0;
	elseif (p <= 9.5e-51)
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p))));
	elseif (p <= 1.3e-37)
		tmp = t_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 <= 3.6e-199)
		tmp = 1.0;
	elseif (p <= 1.15e-147)
		tmp = t_0;
	elseif (p <= 2.75e-101)
		tmp = 1.0;
	elseif (p <= 8.5e-94)
		tmp = t_0;
	elseif (p <= 9.5e-51)
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	elseif (p <= 1.3e-37)
		tmp = t_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, 3.6e-199], 1.0, If[LessEqual[p, 1.15e-147], t$95$0, If[LessEqual[p, 2.75e-101], 1.0, If[LessEqual[p, 8.5e-94], t$95$0, If[LessEqual[p, 9.5e-51], N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[p, 1.3e-37], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 3.6000000000000002e-199 or 1.14999999999999995e-147 < p < 2.74999999999999986e-101

   1. Initial program 78.7%

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

   2. Taylor expanded in x around inf 42.0%

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

   if 3.6000000000000002e-199 < p < 1.14999999999999995e-147 or 2.74999999999999986e-101 < p < 8.50000000000000003e-94 or 9.4999999999999998e-51 < p < 1.2999999999999999e-37

   1. Initial program 44.7%

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

   2. Taylor expanded in x around -inf 31.7%

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

      1. unpow2 31.7%

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

      2. unpow2 31.7%

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

      3. times-frac 42.2%

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

   4. Simplified 42.2%

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

   5. Taylor expanded in p around -inf 59.3%

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

      1. associate-*r/ 59.3%

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

      2. neg-mul-1 59.3%

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

   7. Simplified 59.3%

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

   if 8.50000000000000003e-94 < p < 9.4999999999999998e-51

   1. Initial program 84.1%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l/ 52.6%

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

   4. Simplified 52.6%

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

      1. expm1-log1p-u 51.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(1 + \frac{x \cdot 0.5}{p}\right)}\right)\right)} \]

      2. expm1-udef 52.1%

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

      3. distribute-lft-in 52.1%

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

      4. metadata-eval 52.1%

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

      5. associate-/l* 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. expm1-def 51.8%

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

      2. expm1-log1p 52.6%

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

      3. associate-/r/ 52.6%

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

      4. *-commutative 52.6%

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

      5. associate-*r* 52.6%

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

      6. metadata-eval 52.6%

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

   8. Simplified 52.6%

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

   if 1.2999999999999999e-37 < p

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0 86.5%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.6 \cdot 10^{-199}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-147}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.75 \cdot 10^{-101}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 69.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 2.7 \cdot 10^{-202}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.92 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 6.8 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 2.7e-202)
     1.0
     (if (<= p 1.92e-146)
       t_0
       (if (<= p 6.8e-50) 1.0 (if (<= p 1.3e-37) t_0 (sqrt 0.5)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 2.7e-202) {
		tmp = 1.0;
	} else if (p <= 1.92e-146) {
		tmp = t_0;
	} else if (p <= 6.8e-50) {
		tmp = 1.0;
	} else if (p <= 1.3e-37) {
		tmp = t_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 <= 2.7d-202) then
        tmp = 1.0d0
    else if (p <= 1.92d-146) then
        tmp = t_0
    else if (p <= 6.8d-50) then
        tmp = 1.0d0
    else if (p <= 1.3d-37) then
        tmp = t_0
    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 <= 2.7e-202) {
		tmp = 1.0;
	} else if (p <= 1.92e-146) {
		tmp = t_0;
	} else if (p <= 6.8e-50) {
		tmp = 1.0;
	} else if (p <= 1.3e-37) {
		tmp = t_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 <= 2.7e-202:
		tmp = 1.0
	elif p <= 1.92e-146:
		tmp = t_0
	elif p <= 6.8e-50:
		tmp = 1.0
	elif p <= 1.3e-37:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 2.7e-202)
		tmp = 1.0;
	elseif (p <= 1.92e-146)
		tmp = t_0;
	elseif (p <= 6.8e-50)
		tmp = 1.0;
	elseif (p <= 1.3e-37)
		tmp = t_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 <= 2.7e-202)
		tmp = 1.0;
	elseif (p <= 1.92e-146)
		tmp = t_0;
	elseif (p <= 6.8e-50)
		tmp = 1.0;
	elseif (p <= 1.3e-37)
		tmp = t_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, 2.7e-202], 1.0, If[LessEqual[p, 1.92e-146], t$95$0, If[LessEqual[p, 6.8e-50], 1.0, If[LessEqual[p, 1.3e-37], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.6999999999999999e-202 or 1.91999999999999999e-146 < p < 6.80000000000000029e-50

   1. Initial program 78.7%

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

   2. Taylor expanded in x around inf 42.0%

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

   if 2.6999999999999999e-202 < p < 1.91999999999999999e-146 or 6.80000000000000029e-50 < p < 1.2999999999999999e-37

   1. Initial program 40.2%

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

   2. Taylor expanded in x around -inf 33.9%

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

      1. unpow2 33.9%

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

      2. unpow2 33.9%

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

      3. times-frac 46.3%

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

   4. Simplified 46.3%

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

   5. Taylor expanded in p around -inf 63.4%

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

      1. associate-*r/ 63.4%

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

      2. neg-mul-1 63.4%

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

   7. Simplified 63.4%

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

   if 1.2999999999999999e-37 < p

   1. Initial program 96.2%

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

   2. Taylor expanded in x around 0 86.5%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.7 \cdot 10^{-202}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.92 \cdot 10^{-146}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 6.8 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 68.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 8.5 \cdot 10^{-94} \lor \neg \left(p \leq 1.35 \cdot 10^{-50}\right) \land p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (or (<= p 8.5e-94) (and (not (<= p 1.35e-50)) (<= p 1.3e-37)))
   (/ (- p) x)
   (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((p <= 8.5e-94) || (!(p <= 1.35e-50) && (p <= 1.3e-37))) {
		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 <= 8.5d-94) .or. (.not. (p <= 1.35d-50)) .and. (p <= 1.3d-37)) 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 <= 8.5e-94) || (!(p <= 1.35e-50) && (p <= 1.3e-37))) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (p <= 8.5e-94) or (not (p <= 1.35e-50) and (p <= 1.3e-37)):
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if ((p <= 8.5e-94) || (!(p <= 1.35e-50) && (p <= 1.3e-37)))
		tmp = Float64(Float64(-p) / 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 <= 8.5e-94) || (~((p <= 1.35e-50)) && (p <= 1.3e-37)))
		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[Or[LessEqual[p, 8.5e-94], And[N[Not[LessEqual[p, 1.35e-50]], $MachinePrecision], LessEqual[p, 1.3e-37]]], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 8.50000000000000003e-94 or 1.35e-50 < p < 1.2999999999999999e-37

   1. Initial program 75.2%

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

   2. Taylor expanded in x around -inf 18.9%

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

      1. unpow2 18.9%

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

      2. unpow2 18.9%

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

      3. times-frac 22.3%

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

   4. Simplified 22.3%

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

   5. Taylor expanded in p around -inf 18.7%

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

      1. associate-*r/ 18.7%

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

      2. neg-mul-1 18.7%

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

   7. Simplified 18.7%

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

   if 8.50000000000000003e-94 < p < 1.35e-50 or 1.2999999999999999e-37 < p

   1. Initial program 95.2%

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

   2. Taylor expanded in x around 0 83.3%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 8.5 \cdot 10^{-94} \lor \neg \left(p \leq 1.35 \cdot 10^{-50}\right) \land p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 28.1% 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(Float64(-p) / 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 58.2%

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

   2. Taylor expanded in x around -inf 30.2%

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

      1. unpow2 30.2%

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

      2. unpow2 30.2%

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

      3. times-frac 35.4%

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

   4. Simplified 35.4%

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

   5. Taylor expanded in p around -inf 30.3%

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

      1. associate-*r/ 30.3%

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

      2. neg-mul-1 30.3%

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

   7. Simplified 30.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around -inf 4.4%

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

      1. unpow2 4.4%

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

      2. unpow2 4.4%

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

      3. times-frac 4.7%

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

   4. Simplified 4.7%

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

   5. Taylor expanded in p around 0 3.7%

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

4. Final simplification 15.9%

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

Alternative 8: 6.6% 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 80.7%

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

2. Taylor expanded in x around -inf 16.3%

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

   1. unpow2 16.3%

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

   2. unpow2 16.3%

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

   3. times-frac 18.9%

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

4. Simplified 18.9%

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

5. Taylor expanded in p around 0 16.7%

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

6. Final simplification 16.7%

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

Developer target: 80.1% 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 2023230 
(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))))))))