Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.8%

Time: 10.9s

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: 79.1% 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: 99.8% accurate, 0.5× speedup

Math

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

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 (fma (/ 1.0 (hypot x (* p_m 2.0))) x 1.0)))))

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 * fma((1.0 / hypot(x, (p_m * 2.0))), x, 1.0)));
	}
	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(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * fma(Float64(1.0 / hypot(x, Float64(p_m * 2.0))), x, 1.0)));
	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[(N[(1.0 / N[Sqrt[x ^ 2 + N[(p$95$m * 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 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 15.9%

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

   3. Taylor expanded in x around -inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. distribute-neg-frac2 62.9%

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

      3. associate-*r* 63.1%

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

      4. *-commutative 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in p around 0 62.9%

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

      1. mul-1-neg 62.9%

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

      2. associate-/l* 62.8%

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

      3. distribute-lft-neg-in 62.8%

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

      4. associate-/l* 63.1%

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

   8. Simplified 63.1%

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

      1. distribute-lft-neg-out 63.1%

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

      2. neg-sub0 63.1%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 62.8%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 63.5%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 63.5%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 63.5%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 63.8%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 63.8%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 63.8%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 63.8%

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

       2. distribute-neg-frac2 63.8%

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

   12. Simplified 63.8%

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

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

   1. Initial program 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. +-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 99.9%

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

      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-define 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-define 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 50.5%

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

      12. add-sqr-sqrt 99.9%

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

   4. 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 90.6%

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

Alternative 2: 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(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	end
	return tmp
end

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.9%

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

   3. Taylor expanded in x around -inf 62.9%

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

      1. mul-1-neg 62.9%

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

      2. distribute-neg-frac2 62.9%

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

      3. associate-*r* 63.1%

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

      4. *-commutative 63.1%

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

   5. Simplified 63.1%

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

   6. Taylor expanded in p around 0 62.9%

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

      1. mul-1-neg 62.9%

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

      2. associate-/l* 62.8%

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

      3. distribute-lft-neg-in 62.8%

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

      4. associate-/l* 63.1%

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

   8. Simplified 63.1%

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

      1. distribute-lft-neg-out 63.1%

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

      2. neg-sub0 63.1%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 62.8%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 63.5%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 63.5%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 63.5%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 63.8%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 63.8%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 63.8%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 63.8%

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

       2. distribute-neg-frac2 63.8%

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

   12. Simplified 63.8%

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

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

   1. Initial program 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-define 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 50.5%

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

      7. add-sqr-sqrt 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 90.6%

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

Alternative 3: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 1.5 \cdot 10^{-201}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 7.5 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 5.9 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 1.5e-201)
     t_0
     (if (<= p_m 7.5e-143) 1.0 (if (<= p_m 5.9e-61) t_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 <= 1.5e-201) {
		tmp = t_0;
	} else if (p_m <= 7.5e-143) {
		tmp = 1.0;
	} else if (p_m <= 5.9e-61) {
		tmp = t_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 <= 1.5d-201) then
        tmp = t_0
    else if (p_m <= 7.5d-143) then
        tmp = 1.0d0
    else if (p_m <= 5.9d-61) then
        tmp = t_0
    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 <= 1.5e-201) {
		tmp = t_0;
	} else if (p_m <= 7.5e-143) {
		tmp = 1.0;
	} else if (p_m <= 5.9e-61) {
		tmp = t_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 <= 1.5e-201:
		tmp = t_0
	elif p_m <= 7.5e-143:
		tmp = 1.0
	elif p_m <= 5.9e-61:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 1.5e-201)
		tmp = t_0;
	elseif (p_m <= 7.5e-143)
		tmp = 1.0;
	elseif (p_m <= 5.9e-61)
		tmp = t_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 <= 1.5e-201)
		tmp = t_0;
	elseif (p_m <= 7.5e-143)
		tmp = 1.0;
	elseif (p_m <= 5.9e-61)
		tmp = t_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, 1.5e-201], t$95$0, If[LessEqual[p$95$m, 7.5e-143], 1.0, If[LessEqual[p$95$m, 5.9e-61], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.50000000000000001e-201 or 7.5000000000000003e-143 < p < 5.89999999999999972e-61

   1. Initial program 72.4%

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

   3. Taylor expanded in x around -inf 20.3%

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

      1. mul-1-neg 20.3%

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

      2. distribute-neg-frac2 20.3%

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

      3. associate-*r* 20.4%

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

      4. *-commutative 20.4%

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

   5. Simplified 20.4%

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

   6. Taylor expanded in p around 0 20.3%

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

      1. mul-1-neg 20.3%

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

      2. associate-/l* 20.3%

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

      3. distribute-lft-neg-in 20.3%

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

      4. associate-/l* 20.4%

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

   8. Simplified 20.4%

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

      1. distribute-lft-neg-out 20.4%

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

      2. neg-sub0 20.4%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 20.3%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 20.5%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 20.5%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 20.5%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 20.6%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 20.6%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 20.6%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 20.6%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 20.6%

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

       2. distribute-neg-frac2 20.6%

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

   12. Simplified 20.6%

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

   if 1.50000000000000001e-201 < p < 7.5000000000000003e-143

   1. Initial program 54.4%

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

   3. Taylor expanded in x around inf 46.3%

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

   if 5.89999999999999972e-61 < p

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0 90.9%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-143}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 6.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 6.4e-64) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 6.4e-64) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 6.4d-64) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 6.4e-64) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 6.4e-64:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 6.4e-64)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 6.4e-64)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 6.4e-64], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 6.39999999999999951e-64

   1. Initial program 70.7%

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

   3. Taylor expanded in x around -inf 23.6%

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

      1. mul-1-neg 23.6%

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

      2. distribute-neg-frac2 23.6%

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

      3. associate-*r* 23.7%

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

      4. *-commutative 23.7%

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

   5. Simplified 23.7%

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

   6. Taylor expanded in p around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. associate-/l* 23.5%

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

      3. distribute-lft-neg-in 23.5%

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

      4. associate-/l* 23.7%

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

   8. Simplified 23.7%

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

      1. distribute-lft-neg-out 23.7%

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

      2. neg-sub0 23.7%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 23.5%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 23.8%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 23.8%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 23.8%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 23.9%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 23.9%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 23.9%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 23.9%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 23.9%

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

       2. distribute-neg-frac2 23.9%

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

   12. Simplified 23.9%

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

   if 6.39999999999999951e-64 < p

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0 90.9%

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

Alternative 5: 35.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1e-134) (/ p_m (- x)) 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-134) {
		tmp = p_m / -x;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-134)) then
        tmp = p_m / -x
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-134) {
		tmp = p_m / -x;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1e-134:
		tmp = p_m / -x
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1e-134)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1e-134)
		tmp = p_m / -x;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1e-134], N[(p$95$m / (-x)), $MachinePrecision], 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000004e-134

   1. Initial program 56.9%

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

   3. Taylor expanded in x around -inf 33.7%

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

      1. mul-1-neg 33.7%

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

      2. distribute-neg-frac2 33.7%

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

      3. associate-*r* 33.9%

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

      4. *-commutative 33.9%

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

   5. Simplified 33.9%

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

   6. Taylor expanded in p around 0 33.7%

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

      1. mul-1-neg 33.7%

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

      2. associate-/l* 33.7%

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

      3. distribute-lft-neg-in 33.7%

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

      4. associate-/l* 33.9%

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

   8. Simplified 33.9%

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

      1. distribute-lft-neg-out 33.9%

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

      2. neg-sub0 33.9%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 33.7%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 34.1%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 34.1%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 34.1%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 34.2%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 34.2%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 34.2%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   10. Applied egg-rr 34.2%

       \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   11. Step-by-step derivation

       1. neg-sub0 34.2%

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

       2. distribute-neg-frac2 34.2%

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

   12. Simplified 34.2%

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

   if -1.00000000000000004e-134 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 63.7%

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

   5. Applied egg-rr 19.3%

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

Alternative 6: 16.6% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 4.8 \cdot 10^{-253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 4.8e-253) 0.0 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.8e-253) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 4.8d-253) then
        tmp = 0.0d0
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.8e-253) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 4.8e-253:
		tmp = 0.0
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 4.8e-253)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 4.8e-253)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 4.8e-253], 0.0, 1.5]

Derivation

1. Split input into 2 regimes

2. if p < 4.80000000000000018e-253

   1. Initial program 78.3%

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

   3. Taylor expanded in x around -inf 8.1%

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

      1. neg-mul-1 8.1%

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

   5. Simplified 8.1%

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

   6. Taylor expanded in x around 0 8.1%

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

   if 4.80000000000000018e-253 < p

   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. Taylor expanded in x around 0 65.7%

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

   5. Applied egg-rr 15.7%

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

Alternative 7: 14.9% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 5.4 \cdot 10^{-253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 5.4e-253) 0.0 0.125))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 5.4e-253) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 5.4d-253) then
        tmp = 0.0d0
    else
        tmp = 0.125d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 5.4e-253) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 5.4e-253:
		tmp = 0.0
	else:
		tmp = 0.125
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 5.4e-253)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 5.4e-253)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 5.4e-253], 0.0, 0.125]

Derivation

1. Split input into 2 regimes

2. if p < 5.39999999999999998e-253

   1. Initial program 78.3%

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

   3. Taylor expanded in x around -inf 8.1%

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

      1. neg-mul-1 8.1%

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

   5. Simplified 8.1%

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

   6. Taylor expanded in x around 0 8.1%

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

   if 5.39999999999999998e-253 < p

   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. Taylor expanded in x around 0 65.7%

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

   5. Applied egg-rr 14.0%

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

Alternative 8: 6.3% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in x around -inf 6.4%

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

   1. neg-mul-1 6.4%

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

5. Simplified 6.4%

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

6. Taylor expanded in x around 0 6.4%

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

Developer Target 1: 79.1% 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 2024132 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

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

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