Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.8%

Time: 11.8s

Alternatives: 7

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% 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, 0.4× 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(p_m / Float64(-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 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 18.6%

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

      1. +-commutative 18.6%

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

      2. associate-*l* 18.6%

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

      3. fma-define 18.6%

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

      4. sqr-neg 18.6%

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

      5. fma-define 18.6%

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

      6. associate-*l* 18.6%

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

      7. +-commutative 18.6%

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

      8. distribute-lft-in 18.6%

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

      9. metadata-eval 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in x around -inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. distribute-neg-frac2 62.4%

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

   7. Simplified 62.4%

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

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

   1. Initial program 99.7%

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

      1. add-exp-log 99.7%

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

      2. log1p-define 99.7%

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

      3. div-inv 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-*r* 99.7%

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

      6. fma-undefine 99.7%

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

      7. div-inv 99.7%

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

      8. fma-undefine 99.7%

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

      9. associate-*r* 99.7%

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

      10. add-sqr-sqrt 99.7%

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

      11. hypot-define 99.7%

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

      12. associate-*r* 99.7%

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

      13. *-commutative 99.7%

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

      14. sqrt-prod 99.7%

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

      15. sqrt-prod 48.6%

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

      16. add-sqr-sqrt 99.7%

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

      17. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 91.1%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

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 (+ (/ x (hypot x (* p_m 2.0))) 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 * ((x / hypot(x, (p_m * 2.0))) + 1.0)));
	}
	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 * ((x / Math.hypot(x, (p_m * 2.0))) + 1.0)));
	}
	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 * ((x / math.hypot(x, (p_m * 2.0))) + 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 * Float64(Float64(x / hypot(x, Float64(p_m * 2.0))) + 1.0)));
	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 * ((x / hypot(x, (p_m * 2.0))) + 1.0)));
	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[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 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 18.6%

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

      1. +-commutative 18.6%

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

      2. associate-*l* 18.6%

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

      3. fma-define 18.6%

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

      4. sqr-neg 18.6%

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

      5. fma-define 18.6%

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

      6. associate-*l* 18.6%

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

      7. +-commutative 18.6%

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

      8. distribute-lft-in 18.6%

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

      9. metadata-eval 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in x around -inf 62.4%

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

      1. mul-1-neg 62.4%

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

      2. distribute-neg-frac2 62.4%

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

   7. Simplified 62.4%

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

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

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. fma-define 99.7%

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

      4. sqr-neg 99.7%

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

      5. fma-define 99.7%

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

      6. associate-*l* 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-lft-in 99.7%

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

      9. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. fma-undefine 99.7%

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

      3. associate-*r* 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-rgt1-in 99.7%

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

      6. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 91.1%

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

Alternative 3: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= x -2.3e+38)
     t_0
     (if (<= x -2.2e+20)
       (sqrt 0.5)
       (if (<= x -2.25e-12) t_0 (if (<= x 2.2e-61) (sqrt 0.5) 1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (x <= -2.3e+38) {
		tmp = t_0;
	} else if (x <= -2.2e+20) {
		tmp = sqrt(0.5);
	} else if (x <= -2.25e-12) {
		tmp = t_0;
	} else if (x <= 2.2e-61) {
		tmp = sqrt(0.5);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (x <= (-2.3d+38)) then
        tmp = t_0
    else if (x <= (-2.2d+20)) then
        tmp = sqrt(0.5d0)
    else if (x <= (-2.25d-12)) then
        tmp = t_0
    else if (x <= 2.2d-61) then
        tmp = sqrt(0.5d0)
    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 t_0 = p_m / -x;
	double tmp;
	if (x <= -2.3e+38) {
		tmp = t_0;
	} else if (x <= -2.2e+20) {
		tmp = Math.sqrt(0.5);
	} else if (x <= -2.25e-12) {
		tmp = t_0;
	} else if (x <= 2.2e-61) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if x <= -2.3e+38:
		tmp = t_0
	elif x <= -2.2e+20:
		tmp = math.sqrt(0.5)
	elif x <= -2.25e-12:
		tmp = t_0
	elif x <= 2.2e-61:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (x <= -2.3e+38)
		tmp = t_0;
	elseif (x <= -2.2e+20)
		tmp = sqrt(0.5);
	elseif (x <= -2.25e-12)
		tmp = t_0;
	elseif (x <= 2.2e-61)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	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 (x <= -2.3e+38)
		tmp = t_0;
	elseif (x <= -2.2e+20)
		tmp = sqrt(0.5);
	elseif (x <= -2.25e-12)
		tmp = t_0;
	elseif (x <= 2.2e-61)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	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[x, -2.3e+38], t$95$0, If[LessEqual[x, -2.2e+20], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, -2.25e-12], t$95$0, If[LessEqual[x, 2.2e-61], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e38 or -2.2e20 < x < -2.2499999999999999e-12

   1. Initial program 45.8%

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

      1. +-commutative 45.8%

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

      2. associate-*l* 45.8%

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

      3. fma-define 45.8%

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

      4. sqr-neg 45.8%

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

      5. fma-define 45.8%

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

      6. associate-*l* 45.8%

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

      7. +-commutative 45.8%

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

      8. distribute-lft-in 45.8%

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

      9. metadata-eval 45.8%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in x around -inf 42.6%

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

      1. mul-1-neg 42.6%

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

      2. distribute-neg-frac2 42.6%

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

   7. Simplified 42.6%

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

   if -2.3000000000000001e38 < x < -2.2e20 or -2.2499999999999999e-12 < x < 2.20000000000000009e-61

   1. Initial program 85.2%

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

   3. Taylor expanded in x around 0 78.3%

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

   if 2.20000000000000009e-61 < x

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. log1p-define 100.0%

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

      3. div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. fma-undefine 100.0%

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

      7. div-inv 100.0%

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

      8. fma-undefine 100.0%

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

      9. associate-*r* 100.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. hypot-define 100.0%

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

      12. associate-*r* 100.0%

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

      13. *-commutative 100.0%

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

      14. sqrt-prod 100.0%

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

      15. sqrt-prod 50.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 75.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-12}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p\_m}}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= x -1.5e+38)
     t_0
     (if (<= x -1.4e+20)
       (sqrt (+ 0.5 (* 0.25 (/ x p_m))))
       (if (<= x -1.35e-11) t_0 (if (<= x 2.7e-62) (sqrt 0.5) 1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (x <= -1.5e+38) {
		tmp = t_0;
	} else if (x <= -1.4e+20) {
		tmp = sqrt((0.5 + (0.25 * (x / p_m))));
	} else if (x <= -1.35e-11) {
		tmp = t_0;
	} else if (x <= 2.7e-62) {
		tmp = sqrt(0.5);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (x <= (-1.5d+38)) then
        tmp = t_0
    else if (x <= (-1.4d+20)) then
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p_m))))
    else if (x <= (-1.35d-11)) then
        tmp = t_0
    else if (x <= 2.7d-62) then
        tmp = sqrt(0.5d0)
    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 t_0 = p_m / -x;
	double tmp;
	if (x <= -1.5e+38) {
		tmp = t_0;
	} else if (x <= -1.4e+20) {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p_m))));
	} else if (x <= -1.35e-11) {
		tmp = t_0;
	} else if (x <= 2.7e-62) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if x <= -1.5e+38:
		tmp = t_0
	elif x <= -1.4e+20:
		tmp = math.sqrt((0.5 + (0.25 * (x / p_m))))
	elif x <= -1.35e-11:
		tmp = t_0
	elif x <= 2.7e-62:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (x <= -1.5e+38)
		tmp = t_0;
	elseif (x <= -1.4e+20)
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p_m))));
	elseif (x <= -1.35e-11)
		tmp = t_0;
	elseif (x <= 2.7e-62)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	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 (x <= -1.5e+38)
		tmp = t_0;
	elseif (x <= -1.4e+20)
		tmp = sqrt((0.5 + (0.25 * (x / p_m))));
	elseif (x <= -1.35e-11)
		tmp = t_0;
	elseif (x <= 2.7e-62)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	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[x, -1.5e+38], t$95$0, If[LessEqual[x, -1.4e+20], N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -1.35e-11], t$95$0, If[LessEqual[x, 2.7e-62], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.5000000000000001e38 or -1.4e20 < x < -1.35000000000000002e-11

   1. Initial program 45.8%

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

      1. +-commutative 45.8%

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

      2. associate-*l* 45.8%

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

      3. fma-define 45.8%

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

      4. sqr-neg 45.8%

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

      5. fma-define 45.8%

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

      6. associate-*l* 45.8%

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

      7. +-commutative 45.8%

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

      8. distribute-lft-in 45.8%

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

      9. metadata-eval 45.8%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in x around -inf 42.6%

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

      1. mul-1-neg 42.6%

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

      2. distribute-neg-frac2 42.6%

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

   7. Simplified 42.6%

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

   if -1.5000000000000001e38 < x < -1.4e20

   1. Initial program 79.6%

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

      1. +-commutative 79.6%

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

      2. associate-*l* 79.6%

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

      3. fma-define 79.6%

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

      4. sqr-neg 79.6%

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

      5. fma-define 79.6%

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

      6. associate-*l* 79.6%

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

      7. +-commutative 79.6%

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

      8. distribute-lft-in 79.6%

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

      9. metadata-eval 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in x around 0 68.7%

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

   if -1.35000000000000002e-11 < x < 2.70000000000000019e-62

   1. Initial program 85.9%

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

   3. Taylor expanded in x around 0 79.8%

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

   if 2.70000000000000019e-62 < x

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. log1p-define 100.0%

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

      3. div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. fma-undefine 100.0%

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

      7. div-inv 100.0%

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

      8. fma-undefine 100.0%

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

      9. associate-*r* 100.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. hypot-define 100.0%

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

      12. associate-*r* 100.0%

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

      13. *-commutative 100.0%

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

      14. sqrt-prod 100.0%

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

      15. sqrt-prod 50.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 75.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-11}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.1% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2e-140) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2e-140) {
		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 <= (-2d-140)) 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 <= -2e-140) {
		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 <= -2e-140:
		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 <= -2e-140)
		tmp = Float64(p_m / Float64(-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 <= -2e-140)
		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, -2e-140], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2e-140

   1. Initial program 62.0%

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

      1. +-commutative 62.0%

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

      2. associate-*l* 62.0%

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

      3. fma-define 62.0%

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

      4. sqr-neg 62.0%

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

      5. fma-define 62.0%

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

      6. associate-*l* 62.0%

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

      7. +-commutative 62.0%

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

      8. distribute-lft-in 62.0%

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

      9. metadata-eval 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in x around -inf 30.7%

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

      1. mul-1-neg 30.7%

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

      2. distribute-neg-frac2 30.7%

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

   7. Simplified 30.7%

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

   if -2e-140 < x

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. log1p-define 100.0%

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

      3. div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. fma-undefine 100.0%

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

      7. div-inv 100.0%

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

      8. fma-undefine 100.0%

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

      9. associate-*r* 100.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. hypot-define 100.0%

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

      12. associate-*r* 100.0%

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

      13. *-commutative 100.0%

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

      14. sqrt-prod 100.0%

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

      15. sqrt-prod 50.8%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around inf 59.8%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-140}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.1% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -8.5e+59) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -8.5e+59) {
		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 <= (-8.5d+59)) 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 <= -8.5e+59) {
		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 <= -8.5e+59:
		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 <= -8.5e+59)
		tmp = 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 <= -8.5e+59)
		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, -8.5e+59], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -8.4999999999999999e59

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

      2. associate-*l* 52.4%

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

      3. fma-define 52.4%

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

      4. sqr-neg 52.4%

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

      5. fma-define 52.4%

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

      6. associate-*l* 52.4%

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

      7. +-commutative 52.4%

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

      8. distribute-lft-in 52.4%

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

      9. metadata-eval 52.4%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in x around -inf 56.9%

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

   6. Taylor expanded in p around 0 49.7%

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

   if -8.4999999999999999e59 < x

   1. Initial program 85.8%

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

      1. add-exp-log 85.8%

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

      2. log1p-define 85.8%

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

      3. div-inv 85.8%

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

      4. +-commutative 85.8%

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

      5. associate-*r* 85.8%

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

      6. fma-undefine 85.8%

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

      7. div-inv 85.8%

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

      8. fma-undefine 85.8%

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

      9. associate-*r* 85.8%

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

      10. add-sqr-sqrt 85.8%

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

      11. hypot-define 85.8%

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

      12. associate-*r* 85.8%

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

      13. *-commutative 85.8%

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

      14. sqrt-prod 85.8%

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

      15. sqrt-prod 42.6%

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

      16. add-sqr-sqrt 85.8%

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

      17. metadata-eval 85.8%

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

   4. Applied egg-rr 85.8%

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

   6. Applied egg-rr 85.0%

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

   7. Taylor expanded in x around inf 40.9%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.0% 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 81.0%

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

   1. add-exp-log 81.0%

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

   2. log1p-define 81.0%

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

   3. div-inv 80.3%

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

   4. +-commutative 80.3%

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

   5. associate-*r* 80.3%

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

   6. fma-undefine 80.3%

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

   7. div-inv 81.0%

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

   8. fma-undefine 81.0%

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

   9. associate-*r* 81.0%

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

   10. add-sqr-sqrt 81.0%

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

   11. hypot-define 81.0%

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

   12. associate-*r* 81.0%

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

   13. *-commutative 81.0%

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

   14. sqrt-prod 81.0%

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

   15. sqrt-prod 40.0%

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

   16. add-sqr-sqrt 81.0%

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

   17. metadata-eval 81.0%

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

4. Applied egg-rr 81.0%

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

6. Applied egg-rr 80.2%

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

7. Taylor expanded in x around inf 36.4%

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

8. Final simplification 36.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 79.6% 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 2024095 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (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))))))))