Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.7%

Time: 10.4s

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: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% 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 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf 62.8%

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

      1. mul-1-neg 62.8%

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

      2. associate-/l* 62.9%

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

      3. distribute-rgt-neg-in 62.9%

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

      4. *-commutative 62.9%

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

      5. associate-/l* 63.3%

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

   5. Simplified 63.3%

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

      1. distribute-rgt-neg-out 63.3%

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

      2. neg-sub0 63.3%

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

      3. associate-*r/ 62.9%

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

      4. sqrt-unprod 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. div-inv 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. neg-sub0 63.8%

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

      2. distribute-neg-frac2 63.8%

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

   9. Simplified 63.8%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x 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. +-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. associate-/r/ 100.0%

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

      4. fma-define 100.0%

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

      5. +-commutative 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. hypot-define 100.0%

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

      8. associate-*l* 100.0%

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

      9. sqrt-prod 100.0%

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

      10. metadata-eval 100.0%

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

      11. sqrt-unprod 51.8%

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

      12. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 90.5%

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

Alternative 2: 99.7% 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 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf 62.8%

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

      1. mul-1-neg 62.8%

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

      2. associate-/l* 62.9%

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

      3. distribute-rgt-neg-in 62.9%

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

      4. *-commutative 62.9%

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

      5. associate-/l* 63.3%

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

   5. Simplified 63.3%

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

      1. distribute-rgt-neg-out 63.3%

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

      2. neg-sub0 63.3%

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

      3. associate-*r/ 62.9%

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

      4. sqrt-unprod 63.7%

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

      5. metadata-eval 63.7%

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

      6. metadata-eval 63.7%

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

      7. div-inv 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. neg-sub0 63.8%

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

      2. distribute-neg-frac2 63.8%

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

   9. Simplified 63.8%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x 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-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 51.8%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 90.5%

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

Alternative 3: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 3.7 \cdot 10^{-228}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.62 \cdot 10^{-139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 4.3 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.12 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x}{p\_m} \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 3.7e-228)
     1.0
     (if (<= p_m 1.62e-139)
       t_0
       (if (<= p_m 4.3e-97)
         1.0
         (if (<= p_m 1.12e-44) t_0 (sqrt (+ 0.5 (* (/ x p_m) 0.25)))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 3.7e-228) {
		tmp = 1.0;
	} else if (p_m <= 1.62e-139) {
		tmp = t_0;
	} else if (p_m <= 4.3e-97) {
		tmp = 1.0;
	} else if (p_m <= 1.12e-44) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 + ((x / p_m) * 0.25)));
	}
	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 <= 3.7d-228) then
        tmp = 1.0d0
    else if (p_m <= 1.62d-139) then
        tmp = t_0
    else if (p_m <= 4.3d-97) then
        tmp = 1.0d0
    else if (p_m <= 1.12d-44) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 + ((x / p_m) * 0.25d0)))
    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 <= 3.7e-228) {
		tmp = 1.0;
	} else if (p_m <= 1.62e-139) {
		tmp = t_0;
	} else if (p_m <= 4.3e-97) {
		tmp = 1.0;
	} else if (p_m <= 1.12e-44) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 + ((x / p_m) * 0.25)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 3.7e-228:
		tmp = 1.0
	elif p_m <= 1.62e-139:
		tmp = t_0
	elif p_m <= 4.3e-97:
		tmp = 1.0
	elif p_m <= 1.12e-44:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 + ((x / p_m) * 0.25)))
	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 <= 3.7e-228)
		tmp = 1.0;
	elseif (p_m <= 1.62e-139)
		tmp = t_0;
	elseif (p_m <= 4.3e-97)
		tmp = 1.0;
	elseif (p_m <= 1.12e-44)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x / p_m) * 0.25)));
	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 <= 3.7e-228)
		tmp = 1.0;
	elseif (p_m <= 1.62e-139)
		tmp = t_0;
	elseif (p_m <= 4.3e-97)
		tmp = 1.0;
	elseif (p_m <= 1.12e-44)
		tmp = t_0;
	else
		tmp = sqrt((0.5 + ((x / p_m) * 0.25)));
	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, 3.7e-228], 1.0, If[LessEqual[p$95$m, 1.62e-139], t$95$0, If[LessEqual[p$95$m, 4.3e-97], 1.0, If[LessEqual[p$95$m, 1.12e-44], t$95$0, N[Sqrt[N[(0.5 + N[(N[(x / p$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.7e-228 or 1.62000000000000001e-139 < p < 4.3e-97

   1. Initial program 77.3%

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

   3. Taylor expanded in x around inf 39.4%

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

   if 3.7e-228 < p < 1.62000000000000001e-139 or 4.3e-97 < p < 1.1200000000000001e-44

   1. Initial program 44.9%

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

   3. Taylor expanded in x around -inf 61.2%

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

      1. mul-1-neg 61.2%

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

      2. associate-/l* 61.2%

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

      3. distribute-rgt-neg-in 61.2%

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

      4. *-commutative 61.2%

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

      5. associate-/l* 61.7%

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

   5. Simplified 61.7%

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

      1. distribute-rgt-neg-out 61.7%

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

      2. neg-sub0 61.7%

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

      3. associate-*r/ 61.2%

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

      4. sqrt-unprod 62.0%

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

      5. metadata-eval 62.0%

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

      6. metadata-eval 62.0%

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

      7. div-inv 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. neg-sub0 62.2%

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

      2. distribute-neg-frac2 62.2%

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

   9. Simplified 62.2%

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

   if 1.1200000000000001e-44 < p

   1. Initial program 94.4%

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

      1. add-sqr-sqrt 94.4%

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

      2. hypot-define 94.4%

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

      3. associate-*l* 94.4%

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

      4. sqrt-prod 94.4%

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

      5. metadata-eval 94.4%

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

      6. sqrt-unprod 94.4%

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

      7. add-sqr-sqrt 94.4%

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in x around 0 85.9%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]
   6. Step-by-step derivation

      1. *-commutative 85.9%

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

   7. Simplified 85.9%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{x}{p} \cdot 0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.7 \cdot 10^{-228}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.62 \cdot 10^{-139}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.12 \cdot 10^{-44}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x}{p} \cdot 0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 2.95 \cdot 10^{-225}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.9 \cdot 10^{-140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 9.3 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 2.95e-225)
     1.0
     (if (<= p_m 3.9e-140)
       t_0
       (if (<= p_m 2.9e-95) 1.0 (if (<= p_m 9.3e-45) 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 <= 2.95e-225) {
		tmp = 1.0;
	} else if (p_m <= 3.9e-140) {
		tmp = t_0;
	} else if (p_m <= 2.9e-95) {
		tmp = 1.0;
	} else if (p_m <= 9.3e-45) {
		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 <= 2.95d-225) then
        tmp = 1.0d0
    else if (p_m <= 3.9d-140) then
        tmp = t_0
    else if (p_m <= 2.9d-95) then
        tmp = 1.0d0
    else if (p_m <= 9.3d-45) 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 <= 2.95e-225) {
		tmp = 1.0;
	} else if (p_m <= 3.9e-140) {
		tmp = t_0;
	} else if (p_m <= 2.9e-95) {
		tmp = 1.0;
	} else if (p_m <= 9.3e-45) {
		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 <= 2.95e-225:
		tmp = 1.0
	elif p_m <= 3.9e-140:
		tmp = t_0
	elif p_m <= 2.9e-95:
		tmp = 1.0
	elif p_m <= 9.3e-45:
		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 <= 2.95e-225)
		tmp = 1.0;
	elseif (p_m <= 3.9e-140)
		tmp = t_0;
	elseif (p_m <= 2.9e-95)
		tmp = 1.0;
	elseif (p_m <= 9.3e-45)
		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 <= 2.95e-225)
		tmp = 1.0;
	elseif (p_m <= 3.9e-140)
		tmp = t_0;
	elseif (p_m <= 2.9e-95)
		tmp = 1.0;
	elseif (p_m <= 9.3e-45)
		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, 2.95e-225], 1.0, If[LessEqual[p$95$m, 3.9e-140], t$95$0, If[LessEqual[p$95$m, 2.9e-95], 1.0, If[LessEqual[p$95$m, 9.3e-45], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.95000000000000028e-225 or 3.90000000000000019e-140 < p < 2.90000000000000002e-95

   1. Initial program 77.3%

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

   3. Taylor expanded in x around inf 39.4%

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

   if 2.95000000000000028e-225 < p < 3.90000000000000019e-140 or 2.90000000000000002e-95 < p < 9.30000000000000027e-45

   1. Initial program 44.9%

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

   3. Taylor expanded in x around -inf 61.2%

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

      1. mul-1-neg 61.2%

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

      2. associate-/l* 61.2%

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

      3. distribute-rgt-neg-in 61.2%

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

      4. *-commutative 61.2%

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

      5. associate-/l* 61.7%

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

   5. Simplified 61.7%

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

      1. distribute-rgt-neg-out 61.7%

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

      2. neg-sub0 61.7%

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

      3. associate-*r/ 61.2%

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

      4. sqrt-unprod 62.0%

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

      5. metadata-eval 62.0%

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

      6. metadata-eval 62.0%

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

      7. div-inv 62.2%

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

   7. Applied egg-rr 62.2%

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

      1. neg-sub0 62.2%

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

      2. distribute-neg-frac2 62.2%

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

   9. Simplified 62.2%

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

   if 9.30000000000000027e-45 < p

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 85.3%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.95 \cdot 10^{-225}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 7.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 7.5e-45) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 7.5e-45) {
		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 <= 7.5d-45) 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 <= 7.5e-45) {
		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 <= 7.5e-45:
		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 <= 7.5e-45)
		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 <= 7.5e-45)
		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, 7.5e-45], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 7.5000000000000006e-45

   1. Initial program 71.1%

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

   3. Taylor expanded in x around -inf 23.0%

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

      1. mul-1-neg 23.0%

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

      2. associate-/l* 23.0%

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

      3. distribute-rgt-neg-in 23.0%

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

      4. *-commutative 23.0%

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

      5. associate-/l* 23.2%

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

   5. Simplified 23.2%

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

      1. distribute-rgt-neg-out 23.2%

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

      2. neg-sub0 23.2%

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

      3. associate-*r/ 23.0%

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

      4. sqrt-unprod 23.3%

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

      5. metadata-eval 23.3%

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

      6. metadata-eval 23.3%

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

      7. div-inv 23.3%

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

   7. Applied egg-rr 23.3%

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

      1. neg-sub0 23.3%

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

      2. distribute-neg-frac2 23.3%

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

   9. Simplified 23.3%

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

   if 7.5000000000000006e-45 < p

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 85.3%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1e-311) (/ p_m (- x)) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-311) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	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-311)) then
        tmp = p_m / -x
    else
        tmp = p_m / x
    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-311) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1e-311:
		tmp = p_m / -x
	else:
		tmp = p_m / x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1e-311)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(p_m / x);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1e-311)
		tmp = p_m / -x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1e-311], N[(p$95$m / (-x)), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999948e-312

   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 34.7%

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

      1. mul-1-neg 34.7%

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

      2. associate-/l* 34.7%

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

      3. distribute-rgt-neg-in 34.7%

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

      4. *-commutative 34.7%

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

      5. associate-/l* 35.0%

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

   5. Simplified 35.0%

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

      1. distribute-rgt-neg-out 35.0%

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

      2. neg-sub0 35.0%

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

      3. associate-*r/ 34.7%

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

      4. sqrt-unprod 35.1%

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

      5. metadata-eval 35.1%

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

      6. metadata-eval 35.1%

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

      7. div-inv 35.2%

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

   7. Applied egg-rr 35.2%

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

      1. neg-sub0 35.2%

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

      2. distribute-neg-frac2 35.2%

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

   9. Simplified 35.2%

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

   if -9.99999999999948e-312 < 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 -inf 3.8%

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

      1. mul-1-neg 3.8%

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

      2. associate-/l* 3.8%

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

      3. distribute-rgt-neg-in 3.8%

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

      4. *-commutative 3.8%

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

      5. associate-/l* 3.8%

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

   5. Simplified 3.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 3.7%

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

      3. sqr-neg 3.7%

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

      4. sqrt-unprod 3.7%

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

      5. add-sqr-sqrt 3.7%

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

      6. associate-*r/ 3.7%

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

      7. sqrt-unprod 3.7%

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

      8. metadata-eval 3.7%

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

      9. metadata-eval 3.7%

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

      10. div-inv 3.7%

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

   7. Applied egg-rr 3.7%

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

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.4% accurate, 71.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \frac{p\_m}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[(p$95$m / x), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in x around -inf 19.1%

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

   1. mul-1-neg 19.1%

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

   2. associate-/l* 19.1%

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

   3. distribute-rgt-neg-in 19.1%

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

   4. *-commutative 19.1%

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

   5. associate-/l* 19.2%

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

5. Simplified 19.2%

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

   1. add-sqr-sqrt 17.3%

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

   2. sqrt-unprod 19.2%

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

   3. sqr-neg 19.2%

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

   4. sqrt-unprod 1.9%

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

   5. add-sqr-sqrt 15.3%

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

   6. associate-*r/ 15.2%

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

   7. sqrt-unprod 15.3%

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

   8. metadata-eval 15.3%

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

   9. metadata-eval 15.3%

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

   10. div-inv 15.3%

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

7. Applied egg-rr 15.3%

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

8. Final simplification 15.3%

   \[\leadsto \frac{p}{x} \]
9. Add Preprocessing

Developer target: 79.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024053 
(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))))))))