Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.8%

Time: 9.6s

Alternatives: 9

Speedup: 1.0×

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

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -0.999999996:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p_m \cdot 2\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999999996)
   (/ (- p_m) x)
   (sqrt (* 0.5 (log (exp (+ 1.0 (/ 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)))) <= -0.999999996) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((0.5 * log(exp((1.0 + (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)))) <= -0.999999996) {
		tmp = -p_m / x;
	} else {
		tmp = Math.sqrt((0.5 * Math.log(Math.exp((1.0 + (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)))) <= -0.999999996:
		tmp = -p_m / x
	else:
		tmp = math.sqrt((0.5 * math.log(math.exp((1.0 + (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)))) <= -0.999999996)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * log(exp(Float64(1.0 + Float64(x / hypot(x, Float64(p_m * 2.0))))))));
	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)))) <= -0.999999996)
		tmp = -p_m / x;
	else
		tmp = sqrt((0.5 * log(exp((1.0 + (x / hypot(x, (p_m * 2.0))))))));
	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], -0.999999996], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999996000000002

   1. Initial program 18.4%

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

      1. add-log-exp 18.3%

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

      2. +-commutative 18.3%

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

      3. add-sqr-sqrt 18.3%

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

      4. hypot-def 18.3%

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

      5. associate-*l* 18.3%

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

      6. sqrt-prod 18.3%

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

      7. metadata-eval 18.3%

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

      8. sqrt-unprod 8.6%

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

      9. add-sqr-sqrt 18.3%

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

   4. Applied egg-rr 18.3%

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

   6. Applied egg-rr 18.3%

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

      1. expm1-def 18.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 18.3%

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

      3. associate-/r/ 18.5%

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

      4. *-commutative 18.5%

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

   8. Simplified 18.5%

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

   9. Taylor expanded in x around -inf 59.0%

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

       1. associate-*r/ 59.0%

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

       2. mul-1-neg 59.0%

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

   11. Simplified 59.0%

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

   if -0.999999996000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

      1. add-log-exp 99.8%

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

      2. +-commutative 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. hypot-def 99.8%

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

      5. associate-*l* 99.8%

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

      6. sqrt-prod 99.8%

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

      7. metadata-eval 99.8%

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

      8. sqrt-unprod 49.7%

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

      9. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999996:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -0.999999996:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999999996)
   (/ (- p_m) x)
   (sqrt (+ 0.5 (* x (/ 0.5 (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)))) <= -0.999999996) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((0.5 + (x * (0.5 / 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)))) <= -0.999999996) {
		tmp = -p_m / x;
	} else {
		tmp = Math.sqrt((0.5 + (x * (0.5 / 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)))) <= -0.999999996:
		tmp = -p_m / x
	else:
		tmp = math.sqrt((0.5 + (x * (0.5 / 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)))) <= -0.999999996)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 + Float64(x * Float64(0.5 / hypot(x, Float64(p_m * 2.0))))));
	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)))) <= -0.999999996)
		tmp = -p_m / x;
	else
		tmp = sqrt((0.5 + (x * (0.5 / hypot(x, (p_m * 2.0))))));
	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], -0.999999996], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.999999996000000002

   1. Initial program 18.4%

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

      1. add-log-exp 18.3%

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

      2. +-commutative 18.3%

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

      3. add-sqr-sqrt 18.3%

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

      4. hypot-def 18.3%

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

      5. associate-*l* 18.3%

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

      6. sqrt-prod 18.3%

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

      7. metadata-eval 18.3%

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

      8. sqrt-unprod 8.6%

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

      9. add-sqr-sqrt 18.3%

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

   4. Applied egg-rr 18.3%

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

   6. Applied egg-rr 18.3%

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

      1. expm1-def 18.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 18.3%

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

      3. associate-/r/ 18.5%

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

      4. *-commutative 18.5%

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

   8. Simplified 18.5%

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

   9. Taylor expanded in x around -inf 59.0%

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

       1. associate-*r/ 59.0%

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

       2. mul-1-neg 59.0%

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

   11. Simplified 59.0%

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

   if -0.999999996000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

      1. add-log-exp 99.8%

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

      2. +-commutative 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. hypot-def 99.8%

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

      5. associate-*l* 99.8%

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

      6. sqrt-prod 99.8%

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

      7. metadata-eval 99.8%

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

      8. sqrt-unprod 49.7%

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

      9. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 98.8%

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

      1. expm1-def 98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 99.8%

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

      3. associate-/r/ 99.8%

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

      4. *-commutative 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999996:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.96:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1.96)
   (/ (- 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 <= -1.96) {
		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 <= -1.96) {
		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 <= -1.96:
		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 (x <= -1.96)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p_m * 2.0), x)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.96)
		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[x, -1.96], 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 x < -1.96

   1. Initial program 48.1%

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

      1. add-log-exp 48.1%

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

      2. +-commutative 48.1%

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

      3. add-sqr-sqrt 48.1%

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

      4. hypot-def 48.3%

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

      5. associate-*l* 48.3%

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

      6. sqrt-prod 48.3%

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

      7. metadata-eval 48.3%

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

      8. sqrt-unprod 25.4%

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

      9. add-sqr-sqrt 48.3%

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

   4. Applied egg-rr 48.3%

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

   6. Applied egg-rr 47.7%

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

      1. expm1-def 47.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 48.2%

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

      3. associate-/r/ 48.2%

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

      4. *-commutative 48.2%

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

   8. Simplified 48.2%

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

   9. Taylor expanded in x around -inf 42.3%

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

       1. associate-*r/ 42.3%

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

       2. mul-1-neg 42.3%

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

   11. Simplified 42.3%

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

   if -1.96 < x

   1. Initial program 87.8%

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

      1. add-sqr-sqrt 87.8%

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

      2. hypot-def 87.8%

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

      3. associate-*l* 87.8%

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

      4. sqrt-prod 87.8%

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

      5. metadata-eval 87.8%

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

      6. sqrt-unprod 43.0%

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

      7. add-sqr-sqrt 87.8%

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

   4. Applied egg-rr 87.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.96:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;x \leq -2.65:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p_m}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-28}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= x -2.65)
     t_0
     (if (<= x -3.5e-63)
       (sqrt (+ 0.5 (/ (* x 0.25) p_m)))
       (if (<= x -2.3e-70) t_0 (if (<= x 1e-28) (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 <= -2.65) {
		tmp = t_0;
	} else if (x <= -3.5e-63) {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	} else if (x <= -2.3e-70) {
		tmp = t_0;
	} else if (x <= 1e-28) {
		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 <= (-2.65d0)) then
        tmp = t_0
    else if (x <= (-3.5d-63)) then
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    else if (x <= (-2.3d-70)) then
        tmp = t_0
    else if (x <= 1d-28) 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 <= -2.65) {
		tmp = t_0;
	} else if (x <= -3.5e-63) {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	} else if (x <= -2.3e-70) {
		tmp = t_0;
	} else if (x <= 1e-28) {
		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 <= -2.65:
		tmp = t_0
	elif x <= -3.5e-63:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	elif x <= -2.3e-70:
		tmp = t_0
	elif x <= 1e-28:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (x <= -2.65)
		tmp = t_0;
	elseif (x <= -3.5e-63)
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	elseif (x <= -2.3e-70)
		tmp = t_0;
	elseif (x <= 1e-28)
		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 <= -2.65)
		tmp = t_0;
	elseif (x <= -3.5e-63)
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	elseif (x <= -2.3e-70)
		tmp = t_0;
	elseif (x <= 1e-28)
		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, -2.65], t$95$0, If[LessEqual[x, -3.5e-63], N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -2.3e-70], t$95$0, If[LessEqual[x, 1e-28], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.64999999999999991 or -3.50000000000000003e-63 < x < -2.30000000000000001e-70

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 45.8%

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

      2. +-commutative 45.8%

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

      3. add-sqr-sqrt 45.8%

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

      4. hypot-def 45.9%

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

      5. associate-*l* 45.9%

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

      6. sqrt-prod 45.9%

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

      7. metadata-eval 45.9%

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

      8. sqrt-unprod 23.4%

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

      9. add-sqr-sqrt 45.9%

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

   4. Applied egg-rr 45.9%

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

   6. Applied egg-rr 45.4%

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

      1. expm1-def 45.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 45.9%

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

      3. associate-/r/ 45.9%

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

      4. *-commutative 45.9%

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

   8. Simplified 45.9%

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

   9. Taylor expanded in x around -inf 41.8%

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

       1. associate-*r/ 41.8%

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

       2. mul-1-neg 41.8%

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

   11. Simplified 41.8%

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

   if -2.64999999999999991 < x < -3.50000000000000003e-63

   1. Initial program 73.8%

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

      1. add-log-exp 73.8%

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

      2. +-commutative 73.8%

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

      3. add-sqr-sqrt 73.8%

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

      4. hypot-def 73.8%

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

      5. associate-*l* 73.8%

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

      6. sqrt-prod 73.8%

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

      7. metadata-eval 73.8%

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

      8. sqrt-unprod 42.1%

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

      9. add-sqr-sqrt 73.8%

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

   4. Applied egg-rr 73.8%

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

   6. Applied egg-rr 72.8%

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

      1. expm1-def 72.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 73.8%

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

      3. associate-/r/ 73.8%

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

      4. *-commutative 73.8%

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

   8. Simplified 73.8%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. associate-*r/ 72.6%

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

   11. Simplified 72.6%

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

   if -2.30000000000000001e-70 < x < 9.99999999999999971e-29

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0 70.1%

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

   if 9.99999999999999971e-29 < 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-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. hypot-def 100.0%

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

      5. associate-*l* 100.0%

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

      6. sqrt-prod 100.0%

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

      7. metadata-eval 100.0%

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

      8. sqrt-unprod 50.0%

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

      9. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 99.6%

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

      1. expm1-def 99.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/r/ 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 75.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq 10^{-28}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1.05) (/ (- p_m) x) (if (<= x 3.7e-28) (sqrt 0.5) 1.0)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -p_m / x;
	} else if (x <= 3.7e-28) {
		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) :: tmp
    if (x <= (-1.05d0)) then
        tmp = -p_m / x
    else if (x <= 3.7d-28) 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 tmp;
	if (x <= -1.05) {
		tmp = -p_m / x;
	} else if (x <= 3.7e-28) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.05:
		tmp = -p_m / x
	elif x <= 3.7e-28:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(-p_m) / x);
	elseif (x <= 3.7e-28)
		tmp = sqrt(0.5);
	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 <= -1.05)
		tmp = -p_m / x;
	elseif (x <= 3.7e-28)
		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_] := If[LessEqual[x, -1.05], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[x, 3.7e-28], N[Sqrt[0.5], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 48.1%

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

      1. add-log-exp 48.1%

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

      2. +-commutative 48.1%

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

      3. add-sqr-sqrt 48.1%

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

      4. hypot-def 48.3%

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

      5. associate-*l* 48.3%

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

      6. sqrt-prod 48.3%

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

      7. metadata-eval 48.3%

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

      8. sqrt-unprod 25.4%

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

      9. add-sqr-sqrt 48.3%

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

   4. Applied egg-rr 48.3%

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

   6. Applied egg-rr 47.7%

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

      1. expm1-def 47.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 48.2%

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

      3. associate-/r/ 48.2%

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

      4. *-commutative 48.2%

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

   8. Simplified 48.2%

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

   9. Taylor expanded in x around -inf 42.3%

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

       1. associate-*r/ 42.3%

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

       2. mul-1-neg 42.3%

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

   11. Simplified 42.3%

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

   if -1.05000000000000004 < x < 3.7000000000000002e-28

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0 67.8%

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

   if 3.7000000000000002e-28 < 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-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. hypot-def 100.0%

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

      5. associate-*l* 100.0%

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

      6. sqrt-prod 100.0%

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

      7. metadata-eval 100.0%

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

      8. sqrt-unprod 50.0%

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

      9. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 99.6%

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

      1. expm1-def 99.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/r/ 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 75.9%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.3% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -4.8e-143) (/ (- p_m) x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -4.8e-143) {
		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 <= (-4.8d-143)) 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 <= -4.8e-143) {
		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 <= -4.8e-143:
		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 <= -4.8e-143)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -4.8e-143)
		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, -4.8e-143], N[((-p$95$m) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.7999999999999998e-143

   1. Initial program 58.6%

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

      1. add-log-exp 58.6%

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

      2. +-commutative 58.6%

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

      3. add-sqr-sqrt 58.6%

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

      4. hypot-def 58.6%

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

      5. associate-*l* 58.6%

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

      6. sqrt-prod 58.6%

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

      7. metadata-eval 58.6%

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

      8. sqrt-unprod 32.8%

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

      9. add-sqr-sqrt 58.6%

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

   4. Applied egg-rr 58.6%

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

   6. Applied egg-rr 57.9%

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

      1. expm1-def 57.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 58.6%

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

      3. associate-/r/ 58.7%

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

      4. *-commutative 58.7%

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

   8. Simplified 58.7%

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

   9. Taylor expanded in x around -inf 31.6%

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

       1. associate-*r/ 31.6%

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

       2. mul-1-neg 31.6%

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

   11. Simplified 31.6%

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

   if -4.7999999999999998e-143 < 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-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. hypot-def 100.0%

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

      5. associate-*l* 100.0%

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

      6. sqrt-prod 100.0%

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

      7. metadata-eval 100.0%

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

      8. sqrt-unprod 45.4%

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

      9. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Applied egg-rr 99.3%

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

      1. expm1-def 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/r/ 100.0%

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

      4. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 64.3%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.1% accurate, 24.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(\frac{x}{p_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -9.5e+24) (reciprocal (/ x p_m)) 1.0))

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000001e24

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf 44.0%

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

      1. associate-*r/ 44.0%

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

   5. Simplified 44.0%

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

      1. associate-*r/ 44.0%

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

      2. sqrt-div 60.1%

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

      3. associate-*r* 60.1%

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

      4. metadata-eval 60.1%

         \[\leadsto \frac{\sqrt{\color{blue}{1} \cdot {p}^{2}}}{\sqrt{{x}^{2}}} \]

      5. *-un-lft-identity 60.1%

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

      6. unpow2 60.1%

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

      7. sqrt-prod 33.6%

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

      8. add-sqr-sqrt 44.2%

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

      9. unpow2 44.2%

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

      10. sqrt-prod 0.0%

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

      11. add-sqr-sqrt 45.4%

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

      12. clear-num 45.4%

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

      13. reciprocal-define 29.4%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\frac{x}{p}\right)\right)} \]

   7. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\frac{x}{p}\right)\right)} \]

   if -9.5000000000000001e24 < x

   1. Initial program 85.0%

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

      1. add-log-exp 85.0%

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

      2. +-commutative 85.0%

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

      3. add-sqr-sqrt 85.0%

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

      4. hypot-def 85.0%

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

      5. associate-*l* 85.0%

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

      6. sqrt-prod 85.0%

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

      7. metadata-eval 85.0%

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

      8. sqrt-unprod 42.0%

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

      9. add-sqr-sqrt 85.0%

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

   4. Applied egg-rr 85.0%

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

   6. Applied egg-rr 84.3%

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

      1. expm1-def 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 85.0%

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

      3. associate-/r/ 85.1%

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

      4. *-commutative 85.1%

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

   8. Simplified 85.1%

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

   9. Taylor expanded in x around inf 43.8%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(\frac{x}{p}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.1% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -9.2e+25) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -9.2e+25) {
		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 <= (-9.2d+25)) 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 <= -9.2e+25) {
		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 <= -9.2e+25:
		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 <= -9.2e+25)
		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 <= -9.2e+25)
		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, -9.2e+25], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -9.1999999999999992e25

   1. Initial program 50.9%

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

   3. Taylor expanded in x around -inf 44.0%

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

      1. associate-*r/ 44.0%

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

   5. Simplified 44.0%

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

   6. Taylor expanded in p around 0 45.4%

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

   if -9.1999999999999992e25 < x

   1. Initial program 85.0%

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

      1. add-log-exp 85.0%

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

      2. +-commutative 85.0%

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

      3. add-sqr-sqrt 85.0%

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

      4. hypot-def 85.0%

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

      5. associate-*l* 85.0%

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

      6. sqrt-prod 85.0%

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

      7. metadata-eval 85.0%

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

      8. sqrt-unprod 42.0%

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

      9. add-sqr-sqrt 85.0%

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

   4. Applied egg-rr 85.0%

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

   6. Applied egg-rr 84.3%

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

      1. expm1-def 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

      2. expm1-log1p 85.0%

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

      3. associate-/r/ 85.1%

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

      4. *-commutative 85.1%

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

   8. Simplified 85.1%

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

   9. Taylor expanded in x around inf 43.8%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.2% 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 77.8%

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

   1. add-log-exp 77.8%

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

   2. +-commutative 77.8%

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

   3. add-sqr-sqrt 77.8%

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

   4. hypot-def 77.9%

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

   5. associate-*l* 77.9%

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

   6. sqrt-prod 77.9%

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

   7. metadata-eval 77.9%

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

   8. sqrt-unprod 38.6%

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

   9. add-sqr-sqrt 77.9%

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

4. Applied egg-rr 77.9%

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

6. Applied egg-rr 77.2%

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

   1. expm1-def 77.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, 2 \cdot p\right)}{x}}}\right)\right)} \]

   2. expm1-log1p 77.8%

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

   3. associate-/r/ 77.9%

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

   4. *-commutative 77.9%

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

8. Simplified 77.9%

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

9. Taylor expanded in x around inf 36.5%

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

10. Final simplification 36.5%

    \[\leadsto 1 \]
11. 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 2024024 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

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