Given's Rotation SVD example

Percentage Accurate: 78.5% → 99.9%

Time: 11.3s

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: 78.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;3 \cdot \left({\left(\frac{p_m}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.6)
   (- (* 3.0 (* (pow (/ p_m x) 3.0) (/ (sqrt 0.5) (sqrt 2.0)))) (/ 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)))) <= -0.6) {
		tmp = (3.0 * (pow((p_m / x), 3.0) * (sqrt(0.5) / sqrt(2.0)))) - (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)))) <= -0.6) {
		tmp = (3.0 * (Math.pow((p_m / x), 3.0) * (Math.sqrt(0.5) / Math.sqrt(2.0)))) - (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)))) <= -0.6:
		tmp = (3.0 * (math.pow((p_m / x), 3.0) * (math.sqrt(0.5) / math.sqrt(2.0)))) - (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)))) <= -0.6)
		tmp = Float64(Float64(3.0 * Float64((Float64(p_m / x) ^ 3.0) * Float64(sqrt(0.5) / sqrt(2.0)))) - 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 / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.6)
		tmp = (3.0 * (((p_m / x) ^ 3.0) * (sqrt(0.5) / sqrt(2.0)))) - (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], -0.6], N[(N[(3.0 * N[(N[Power[N[(p$95$m / x), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(p$95$m / x), $MachinePrecision]), $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)))) < -0.599999999999999978

   1. Initial program 14.4%

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

      1. add-sqr-sqrt 14.4%

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

      2. hypot-def 14.4%

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

      3. associate-*l* 14.4%

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

      4. sqrt-prod 14.4%

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

      5. metadata-eval 14.4%

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

      6. sqrt-unprod 9.4%

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

      7. add-sqr-sqrt 14.4%

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

   4. Applied egg-rr 14.4%

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

   5. Taylor expanded in x around -inf 39.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 39.5%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 39.5%

         \[\leadsto -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 39.5%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      4. associate-*r/ 39.5%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      5. times-frac 48.5%

         \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{{x}^{3} \cdot \sqrt{2}}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      6. distribute-rgt-out 48.5%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \color{blue}{\left({p}^{4} \cdot \left(-4 + -2\right)\right)}}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      7. metadata-eval 48.5%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot \color{blue}{-6}\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      8. associate-/l* 48.5%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \color{blue}{\frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   7. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   8. Taylor expanded in p around 0 48.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + 3 \cdot \frac{{p}^{3} \cdot \sqrt{0.5}}{{x}^{3} \cdot \sqrt{2}}} \]
   9. Step-by-step derivation

      1. +-commutative 48.5%

         \[\leadsto \color{blue}{3 \cdot \frac{{p}^{3} \cdot \sqrt{0.5}}{{x}^{3} \cdot \sqrt{2}} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 48.5%

         \[\leadsto 3 \cdot \frac{{p}^{3} \cdot \sqrt{0.5}}{{x}^{3} \cdot \sqrt{2}} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 48.5%

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

      4. times-frac 48.5%

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

      5. cube-div 52.3%

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

      6. associate-/l* 52.3%

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

      7. associate-/r* 52.6%

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

   10. Simplified 52.6%

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

       1. associate-/l/ 52.3%

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

       2. metadata-eval 52.3%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{0.25}}}}} \]

       3. metadata-eval 52.3%

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

       4. sqrt-undiv 52.3%

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

       5. sqrt-unprod 53.2%

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

       6. sqrt-undiv 53.2%

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

       7. metadata-eval 53.2%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2 \cdot \sqrt{\color{blue}{0.25}}}}} \]

       8. metadata-eval 53.2%

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

       9. metadata-eval 53.2%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{\color{blue}{1}}}} \]

       10. metadata-eval 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\color{blue}{1}}} \]

       11. div-inv 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{x \cdot \frac{1}{1}}} \]

       12. metadata-eval 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x \cdot \color{blue}{1}} \]

       13. *-commutative 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{1 \cdot x}} \]

       14. *-un-lft-identity 53.2%

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

       15. expm1-log1p-u 22.1%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       16. expm1-udef 3.6%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]

   12. Applied egg-rr 3.6%

       \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]
   13. Step-by-step derivation

       1. expm1-def 22.1%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       2. expm1-log1p 53.2%

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

   14. Simplified 53.2%

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

   if -0.599999999999999978 < (/.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-sqr-sqrt 99.8%

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

      2. hypot-def 99.8%

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

      3. associate-*l* 99.8%

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

      4. sqrt-prod 99.8%

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

      5. metadata-eval 99.8%

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

      6. sqrt-unprod 57.5%

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

      7. add-sqr-sqrt 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 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.6)
   (/ (- 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)))) <= -0.6) {
		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)))) <= -0.6) {
		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)))) <= -0.6:
		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)))) <= -0.6)
		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 / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.6)
		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], -0.6], 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)))) < -0.599999999999999978

   1. Initial program 14.4%

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

      1. add-sqr-sqrt 14.4%

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

      2. hypot-def 14.4%

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

      3. associate-*l* 14.4%

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

      4. sqrt-prod 14.4%

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

      5. metadata-eval 14.4%

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

      6. sqrt-unprod 9.4%

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

      7. add-sqr-sqrt 14.4%

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

   4. Applied egg-rr 14.4%

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

   5. Taylor expanded in x around -inf 39.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 39.5%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 39.5%

         \[\leadsto -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 39.5%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      4. associate-*r/ 39.5%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      5. times-frac 48.5%

         \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{{x}^{3} \cdot \sqrt{2}}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      6. distribute-rgt-out 48.5%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \color{blue}{\left({p}^{4} \cdot \left(-4 + -2\right)\right)}}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      7. metadata-eval 48.5%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot \color{blue}{-6}\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      8. associate-/l* 48.5%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \color{blue}{\frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   7. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   8. Taylor expanded in p around 0 52.3%

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

      1. mul-1-neg 52.3%

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

      2. associate-/l* 52.3%

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

      3. distribute-neg-frac 52.3%

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

      4. associate-/r* 52.5%

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

   10. Simplified 52.5%

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

       1. associate-/l/ 52.3%

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

       2. metadata-eval 52.3%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{0.25}}}}} \]

       3. metadata-eval 52.3%

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

       4. sqrt-undiv 52.3%

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

       5. sqrt-unprod 53.2%

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

       6. sqrt-undiv 53.2%

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

       7. metadata-eval 53.2%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2 \cdot \sqrt{\color{blue}{0.25}}}}} \]

       8. metadata-eval 53.2%

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

       9. metadata-eval 53.2%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{\color{blue}{1}}}} \]

       10. metadata-eval 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\color{blue}{1}}} \]

       11. div-inv 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{x \cdot \frac{1}{1}}} \]

       12. metadata-eval 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x \cdot \color{blue}{1}} \]

       13. *-commutative 53.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{1 \cdot x}} \]

       14. *-un-lft-identity 53.2%

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

       15. expm1-log1p-u 22.1%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       16. expm1-udef 3.6%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]

   12. Applied egg-rr 3.6%

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

       1. expm1-def 22.1%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       2. expm1-log1p 53.2%

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

   14. Simplified 53.2%

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

   if -0.599999999999999978 < (/.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-sqr-sqrt 99.8%

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

      2. hypot-def 99.8%

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

      3. associate-*l* 99.8%

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

      4. sqrt-prod 99.8%

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

      5. metadata-eval 99.8%

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

      6. sqrt-unprod 57.5%

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

      7. add-sqr-sqrt 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 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 3.6 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.52 \cdot 10^{-238}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{elif}\;p_m \leq 4 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 3.6e-257)
   1.0
   (if (<= p_m 1.52e-238) (/ (- p_m) x) (if (<= p_m 4e-40) 1.0 (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.6e-257) {
		tmp = 1.0;
	} else if (p_m <= 1.52e-238) {
		tmp = -p_m / x;
	} else if (p_m <= 4e-40) {
		tmp = 1.0;
	} 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) :: tmp
    if (p_m <= 3.6d-257) then
        tmp = 1.0d0
    else if (p_m <= 1.52d-238) then
        tmp = -p_m / x
    else if (p_m <= 4d-40) then
        tmp = 1.0d0
    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 <= 3.6e-257) {
		tmp = 1.0;
	} else if (p_m <= 1.52e-238) {
		tmp = -p_m / x;
	} else if (p_m <= 4e-40) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 3.6e-257:
		tmp = 1.0
	elif p_m <= 1.52e-238:
		tmp = -p_m / x
	elif p_m <= 4e-40:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 3.6e-257)
		tmp = 1.0;
	elseif (p_m <= 1.52e-238)
		tmp = Float64(Float64(-p_m) / x);
	elseif (p_m <= 4e-40)
		tmp = 1.0;
	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 <= 3.6e-257)
		tmp = 1.0;
	elseif (p_m <= 1.52e-238)
		tmp = -p_m / x;
	elseif (p_m <= 4e-40)
		tmp = 1.0;
	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, 3.6e-257], 1.0, If[LessEqual[p$95$m, 1.52e-238], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[p$95$m, 4e-40], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.60000000000000007e-257 or 1.5200000000000001e-238 < p < 3.9999999999999997e-40

   1. Initial program 77.1%

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

   3. Taylor expanded in x around inf 44.4%

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

   if 3.60000000000000007e-257 < p < 1.5200000000000001e-238

   1. Initial program 36.6%

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

      1. add-sqr-sqrt 36.6%

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

      2. hypot-def 36.6%

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

      3. associate-*l* 36.6%

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

      4. sqrt-prod 36.6%

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

      5. metadata-eval 36.6%

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

      6. sqrt-unprod 36.6%

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

      7. add-sqr-sqrt 36.6%

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

   4. Applied egg-rr 36.6%

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

   5. Taylor expanded in x around -inf 33.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 33.3%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 33.3%

         \[\leadsto -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 33.3%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      4. associate-*r/ 33.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      5. times-frac 65.8%

         \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{{x}^{3} \cdot \sqrt{2}}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      6. distribute-rgt-out 65.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \color{blue}{\left({p}^{4} \cdot \left(-4 + -2\right)\right)}}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      7. metadata-eval 65.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot \color{blue}{-6}\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      8. associate-/l* 66.1%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \color{blue}{\frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   7. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   8. Taylor expanded in p around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. associate-/l* 99.0%

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

      3. distribute-neg-frac 99.0%

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

      4. associate-/r* 99.0%

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

   10. Simplified 99.0%

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

       1. associate-/l/ 99.0%

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

       2. metadata-eval 99.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{0.25}}}}} \]

       3. metadata-eval 99.0%

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

       4. sqrt-undiv 99.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqrt-undiv 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2 \cdot \sqrt{\color{blue}{0.25}}}}} \]

       8. metadata-eval 100.0%

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

       9. metadata-eval 100.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{\color{blue}{1}}}} \]

       10. metadata-eval 100.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\color{blue}{1}}} \]

       11. div-inv 100.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{x \cdot \frac{1}{1}}} \]

       12. metadata-eval 100.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x \cdot \color{blue}{1}} \]

       13. *-commutative 100.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{1 \cdot x}} \]

       14. *-un-lft-identity 100.0%

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

       15. expm1-log1p-u 66.7%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       16. expm1-udef 0.7%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]

   12. Applied egg-rr 0.7%

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

       1. expm1-def 66.7%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       2. expm1-log1p 100.0%

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

   14. Simplified 100.0%

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

   if 3.9999999999999997e-40 < p

   1. Initial program 92.0%

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

   3. Taylor expanded in x around 0 86.8%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.6 \cdot 10^{-257}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.52 \cdot 10^{-238}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4 \cdot 10^{-40}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 1.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.5e-82) (/ (- p_m) x) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.5e-82) {
		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 <= 1.5d-82) 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 <= 1.5e-82) {
		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 <= 1.5e-82:
		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 <= 1.5e-82)
		tmp = Float64(Float64(-p_m) / 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 <= 1.5e-82)
		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, 1.5e-82], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.4999999999999999e-82

   1. Initial program 74.9%

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

      1. add-sqr-sqrt 74.9%

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

      2. hypot-def 74.9%

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

      3. associate-*l* 74.9%

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

      4. sqrt-prod 74.9%

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

      5. metadata-eval 74.9%

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

      6. sqrt-unprod 22.6%

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

      7. add-sqr-sqrt 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in x around -inf 10.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 10.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 10.8%

         \[\leadsto -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 10.8%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      4. associate-*r/ 10.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      5. times-frac 14.2%

         \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{{x}^{3} \cdot \sqrt{2}}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      6. distribute-rgt-out 14.2%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \color{blue}{\left({p}^{4} \cdot \left(-4 + -2\right)\right)}}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      7. metadata-eval 14.2%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot \color{blue}{-6}\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      8. associate-/l* 14.2%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \color{blue}{\frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   7. Simplified 14.2%

      \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   8. Taylor expanded in p around 0 16.4%

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

      1. mul-1-neg 16.4%

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

      2. associate-/l* 16.4%

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

      3. distribute-neg-frac 16.4%

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

      4. associate-/r* 16.4%

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

   10. Simplified 16.4%

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

       1. associate-/l/ 16.0%

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

       2. metadata-eval 16.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{0.25}}}}} \]

       3. metadata-eval 16.0%

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

       4. sqrt-undiv 16.0%

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

       5. sqrt-unprod 16.2%

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

       6. sqrt-undiv 16.2%

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

       7. metadata-eval 16.2%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2 \cdot \sqrt{\color{blue}{0.25}}}}} \]

       8. metadata-eval 16.2%

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

       9. metadata-eval 16.2%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{\color{blue}{1}}}} \]

       10. metadata-eval 16.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\color{blue}{1}}} \]

       11. div-inv 16.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{x \cdot \frac{1}{1}}} \]

       12. metadata-eval 16.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x \cdot \color{blue}{1}} \]

       13. *-commutative 16.2%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{1 \cdot x}} \]

       14. *-un-lft-identity 16.2%

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

       15. expm1-log1p-u 8.5%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       16. expm1-udef 2.3%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]

   12. Applied egg-rr 2.9%

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

       1. expm1-def 8.5%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       2. expm1-log1p 16.2%

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

   14. Simplified 16.6%

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

   if 1.4999999999999999e-82 < p

   1. Initial program 93.0%

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

   3. Taylor expanded in x around 0 82.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.6% accurate, 21.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{p_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5e-310) (/ (- p_m) x) (/ 1.0 (/ x p_m))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 / (x / p_m);
	}
	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 <= (-5d-310)) then
        tmp = -p_m / x
    else
        tmp = 1.0d0 / (x / p_m)
    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 <= -5e-310) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 / (x / p_m);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e-310:
		tmp = -p_m / x
	else:
		tmp = 1.0 / (x / p_m)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = Float64(1.0 / Float64(x / p_m));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -p_m / x;
	else
		tmp = 1.0 / (x / p_m);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e-310], N[((-p$95$m) / x), $MachinePrecision], N[(1.0 / N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 61.6%

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

      1. add-sqr-sqrt 61.6%

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

      2. hypot-def 61.6%

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

      3. associate-*l* 61.6%

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

      4. sqrt-prod 61.6%

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

      5. metadata-eval 61.6%

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

      6. sqrt-unprod 34.6%

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

      7. add-sqr-sqrt 61.6%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in x around -inf 18.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 18.4%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 18.4%

         \[\leadsto -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 18.4%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      4. associate-*r/ 18.4%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      5. times-frac 22.8%

         \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{{x}^{3} \cdot \sqrt{2}}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      6. distribute-rgt-out 22.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \color{blue}{\left({p}^{4} \cdot \left(-4 + -2\right)\right)}}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      7. metadata-eval 22.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot \color{blue}{-6}\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      8. associate-/l* 22.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \color{blue}{\frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   7. Simplified 22.8%

      \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   8. Taylor expanded in p around 0 25.4%

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

      1. mul-1-neg 25.4%

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

      2. associate-/l* 25.4%

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

      3. distribute-neg-frac 25.4%

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

      4. associate-/r* 25.5%

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

   10. Simplified 25.5%

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

       1. associate-/l/ 24.6%

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

       2. metadata-eval 24.6%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{0.25}}}}} \]

       3. metadata-eval 24.6%

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

       4. sqrt-undiv 24.6%

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

       5. sqrt-unprod 25.0%

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

       6. sqrt-undiv 25.0%

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

       7. metadata-eval 25.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2 \cdot \sqrt{\color{blue}{0.25}}}}} \]

       8. metadata-eval 25.0%

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

       9. metadata-eval 25.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{\color{blue}{1}}}} \]

       10. metadata-eval 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\color{blue}{1}}} \]

       11. div-inv 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{x \cdot \frac{1}{1}}} \]

       12. metadata-eval 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x \cdot \color{blue}{1}} \]

       13. *-commutative 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{1 \cdot x}} \]

       14. *-un-lft-identity 25.0%

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

       15. expm1-log1p-u 10.5%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       16. expm1-udef 2.1%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]

   12. Applied egg-rr 2.3%

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

       1. expm1-def 10.5%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       2. expm1-log1p 25.0%

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

   14. Simplified 25.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. fma-def 99.7%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around -inf 1.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{0.5}}{x} \cdot \sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 1.8%

         \[\leadsto \color{blue}{-\frac{\sqrt{0.5}}{x} \cdot \sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}} \]

      2. distribute-rgt-neg-in 1.8%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{x} \cdot \left(-\sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}\right)} \]

      3. distribute-rgt-out 1.8%

         \[\leadsto \frac{\sqrt{0.5}}{x} \cdot \left(-\sqrt{\color{blue}{{p}^{2} \cdot \left(0.6666666666666666 + 1.3333333333333333\right)}}\right) \]

      4. metadata-eval 1.8%

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

      5. rem-square-sqrt 1.8%

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

      6. unpow2 1.8%

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

      7. *-commutative 1.8%

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

      8. unpow2 1.8%

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

      9. rem-square-sqrt 1.8%

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

   7. Simplified 1.8%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 4.6%

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

      3. sqr-neg 4.6%

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

      4. add-sqr-sqrt 4.6%

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

      5. sqrt-unprod 4.6%

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

      6. unpow2 4.6%

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

      7. sqrt-prod 3.0%

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

      8. add-sqr-sqrt 3.6%

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

      9. associate-*r* 3.6%

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

      10. clear-num 3.6%

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

      11. associate-/r/ 3.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x}{\sqrt{0.5}}}{\sqrt{2}}}} \cdot p \]

      12. associate-/r/ 3.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{x}{\sqrt{0.5}}}{\sqrt{2}}}{p}}} \]

      13. clear-num 3.6%

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

      14. clear-num 3.6%

          \[\leadsto \frac{p}{\color{blue}{\frac{1}{\frac{\sqrt{2}}{\frac{x}{\sqrt{0.5}}}}}} \]

      15. clear-num 3.6%

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

      16. associate-/l/ 3.6%

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

   9. Applied egg-rr 3.6%

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

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.6% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5e-310) (/ (- p_m) x) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		tmp = Float64(Float64(-p_m) / 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 <= -5e-310)
		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, -5e-310], N[((-p$95$m) / x), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 61.6%

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

      1. add-sqr-sqrt 61.6%

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

      2. hypot-def 61.6%

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

      3. associate-*l* 61.6%

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

      4. sqrt-prod 61.6%

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

      5. metadata-eval 61.6%

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

      6. sqrt-unprod 34.6%

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

      7. add-sqr-sqrt 61.6%

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in x around -inf 18.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} + -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 18.4%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + -1 \cdot \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      2. mul-1-neg 18.4%

         \[\leadsto -0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} + \color{blue}{\left(-\frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}\right)} \]

      3. unsub-neg 18.4%

         \[\leadsto \color{blue}{-0.5 \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]

      4. associate-*r/ 18.4%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)\right)}{p \cdot \left({x}^{3} \cdot \sqrt{2}\right)}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      5. times-frac 22.8%

         \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left(-4 \cdot {p}^{4} + -2 \cdot {p}^{4}\right)}{{x}^{3} \cdot \sqrt{2}}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      6. distribute-rgt-out 22.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \color{blue}{\left({p}^{4} \cdot \left(-4 + -2\right)\right)}}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      7. metadata-eval 22.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot \color{blue}{-6}\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]

      8. associate-/l* 22.8%

         \[\leadsto \frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \color{blue}{\frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   7. Simplified 22.8%

      \[\leadsto \color{blue}{\frac{-0.5}{p} \cdot \frac{\sqrt{0.5} \cdot \left({p}^{4} \cdot -6\right)}{{x}^{3} \cdot \sqrt{2}} - \frac{p}{\frac{x}{\sqrt{0.5} \cdot \sqrt{2}}}} \]

   8. Taylor expanded in p around 0 25.4%

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

      1. mul-1-neg 25.4%

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

      2. associate-/l* 25.4%

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

      3. distribute-neg-frac 25.4%

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

      4. associate-/r* 25.5%

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

   10. Simplified 25.5%

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

       1. associate-/l/ 24.6%

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

       2. metadata-eval 24.6%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2} \cdot \sqrt{\color{blue}{\sqrt{0.25}}}}} \]

       3. metadata-eval 24.6%

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

       4. sqrt-undiv 24.6%

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

       5. sqrt-unprod 25.0%

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

       6. sqrt-undiv 25.0%

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

       7. metadata-eval 25.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{2 \cdot \sqrt{\color{blue}{0.25}}}}} \]

       8. metadata-eval 25.0%

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

       9. metadata-eval 25.0%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\sqrt{\color{blue}{1}}}} \]

       10. metadata-eval 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\frac{x}{\color{blue}{1}}} \]

       11. div-inv 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{x \cdot \frac{1}{1}}} \]

       12. metadata-eval 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{x \cdot \color{blue}{1}} \]

       13. *-commutative 25.0%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{1 \cdot x}} \]

       14. *-un-lft-identity 25.0%

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

       15. expm1-log1p-u 10.5%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       16. expm1-udef 2.1%

           \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1}} \]

   12. Applied egg-rr 2.3%

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

       1. expm1-def 10.5%

          \[\leadsto 3 \cdot \left({\left(\frac{p}{x}\right)}^{3} \cdot \frac{\sqrt{0.5}}{\sqrt{2}}\right) - \frac{p}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}} \]

       2. expm1-log1p 25.0%

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

   14. Simplified 25.8%

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

   if -4.999999999999985e-310 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. fma-def 99.7%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around -inf 1.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{0.5}}{x} \cdot \sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 1.8%

         \[\leadsto \color{blue}{-\frac{\sqrt{0.5}}{x} \cdot \sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}} \]

      2. distribute-rgt-neg-in 1.8%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{x} \cdot \left(-\sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}\right)} \]

      3. distribute-rgt-out 1.8%

         \[\leadsto \frac{\sqrt{0.5}}{x} \cdot \left(-\sqrt{\color{blue}{{p}^{2} \cdot \left(0.6666666666666666 + 1.3333333333333333\right)}}\right) \]

      4. metadata-eval 1.8%

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

      5. rem-square-sqrt 1.8%

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

      6. unpow2 1.8%

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

      7. *-commutative 1.8%

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

      8. unpow2 1.8%

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

      9. rem-square-sqrt 1.8%

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

   7. Simplified 1.8%

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

      1. add-sqr-sqrt 0.8%

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

      2. sqrt-unprod 4.6%

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

      3. sqr-neg 4.6%

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

      4. add-sqr-sqrt 4.6%

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

      5. sqrt-unprod 4.6%

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

      6. unpow2 4.6%

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

      7. sqrt-prod 3.0%

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

      8. add-sqr-sqrt 3.6%

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

      9. associate-*r* 3.6%

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

      10. clear-num 3.6%

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

      11. associate-/r/ 3.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x}{\sqrt{0.5}}}{\sqrt{2}}}} \cdot p \]

      12. associate-/r/ 3.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{x}{\sqrt{0.5}}}{\sqrt{2}}}{p}}} \]

      13. clear-num 3.6%

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

      14. clear-num 3.6%

          \[\leadsto \frac{p}{\color{blue}{\frac{1}{\frac{\sqrt{2}}{\frac{x}{\sqrt{0.5}}}}}} \]

      15. clear-num 3.6%

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

      16. associate-/l/ 3.6%

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

   9. Applied egg-rr 3.0%

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

       1. expm1-def 3.4%

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

       2. expm1-log1p 3.6%

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

   11. Simplified 3.6%

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

4. Final simplification 14.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.6% 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 81.1%

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

   1. +-commutative 81.1%

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

   2. add-cube-cbrt 81.0%

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

   3. fma-def 81.0%

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

4. Applied egg-rr 81.2%

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

5. Taylor expanded in x around -inf 14.4%

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{0.5}}{x} \cdot \sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 14.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{0.5}}{x} \cdot \sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}} \]

   2. distribute-rgt-neg-in 14.4%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{x} \cdot \left(-\sqrt{0.6666666666666666 \cdot {p}^{2} + 1.3333333333333333 \cdot {p}^{2}}\right)} \]

   3. distribute-rgt-out 14.4%

      \[\leadsto \frac{\sqrt{0.5}}{x} \cdot \left(-\sqrt{\color{blue}{{p}^{2} \cdot \left(0.6666666666666666 + 1.3333333333333333\right)}}\right) \]

   4. metadata-eval 14.4%

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

   5. rem-square-sqrt 14.3%

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

   6. unpow2 14.3%

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

   7. *-commutative 14.3%

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

   8. unpow2 14.3%

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

   9. rem-square-sqrt 14.4%

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

7. Simplified 14.4%

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

   1. add-sqr-sqrt 2.8%

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

   2. sqrt-unprod 5.3%

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

   3. sqr-neg 5.3%

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

   4. add-sqr-sqrt 5.3%

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

   5. sqrt-unprod 5.3%

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

   6. unpow2 5.3%

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

   7. sqrt-prod 3.5%

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

   8. add-sqr-sqrt 15.1%

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

   9. associate-*r* 15.1%

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

   10. clear-num 15.1%

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

   11. associate-/r/ 15.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x}{\sqrt{0.5}}}{\sqrt{2}}}} \cdot p \]

   12. associate-/r/ 15.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\frac{x}{\sqrt{0.5}}}{\sqrt{2}}}{p}}} \]

   13. clear-num 15.1%

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

   14. clear-num 15.1%

       \[\leadsto \frac{p}{\color{blue}{\frac{1}{\frac{\sqrt{2}}{\frac{x}{\sqrt{0.5}}}}}} \]

   15. clear-num 15.1%

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

   16. associate-/l/ 15.0%

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

9. Applied egg-rr 5.5%

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

    1. expm1-def 15.0%

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

    2. expm1-log1p 15.2%

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

11. Simplified 15.2%

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

12. Final simplification 15.2%

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

Developer target: 78.5% 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 2024020 
(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))))))))