Given's Rotation SVD example

Percentage Accurate: 79.2% → 91.0%

Time: 7.9s

Alternatives: 6

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% 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: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(2, {\left(\frac{p}{x}\right)}^{2}, {\left(\frac{p}{x}\right)}^{4} \cdot -6\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.98)
   (sqrt (* 0.5 (fma 2.0 (pow (/ p x) 2.0) (* (pow (/ p x) 4.0) -6.0))))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = sqrt((0.5 * fma(2.0, pow((p / x), 2.0), (pow((p / x), 4.0) * -6.0))));
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Julia

function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.98)
		tmp = sqrt(Float64(0.5 * fma(2.0, (Float64(p / x) ^ 2.0), Float64((Float64(p / x) ^ 4.0) * -6.0))));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

Wolfram

code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.98], N[Sqrt[N[(0.5 * N[(2.0 * N[Power[N[(p / x), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[(p / x), $MachinePrecision], 4.0], $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 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)))) < -0.97999999999999998

   1. Initial program 22.0%

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

      1. +-commutative 22.0%

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

      2. clear-num 22.0%

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

      3. associate-/r/ 18.4%

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

      4. fma-def 3.0%

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

      5. +-commutative 3.0%

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

      6. add-sqr-sqrt 3.0%

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

      7. hypot-def 3.0%

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

      8. associate-*l* 3.0%

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

      9. sqrt-prod 3.0%

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

      10. metadata-eval 3.0%

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

      11. sqrt-unprod 2.0%

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

      12. add-sqr-sqrt 3.0%

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

   3. Applied egg-rr 3.0%

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

   4. Taylor expanded in x around -inf 49.3%

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

      1. fma-def 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

      4. times-frac 54.8%

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

      5. unpow2 54.8%

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

      6. distribute-rgt-out 54.8%

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

      7. metadata-eval 54.8%

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

      8. pow-sqr 54.8%

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

      9. metadata-eval 54.8%

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

      10. pow-sqr 54.8%

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

      11. times-frac 68.7%

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

      12. unpow2 68.7%

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

      13. unpow2 68.7%

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

      14. times-frac 68.7%

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

      15. unpow2 68.7%

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

      16. unpow2 68.7%

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

      17. unpow2 68.7%

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

      18. times-frac 68.7%

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

      19. unpow2 68.7%

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

      20. pow-sqr 68.7%

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

      21. metadata-eval 68.7%

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

      22. metadata-eval 68.7%

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

   6. Simplified 68.7%

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

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-def 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 51.9%

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

      7. add-sqr-sqrt 100.0%

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

   3. 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 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(2, {\left(\frac{p}{x}\right)}^{2}, {\left(\frac{p}{x}\right)}^{4} \cdot -6\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.98)
   (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = Math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98:
		tmp = math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))))
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.98)
		tmp = sqrt(Float64(0.5 * Float64(2.0 * Float64(Float64(p / x) * Float64(p / x)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98)
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.98], N[Sqrt[N[(0.5 * N[(2.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 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)))) < -0.97999999999999998

   1. Initial program 22.0%

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

   2. Taylor expanded in x around -inf 54.8%

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

      1. unpow2 54.8%

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

      2. unpow2 54.8%

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

      3. times-frac 68.6%

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

   4. Simplified 68.6%

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

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-def 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 51.9%

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

      7. add-sqr-sqrt 100.0%

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

   3. 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 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 3: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.1 \cdot 10^{-306}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-105}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p -4.4e-11)
   (sqrt 0.5)
   (if (<= p -1.1e-306)
     1.0
     (if (<= p 9e-105) (/ (- p) x) (if (<= p 1.05e-19) 1.0 (sqrt 0.5))))))

C

double code(double p, double x) {
	double tmp;
	if (p <= -4.4e-11) {
		tmp = sqrt(0.5);
	} else if (p <= -1.1e-306) {
		tmp = 1.0;
	} else if (p <= 9e-105) {
		tmp = -p / x;
	} else if (p <= 1.05e-19) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= (-4.4d-11)) then
        tmp = sqrt(0.5d0)
    else if (p <= (-1.1d-306)) then
        tmp = 1.0d0
    else if (p <= 9d-105) then
        tmp = -p / x
    else if (p <= 1.05d-19) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= -4.4e-11) {
		tmp = Math.sqrt(0.5);
	} else if (p <= -1.1e-306) {
		tmp = 1.0;
	} else if (p <= 9e-105) {
		tmp = -p / x;
	} else if (p <= 1.05e-19) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -4.4e-11:
		tmp = math.sqrt(0.5)
	elif p <= -1.1e-306:
		tmp = 1.0
	elif p <= 9e-105:
		tmp = -p / x
	elif p <= 1.05e-19:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -4.4e-11)
		tmp = sqrt(0.5);
	elseif (p <= -1.1e-306)
		tmp = 1.0;
	elseif (p <= 9e-105)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 1.05e-19)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -4.4e-11)
		tmp = sqrt(0.5);
	elseif (p <= -1.1e-306)
		tmp = 1.0;
	elseif (p <= 9e-105)
		tmp = -p / x;
	elseif (p <= 1.05e-19)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -4.4e-11], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, -1.1e-306], 1.0, If[LessEqual[p, 9e-105], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 1.05e-19], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if p < -4.4000000000000003e-11 or 1.0499999999999999e-19 < p

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 87.6%

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

   if -4.4000000000000003e-11 < p < -1.10000000000000008e-306 or 8.9999999999999995e-105 < p < 1.0499999999999999e-19

   1. Initial program 68.6%

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

   2. Taylor expanded in x around inf 54.5%

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

   if -1.10000000000000008e-306 < p < 8.9999999999999995e-105

   1. Initial program 57.0%

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

   2. Taylor expanded in x around -inf 28.1%

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

      1. unpow2 28.1%

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

      2. unpow2 28.1%

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

      3. times-frac 39.2%

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

   4. Simplified 39.2%

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

   5. Taylor expanded in p around -inf 62.6%

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

      1. neg-mul-1 62.6%

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

      2. distribute-frac-neg 62.6%

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

   7. Simplified 62.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.1 \cdot 10^{-306}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-105}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 67.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -4.6 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-101}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p -4.6e-71)
   (sqrt 0.5)
   (if (<= p -1.15e-290) (/ p x) (if (<= p 9e-101) (/ (- p) x) (sqrt 0.5)))))

C

double code(double p, double x) {
	double tmp;
	if (p <= -4.6e-71) {
		tmp = sqrt(0.5);
	} else if (p <= -1.15e-290) {
		tmp = p / x;
	} else if (p <= 9e-101) {
		tmp = -p / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= (-4.6d-71)) then
        tmp = sqrt(0.5d0)
    else if (p <= (-1.15d-290)) then
        tmp = p / x
    else if (p <= 9d-101) then
        tmp = -p / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= -4.6e-71) {
		tmp = Math.sqrt(0.5);
	} else if (p <= -1.15e-290) {
		tmp = p / x;
	} else if (p <= 9e-101) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -4.6e-71:
		tmp = math.sqrt(0.5)
	elif p <= -1.15e-290:
		tmp = p / x
	elif p <= 9e-101:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -4.6e-71)
		tmp = sqrt(0.5);
	elseif (p <= -1.15e-290)
		tmp = Float64(p / x);
	elseif (p <= 9e-101)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -4.6e-71)
		tmp = sqrt(0.5);
	elseif (p <= -1.15e-290)
		tmp = p / x;
	elseif (p <= 9e-101)
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -4.6e-71], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, -1.15e-290], N[(p / x), $MachinePrecision], If[LessEqual[p, 9e-101], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -4.5999999999999997e-71 or 8.9999999999999997e-101 < p

   1. Initial program 92.2%

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

   2. Taylor expanded in x around 0 78.8%

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

   if -4.5999999999999997e-71 < p < -1.15e-290

   1. Initial program 60.7%

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

   2. Taylor expanded in x around -inf 27.3%

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

      1. unpow2 27.3%

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

      2. unpow2 27.3%

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

      3. times-frac 34.5%

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

   4. Simplified 34.5%

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

   5. Taylor expanded in p around 0 49.4%

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

   if -1.15e-290 < p < 8.9999999999999997e-101

   1. Initial program 57.5%

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

   2. Taylor expanded in x around -inf 26.9%

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

      1. unpow2 26.9%

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

      2. unpow2 26.9%

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

      3. times-frac 37.6%

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

   4. Simplified 37.6%

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

   5. Taylor expanded in p around -inf 61.4%

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

      1. neg-mul-1 61.4%

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

      2. distribute-frac-neg 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -4.6 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-101}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 26.5% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -5.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= p -5.7e-291) (/ p x) (/ (- p) x)))

C

double code(double p, double x) {
	double tmp;
	if (p <= -5.7e-291) {
		tmp = p / x;
	} else {
		tmp = -p / x;
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= (-5.7d-291)) then
        tmp = p / x
    else
        tmp = -p / x
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= -5.7e-291) {
		tmp = p / x;
	} else {
		tmp = -p / x;
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -5.7e-291:
		tmp = p / x
	else:
		tmp = -p / x
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -5.7e-291)
		tmp = Float64(p / x);
	else
		tmp = Float64(Float64(-p) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -5.7e-291)
		tmp = p / x;
	else
		tmp = -p / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -5.7e-291], N[(p / x), $MachinePrecision], N[((-p) / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -5.70000000000000034e-291

   1. Initial program 79.2%

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

   2. Taylor expanded in x around -inf 19.3%

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

      1. unpow2 19.3%

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

      2. unpow2 19.3%

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

      3. times-frac 22.3%

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

   4. Simplified 22.3%

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

   5. Taylor expanded in p around 0 26.9%

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

   if -5.70000000000000034e-291 < p

   1. Initial program 76.5%

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

   2. Taylor expanded in x around -inf 19.0%

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

      1. unpow2 19.0%

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

      2. unpow2 19.0%

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

      3. times-frac 24.1%

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

   4. Simplified 24.1%

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

   5. Taylor expanded in p around -inf 33.9%

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

      1. neg-mul-1 33.9%

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

      2. distribute-frac-neg 33.9%

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

   7. Simplified 33.9%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -5.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 6: 16.8% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ \frac{p}{x} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (/ p x))

C

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

Fortran

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

Java

public static double code(double p, double x) {
	return p / x;
}

Python

def code(p, x):
	return p / x

Julia

function code(p, x)
	return Float64(p / x)
end

MATLAB

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

Wolfram

code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

2. Taylor expanded in x around -inf 19.2%

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

   1. unpow2 19.2%

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

   2. unpow2 19.2%

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

   3. times-frac 23.3%

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

4. Simplified 23.3%

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

5. Taylor expanded in p around 0 17.8%

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

6. Final simplification 17.8%

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

Developer target: 79.2% 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 2023182 
(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))))))))