Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.9%

Time: 8.9s

Alternatives: 5

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{p}{x}\right)}^{3}, 1.5, \frac{-p}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.6)
   (fma (pow (/ p x) 3.0) 1.5 (/ (- p) x))
   (sqrt (* (+ 1.0 (/ x (hypot x (* p 2.0)))) 0.5))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.6) {
		tmp = fma(pow((p / x), 3.0), 1.5, (-p / x));
	} else {
		tmp = sqrt(((1.0 + (x / hypot(x, (p * 2.0)))) * 0.5));
	}
	return tmp;
}

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.6)
		tmp = fma((Float64(p / x) ^ 3.0), 1.5, Float64(Float64(-p) / x));
	else
		tmp = sqrt(Float64(Float64(1.0 + Float64(x / hypot(x, Float64(p * 2.0)))) * 0.5));
	end
	return tmp
end

Wolfram

NOTE: p should be positive before calling this function
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], -0.6], N[(N[Power[N[(p / x), $MachinePrecision], 3.0], $MachinePrecision] * 1.5 + N[((-p) / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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 18.3%

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

      1. distribute-lft-in 18.3%

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

      2. metadata-eval 18.3%

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

      3. fma-def 18.3%

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

   3. Simplified 18.3%

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

      1. *-commutative 18.3%

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

      2. fma-def 18.3%

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

      3. distribute-rgt1-in 18.3%

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

      4. +-commutative 18.3%

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

   5. Applied egg-rr 18.3%

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

   6. Taylor expanded in x around -inf 41.6%

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

      1. +-commutative 41.6%

         \[\leadsto \sqrt{\color{blue}{0.5 \cdot \frac{-2 \cdot {p}^{4} + -4 \cdot {p}^{4}}{{x}^{4}} + \frac{{p}^{2}}{{x}^{2}}}} \]

      2. fma-def 41.6%

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

   8. Simplified 57.5%

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

   9. Taylor expanded in x around -inf 54.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + 1.5 \cdot \frac{{p}^{3}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. +-commutative 54.9%

          \[\leadsto \color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{3}} + -1 \cdot \frac{p}{x}} \]

       2. *-commutative 54.9%

          \[\leadsto \color{blue}{\frac{{p}^{3}}{{x}^{3}} \cdot 1.5} + -1 \cdot \frac{p}{x} \]

       3. fma-def 54.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{p}^{3}}{{x}^{3}}, 1.5, -1 \cdot \frac{p}{x}\right)} \]

       4. cube-div 59.5%

          \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\frac{p}{x}\right)}^{3}}, 1.5, -1 \cdot \frac{p}{x}\right) \]

       5. associate-*r/ 59.5%

          \[\leadsto \mathsf{fma}\left({\left(\frac{p}{x}\right)}^{3}, 1.5, \color{blue}{\frac{-1 \cdot p}{x}}\right) \]

       6. neg-mul-1 59.5%

          \[\leadsto \mathsf{fma}\left({\left(\frac{p}{x}\right)}^{3}, 1.5, \frac{\color{blue}{-p}}{x}\right) \]

   11. Simplified 59.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{p}{x}\right)}^{3}, 1.5, \frac{-p}{x}\right)} \]

   if -0.599999999999999978 < (/.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. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{p}{x}\right)}^{3}, 1.5, \frac{-p}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right) \cdot 0.5}\\ \end{array} \]

Alternative 2: 68.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 7.6 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.5 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 7.6e-200)
     t_0
     (if (<= p 4.3e-185)
       1.0
       (if (<= p 2.5e-154) t_0 (if (<= p 1.06e-54) 1.0 (sqrt 0.5)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 7.6e-200) {
		tmp = t_0;
	} else if (p <= 4.3e-185) {
		tmp = 1.0;
	} else if (p <= 2.5e-154) {
		tmp = t_0;
	} else if (p <= 1.06e-54) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (p <= 7.6d-200) then
        tmp = t_0
    else if (p <= 4.3d-185) then
        tmp = 1.0d0
    else if (p <= 2.5d-154) then
        tmp = t_0
    else if (p <= 1.06d-54) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 7.6e-200) {
		tmp = t_0;
	} else if (p <= 4.3e-185) {
		tmp = 1.0;
	} else if (p <= 2.5e-154) {
		tmp = t_0;
	} else if (p <= 1.06e-54) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= 7.6e-200:
		tmp = t_0
	elif p <= 4.3e-185:
		tmp = 1.0
	elif p <= 2.5e-154:
		tmp = t_0
	elif p <= 1.06e-54:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 7.6e-200)
		tmp = t_0;
	elseif (p <= 4.3e-185)
		tmp = 1.0;
	elseif (p <= 2.5e-154)
		tmp = t_0;
	elseif (p <= 1.06e-54)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= 7.6e-200)
		tmp = t_0;
	elseif (p <= 4.3e-185)
		tmp = 1.0;
	elseif (p <= 2.5e-154)
		tmp = t_0;
	elseif (p <= 1.06e-54)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[((-p) / x), $MachinePrecision]}, If[LessEqual[p, 7.6e-200], t$95$0, If[LessEqual[p, 4.3e-185], 1.0, If[LessEqual[p, 2.5e-154], t$95$0, If[LessEqual[p, 1.06e-54], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.6000000000000001e-200 or 4.3000000000000001e-185 < p < 2.5000000000000001e-154

   1. Initial program 74.6%

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

      1. distribute-lft-in 74.6%

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

      2. metadata-eval 74.6%

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

      3. fma-def 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in x around -inf 18.7%

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

      1. associate-*r/ 18.7%

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

      2. mul-1-neg 18.7%

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

   6. Simplified 18.7%

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

   if 7.6000000000000001e-200 < p < 4.3000000000000001e-185 or 2.5000000000000001e-154 < p < 1.0600000000000001e-54

   1. Initial program 79.3%

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

      1. distribute-lft-in 79.3%

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

      2. metadata-eval 79.3%

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

      3. fma-def 79.3%

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

      4. add-log-exp 79.2%

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

      5. *-un-lft-identity 79.2%

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

      6. log-prod 79.2%

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

      7. metadata-eval 79.2%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}\right) \]

      8. add-log-exp 79.3%

         \[\leadsto 0 + \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}} \]

      9. +-commutative 79.3%

         \[\leadsto 0 + \sqrt{\color{blue}{0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 0.5}} \]

   3. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]
   4. Step-by-step derivation

      1. +-lft-identity 79.3%

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

      2. fma-udef 79.3%

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

      3. associate-*r/ 79.3%

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

      4. *-commutative 79.3%

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

      5. associate-*r/ 79.3%

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

      6. metadata-eval 79.3%

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

      7. associate-*l/ 79.3%

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

      8. fma-udef 78.5%

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

      9. associate-*l/ 78.5%

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

      10. metadata-eval 78.5%

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

   5. Simplified 78.5%

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

   6. Taylor expanded in x around inf 65.5%

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

   if 1.0600000000000001e-54 < p

   1. Initial program 89.9%

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

   2. Taylor expanded in x around 0 84.2%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.6 \cdot 10^{-200}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4.3 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.06 \cdot 10^{-54}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 56.2% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -3.2e-207) (/ (- p) x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -3.2e-207) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.2d-207)) then
        tmp = -p / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -3.2e-207) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -3.2e-207:
		tmp = -p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -3.2e-207)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -3.2e-207)
		tmp = -p / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -3.2e-207], N[((-p) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000003e-207

   1. Initial program 55.4%

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

      1. distribute-lft-in 55.4%

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

      2. metadata-eval 55.4%

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

      3. fma-def 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around -inf 33.7%

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

      1. associate-*r/ 33.7%

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

      2. mul-1-neg 33.7%

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

   6. Simplified 33.7%

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

   if -3.2000000000000003e-207 < 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. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. fma-def 100.0%

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

      4. add-log-exp 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. log-prod 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}\right) \]

      8. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}} \]

      9. +-commutative 100.0%

         \[\leadsto 0 + \sqrt{\color{blue}{0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 0.5}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]
   4. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. fma-udef 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-*l/ 100.0%

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

      8. fma-udef 100.0%

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

      9. associate-*l/ 100.0%

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

      10. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 57.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 37.6% accurate, 42.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -3.6e+105) (/ p x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -3.6e+105) {
		tmp = p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d+105)) then
        tmp = p / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -3.6e+105) {
		tmp = p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -3.6e+105:
		tmp = p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -3.6e+105)
		tmp = Float64(p / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -3.6e+105)
		tmp = p / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -3.6e+105], N[(p / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999999e105

   1. Initial program 52.7%

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

      1. distribute-lft-in 52.7%

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

      2. metadata-eval 52.7%

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

      3. fma-def 52.7%

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

   3. Simplified 52.7%

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

      1. *-commutative 52.7%

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

      2. fma-def 52.7%

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

      3. distribute-rgt1-in 52.7%

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

      4. +-commutative 52.7%

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

   5. Applied egg-rr 52.7%

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

   6. Taylor expanded in x around -inf 48.6%

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

      1. unpow2 48.6%

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

      2. unpow2 48.6%

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

      3. associate-*r/ 48.9%

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

   8. Simplified 48.9%

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

   9. Taylor expanded in p around 0 57.3%

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

   if -3.5999999999999999e105 < x

   1. Initial program 82.2%

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

      1. distribute-lft-in 82.2%

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

      2. metadata-eval 82.2%

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

      3. fma-def 82.3%

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

      4. add-log-exp 82.2%

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

      5. *-un-lft-identity 82.2%

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

      6. log-prod 82.2%

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

      7. metadata-eval 82.2%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}\right) \]

      8. add-log-exp 82.3%

         \[\leadsto 0 + \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}} \]

      9. +-commutative 82.3%

         \[\leadsto 0 + \sqrt{\color{blue}{0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 0.5}} \]

   3. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]
   4. Step-by-step derivation

      1. +-lft-identity 82.3%

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

      2. fma-udef 82.3%

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

      3. associate-*r/ 82.3%

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

      4. *-commutative 82.3%

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

      5. associate-*r/ 82.3%

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

      6. metadata-eval 82.3%

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

      7. associate-*l/ 82.3%

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

      8. fma-udef 81.3%

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

      9. associate-*l/ 81.3%

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

      10. metadata-eval 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in x around inf 39.4%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 36.9% accurate, 215.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ 1 \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 1.0)

C

p = abs(p);
double code(double p, double x) {
	return 1.0;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	return 1.0;
}

Python

p = abs(p)
def code(p, x):
	return 1.0

Julia

p = abs(p)
function code(p, x)
	return 1.0
end

MATLAB

p = abs(p)
function tmp = code(p, x)
	tmp = 1.0;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := 1.0

Derivation

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. Step-by-step derivation

   1. distribute-lft-in 79.2%

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

   2. metadata-eval 79.2%

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

   3. fma-def 79.2%

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

   4. add-log-exp 79.2%

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

   5. *-un-lft-identity 79.2%

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

   6. log-prod 79.2%

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

   7. metadata-eval 79.2%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}}\right) \]

   8. add-log-exp 79.2%

      \[\leadsto 0 + \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}}}} \]

   9. +-commutative 79.2%

      \[\leadsto 0 + \sqrt{\color{blue}{0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}} + 0.5}} \]

3. Applied egg-rr 79.2%

   \[\leadsto \color{blue}{0 + \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5\right)}} \]
4. Step-by-step derivation

   1. +-lft-identity 79.2%

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

   2. fma-udef 79.2%

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

   3. associate-*r/ 79.2%

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

   4. *-commutative 79.2%

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

   5. associate-*r/ 79.2%

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

   6. metadata-eval 79.2%

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

   7. associate-*l/ 79.2%

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

   8. fma-udef 75.5%

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

   9. associate-*l/ 75.5%

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

   10. metadata-eval 75.5%

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

5. Simplified 75.5%

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

6. Taylor expanded in x around inf 36.2%

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

7. Final simplification 36.2%

   \[\leadsto 1 \]

Developer target: 79.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023192 
(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))))))))