Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.7%

Time: 9.3s

Alternatives: 4

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.6)
   (/ p_m (- x))
   (sqrt (+ 0.5 (* 0.5 (/ x (hypot x (* p_m 2.0))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.6) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.6) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (x / Math.hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.6:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + (0.5 * (x / math.hypot(x, (p_m * 2.0))))))
	return tmp

Julia

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

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.6)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p_m * 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.6], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.599999999999999978

   1. Initial program 19.5%

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

      1. +-commutative 19.5%

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

      2. sqr-neg 19.5%

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

      3. associate-*l* 19.5%

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

      4. sqr-neg 19.5%

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

      5. fma-define 19.5%

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

      6. sqr-neg 19.5%

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

      7. fma-define 19.5%

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

      8. associate-*l* 19.5%

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

      9. +-commutative 19.5%

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

   3. Simplified 19.5%

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

   5. Taylor expanded in x around -inf 57.0%

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

      1. mul-1-neg 57.0%

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

      2. distribute-neg-frac2 57.0%

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

   7. Simplified 57.0%

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

   if -0.599999999999999978 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqr-neg 99.9%

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

      5. fma-define 99.9%

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

      6. sqr-neg 99.9%

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

      7. fma-define 99.9%

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

      8. associate-*l* 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-undefine 99.9%

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

      3. associate-*r* 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-define 99.9%

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

      6. associate-*r* 99.9%

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

      7. *-commutative 99.9%

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

      8. sqrt-prod 99.9%

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

      9. sqrt-prod 53.6%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 91.8%

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

Alternative 2: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.05 \cdot 10^{-272}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.9 \cdot 10^{-249}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 3.8 \cdot 10^{-96}:\\ \;\;\;\;1 + \left(\frac{p\_m}{x} \cdot \frac{p\_m}{x}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.05e-272)
   1.0
   (if (<= p_m 1.9e-249)
     (/ p_m (- x))
     (if (<= p_m 3.8e-96)
       (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))
       (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.05e-272) {
		tmp = 1.0;
	} else if (p_m <= 1.9e-249) {
		tmp = p_m / -x;
	} else if (p_m <= 3.8e-96) {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	} 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 <= 2.05d-272) then
        tmp = 1.0d0
    else if (p_m <= 1.9d-249) then
        tmp = p_m / -x
    else if (p_m <= 3.8d-96) then
        tmp = 1.0d0 + (((p_m / x) * (p_m / x)) * (-0.5d0))
    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 <= 2.05e-272) {
		tmp = 1.0;
	} else if (p_m <= 1.9e-249) {
		tmp = p_m / -x;
	} else if (p_m <= 3.8e-96) {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.05e-272:
		tmp = 1.0
	elif p_m <= 1.9e-249:
		tmp = p_m / -x
	elif p_m <= 3.8e-96:
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.05e-272)
		tmp = 1.0;
	elseif (p_m <= 1.9e-249)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 3.8e-96)
		tmp = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5));
	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 <= 2.05e-272)
		tmp = 1.0;
	elseif (p_m <= 1.9e-249)
		tmp = p_m / -x;
	elseif (p_m <= 3.8e-96)
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	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, 2.05e-272], 1.0, If[LessEqual[p$95$m, 1.9e-249], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 3.8e-96], N[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if p < 2.0499999999999999e-272

   1. Initial program 83.5%

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

      1. +-commutative 83.5%

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

      2. sqr-neg 83.5%

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

      3. associate-*l* 83.5%

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

      4. sqr-neg 83.5%

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

      5. fma-define 83.5%

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

      6. sqr-neg 83.5%

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

      7. fma-define 83.5%

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

      8. associate-*l* 83.5%

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

      9. +-commutative 83.5%

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

   3. Simplified 83.5%

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

      1. div-inv 83.4%

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

      2. fma-undefine 83.4%

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

      3. associate-*r* 83.4%

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

      4. add-sqr-sqrt 83.4%

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

      5. hypot-define 83.4%

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

      6. associate-*r* 83.4%

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

      7. *-commutative 83.4%

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

      8. sqrt-prod 83.4%

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

      9. sqrt-prod 6.9%

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

      10. add-sqr-sqrt 83.6%

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

      11. metadata-eval 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. associate-*r/ 83.7%

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

      2. *-rgt-identity 83.7%

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

      3. *-commutative 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in x around inf 32.8%

      \[\leadsto \sqrt{\color{blue}{1 + -1 \cdot \frac{{p}^{2}}{{x}^{2}}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 32.8%

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

       2. unsub-neg 32.8%

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

       3. unpow2 32.8%

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

       4. unpow2 32.8%

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

       5. times-frac 32.8%

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

       6. unpow2 32.8%

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

   11. Simplified 32.8%

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

       1. pow2 32.8%

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

   13. Applied egg-rr 32.8%

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

   14. Taylor expanded in p around 0 41.8%

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

   if 2.0499999999999999e-272 < p < 1.9e-249

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. sqr-neg 68.1%

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

      3. associate-*l* 68.1%

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

      4. sqr-neg 68.1%

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

      5. fma-define 68.1%

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

      6. sqr-neg 68.1%

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

      7. fma-define 68.1%

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

      8. associate-*l* 68.1%

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

      9. +-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around -inf 51.3%

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

      1. mul-1-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

   7. Simplified 51.3%

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

   if 1.9e-249 < p < 3.8000000000000001e-96

   1. Initial program 75.6%

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

      1. +-commutative 75.6%

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

      2. sqr-neg 75.6%

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

      3. associate-*l* 75.6%

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

      4. sqr-neg 75.6%

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

      5. fma-define 75.6%

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

      6. sqr-neg 75.6%

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

      7. fma-define 75.6%

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

      8. associate-*l* 75.6%

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

      9. +-commutative 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in x around inf 71.6%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 71.6%

         \[\leadsto 1 + \color{blue}{\frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]

   7. Simplified 71.6%

      \[\leadsto \color{blue}{1 + \frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]
   8. Step-by-step derivation

      1. unpow2 71.6%

         \[\leadsto 1 + \frac{{p}^{2}}{\color{blue}{x \cdot x}} \cdot -0.5 \]

      2. unpow2 71.6%

         \[\leadsto 1 + \frac{\color{blue}{p \cdot p}}{x \cdot x} \cdot -0.5 \]

      3. times-frac 71.6%

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

   9. Applied egg-rr 71.6%

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

   if 3.8000000000000001e-96 < p

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0 83.2%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.05 \cdot 10^{-272}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.9 \cdot 10^{-249}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-96}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.1% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -7.4e-109) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -7.4e-109) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7.4d-109)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -7.4e-109) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -7.4e-109:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -7.4e-109)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -7.4e-109)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -7.4e-109], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.39999999999999961e-109

   1. Initial program 63.4%

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

      1. +-commutative 63.4%

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

      2. sqr-neg 63.4%

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

      3. associate-*l* 63.4%

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

      4. sqr-neg 63.4%

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

      5. fma-define 63.4%

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

      6. sqr-neg 63.4%

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

      7. fma-define 63.4%

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

      8. associate-*l* 63.4%

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

      9. +-commutative 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around -inf 27.2%

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

      1. mul-1-neg 27.2%

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

      2. distribute-neg-frac2 27.2%

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

   7. Simplified 27.2%

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

   if -7.39999999999999961e-109 < x

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

      2. sqr-neg 98.9%

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

      3. associate-*l* 98.9%

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

      4. sqr-neg 98.9%

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

      5. fma-define 98.9%

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

      6. sqr-neg 98.9%

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

      7. fma-define 98.9%

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

      8. associate-*l* 98.9%

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

      9. +-commutative 98.9%

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

   3. Simplified 98.9%

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

      1. div-inv 98.9%

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

      2. fma-undefine 98.9%

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

      3. associate-*r* 98.9%

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

      4. add-sqr-sqrt 98.9%

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

      5. hypot-define 98.9%

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

      6. associate-*r* 98.9%

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

      7. *-commutative 98.9%

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

      8. sqrt-prod 98.9%

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

      9. sqrt-prod 52.2%

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

      10. add-sqr-sqrt 98.9%

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

      11. metadata-eval 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

      3. *-commutative 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in x around inf 46.8%

      \[\leadsto \sqrt{\color{blue}{1 + -1 \cdot \frac{{p}^{2}}{{x}^{2}}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 46.8%

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

       2. unsub-neg 46.8%

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

       3. unpow2 46.8%

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

       4. unpow2 46.8%

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

       5. times-frac 46.8%

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

       6. unpow2 46.8%

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

   11. Simplified 46.8%

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

       1. pow2 46.8%

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

   13. Applied egg-rr 46.8%

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

   14. Taylor expanded in p around 0 56.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.8% accurate, 215.0× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

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

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

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

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 84.5%

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

   1. +-commutative 84.5%

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

   2. sqr-neg 84.5%

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

   3. associate-*l* 84.5%

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

   4. sqr-neg 84.5%

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

   5. fma-define 84.5%

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

   6. sqr-neg 84.5%

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

   7. fma-define 84.5%

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

   8. associate-*l* 84.5%

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

   9. +-commutative 84.5%

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

3. Simplified 84.5%

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

   1. div-inv 84.1%

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

   2. fma-undefine 84.1%

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

   3. associate-*r* 84.1%

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

   4. add-sqr-sqrt 84.1%

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

   5. hypot-define 84.1%

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

   6. associate-*r* 84.1%

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

   7. *-commutative 84.1%

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

   8. sqrt-prod 84.1%

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

   9. sqrt-prod 44.6%

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

   10. add-sqr-sqrt 84.2%

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

   11. metadata-eval 84.2%

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

6. Applied egg-rr 84.2%

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

   1. associate-*r/ 84.6%

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

   2. *-rgt-identity 84.6%

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

   3. *-commutative 84.6%

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

8. Simplified 84.6%

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

9. Taylor expanded in x around inf 28.8%

   \[\leadsto \sqrt{\color{blue}{1 + -1 \cdot \frac{{p}^{2}}{{x}^{2}}}} \]
10. Step-by-step derivation

    1. mul-1-neg 28.8%

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

    2. unsub-neg 28.8%

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

    3. unpow2 28.8%

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

    4. unpow2 28.8%

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

    5. times-frac 28.8%

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

    6. unpow2 28.8%

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

11. Simplified 28.8%

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

    1. pow2 28.8%

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

13. Applied egg-rr 28.8%

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

14. Taylor expanded in p around 0 39.0%

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

15. Final simplification 39.0%

    \[\leadsto 1 \]
16. Add Preprocessing

Developer target: 79.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024077 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

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