Given's Rotation SVD example

Percentage Accurate: 79.2% → 99.7%

Time: 10.1s

Alternatives: 7

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(t\_0 + 1\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.5) (- 0.0 (/ p_m x)) (sqrt (* 0.5 (+ t_0 1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = sqrt((0.5 * (t_0 + 1.0)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / sqrt(((p_m * (4.0d0 * p_m)) + (x * x)))
    if (t_0 <= (-0.5d0)) then
        tmp = 0.0d0 - (p_m / x)
    else
        tmp = sqrt((0.5d0 * (t_0 + 1.0d0)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = Math.sqrt((0.5 * (t_0 + 1.0)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))
	tmp = 0
	if t_0 <= -0.5:
		tmp = 0.0 - (p_m / x)
	else:
		tmp = math.sqrt((0.5 * (t_0 + 1.0)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(0.0 - Float64(p_m / x));
	else
		tmp = sqrt(Float64(0.5 * Float64(t_0 + 1.0)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = 0.0 - (p_m / x);
	else
		tmp = sqrt((0.5 * (t_0 + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(t$95$0 + 1.0), $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)))) < -0.5

   1. Initial program 13.8%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      4. /-lowering-/.f64 55.8

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

   8. Simplified 55.8%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) 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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;0 - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+95}:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + \frac{2 \cdot \left(p\_m \cdot p\_m\right)}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -3.7e+95)
   (- 0.0 (/ p_m x))
   (sqrt (* 0.5 (+ 1.0 (/ x (+ x (/ (* 2.0 (* p_m p_m)) x))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.7e+95) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (x + ((2.0 * (p_m * p_m)) / x))))));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.7d+95)) then
        tmp = 0.0d0 - (p_m / x)
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (x + ((2.0d0 * (p_m * p_m)) / x))))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -3.7e+95) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (x + ((2.0 * (p_m * p_m)) / x))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.7e+95:
		tmp = 0.0 - (p_m / x)
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (x + ((2.0 * (p_m * p_m)) / x))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.7e+95)
		tmp = Float64(0.0 - Float64(p_m / x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(x + Float64(Float64(2.0 * Float64(p_m * p_m)) / x))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -3.7e+95)
		tmp = 0.0 - (p_m / x);
	else
		tmp = sqrt((0.5 * (1.0 + (x / (x + ((2.0 * (p_m * p_m)) / x))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.7e+95], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(x + N[(N[(2.0 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000001e95

   1. Initial program 52.5%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 45.6

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

   5. Simplified 45.6%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      4. /-lowering-/.f64 49.0

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

   if -3.7000000000000001e95 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in p around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. +-lowering-+.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 81.8

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

   5. Simplified 81.8%

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

Alternative 3: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.3 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 4.8 \cdot 10^{-10}:\\ \;\;\;\;p\_m \cdot \left|\frac{1}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.3e-73)
   1.0
   (if (<= p_m 4.8e-10) (* p_m (fabs (/ 1.0 x))) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.3e-73) {
		tmp = 1.0;
	} else if (p_m <= 4.8e-10) {
		tmp = p_m * fabs((1.0 / x));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.3d-73) then
        tmp = 1.0d0
    else if (p_m <= 4.8d-10) then
        tmp = p_m * abs((1.0d0 / x))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.3e-73) {
		tmp = 1.0;
	} else if (p_m <= 4.8e-10) {
		tmp = p_m * Math.abs((1.0 / x));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.3e-73:
		tmp = 1.0
	elif p_m <= 4.8e-10:
		tmp = p_m * math.fabs((1.0 / x))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.3e-73)
		tmp = 1.0;
	elseif (p_m <= 4.8e-10)
		tmp = Float64(p_m * abs(Float64(1.0 / x)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.3e-73)
		tmp = 1.0;
	elseif (p_m <= 4.8e-10)
		tmp = p_m * abs((1.0 / x));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.3e-73], 1.0, If[LessEqual[p$95$m, 4.8e-10], N[(p$95$m * N[Abs[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.3e-73

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Simplified 43.8%

      \[\leadsto \sqrt{\color{blue}{1}} \]
   6. Step-by-step derivation

      1. metadata-eval 43.8

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

   7. Applied egg-rr 43.8%

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

   if 1.3e-73 < p < 4.8e-10

   1. Initial program 31.6%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 46.1

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

   5. Simplified 46.1%

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

      1. times-frac N/A

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

      2. rem-sqrt-square N/A

         \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

      3. div-inv N/A

         \[\leadsto \left|\color{blue}{p \cdot \frac{1}{x}}\right| \]

      4. fabs-mul N/A

         \[\leadsto \color{blue}{\left|p\right| \cdot \left|\frac{1}{x}\right|} \]

      5. rem-sqrt-square N/A

         \[\leadsto \color{blue}{\sqrt{p \cdot p}} \cdot \left|\frac{1}{x}\right| \]

      6. sqrt-prod N/A

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

      7. rem-square-sqrt N/A

         \[\leadsto \color{blue}{p} \cdot \left|\frac{1}{x}\right| \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{p \cdot \left|\frac{1}{x}\right|} \]

      9. fabs-lowering-fabs.f64 N/A

         \[\leadsto p \cdot \color{blue}{\left|\frac{1}{x}\right|} \]

      10. /-lowering-/.f64 73.3

          \[\leadsto p \cdot \left|\color{blue}{\frac{1}{x}}\right| \]

   7. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{p \cdot \left|\frac{1}{x}\right|} \]

   if 4.8e-10 < p

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   4. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 87.0

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

   5. Simplified 87.0%

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

Alternative 4: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.4e-74) 1.0 (if (<= p_m 6.5e-10) (- 0.0 (/ p_m x)) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.4e-74) {
		tmp = 1.0;
	} else if (p_m <= 6.5e-10) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.4d-74) then
        tmp = 1.0d0
    else if (p_m <= 6.5d-10) then
        tmp = 0.0d0 - (p_m / x)
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.4e-74) {
		tmp = 1.0;
	} else if (p_m <= 6.5e-10) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.4e-74:
		tmp = 1.0
	elif p_m <= 6.5e-10:
		tmp = 0.0 - (p_m / x)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.4e-74)
		tmp = 1.0;
	elseif (p_m <= 6.5e-10)
		tmp = Float64(0.0 - Float64(p_m / x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.4e-74)
		tmp = 1.0;
	elseif (p_m <= 6.5e-10)
		tmp = 0.0 - (p_m / x);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.4e-74], 1.0, If[LessEqual[p$95$m, 6.5e-10], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.39999999999999994e-74

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Simplified 43.8%

      \[\leadsto \sqrt{\color{blue}{1}} \]
   6. Step-by-step derivation

      1. metadata-eval 43.8

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

   7. Applied egg-rr 43.8%

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

   if 1.39999999999999994e-74 < p < 6.5000000000000003e-10

   1. Initial program 31.6%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      4. /-lowering-/.f64 72.7

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

   8. Simplified 72.7%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

   if 6.5000000000000003e-10 < p

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   4. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 87.0

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

   5. Simplified 87.0%

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

Alternative 5: 56.0% accurate, 21.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-178}:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -7.2e-178) (- 0.0 (/ p_m x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -7.2e-178) {
		tmp = 0.0 - (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 <= (-7.2d-178)) then
        tmp = 0.0d0 - (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 <= -7.2e-178) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -7.2e-178:
		tmp = 0.0 - (p_m / x)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -7.2e-178)
		tmp = Float64(0.0 - Float64(p_m / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -7.2e-178)
		tmp = 0.0 - (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, -7.2e-178], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999987e-178

   1. Initial program 61.2%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 27.3

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

   5. Simplified 27.3%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

      4. /-lowering-/.f64 27.0

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

   8. Simplified 27.0%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]

   if -7.19999999999999987e-178 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Simplified 63.9%

      \[\leadsto \sqrt{\color{blue}{1}} \]
   6. Step-by-step derivation

      1. metadata-eval 63.9

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

   7. Applied egg-rr 63.9%

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

Alternative 6: 36.1% 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 80.1%

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

3. Taylor expanded in x around inf

   \[\leadsto \sqrt{\color{blue}{1}} \]
4. Step-by-step derivation

5. Simplified 38.0%

   \[\leadsto \sqrt{\color{blue}{1}} \]
6. Step-by-step derivation

   1. metadata-eval 38.0

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

7. Applied egg-rr 38.0%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Alternative 7: 6.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 0 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 0.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 0.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 0.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 0.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 0.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 0.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 0.0

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in x around -inf

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

5. Simplified 5.4%

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{-1}\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{0}} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\color{blue}{0}} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

   4. pow1/2 N/A

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

   5. metadata-eval N/A

      \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

   6. metadata-eval 5.4

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 5.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 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 2024191 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

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