Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.7%

Time: 9.2s

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: 79.7% 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

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ p_m (- x))
   (sqrt (+ 0.5 (* (/ x (hypot x (* p_m 2.0))) 0.5)))))

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)))) <= -1.0) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + ((x / hypot(x, (p_m * 2.0))) * 0.5)));
	}
	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)))) <= -1.0) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + ((x / Math.hypot(x, (p_m * 2.0))) * 0.5)));
	}
	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)))) <= -1.0:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + ((x / math.hypot(x, (p_m * 2.0))) * 0.5)))
	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)))) <= -1.0)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x / hypot(x, Float64(p_m * 2.0))) * 0.5)));
	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)))) <= -1.0)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + ((x / hypot(x, (p_m * 2.0))) * 0.5)));
	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], -1.0], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $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)))) < -1

   1. Initial program 14.4%

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

   3. Taylor expanded in x around -inf 54.6%

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

   4. Taylor expanded in p around -inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. neg-mul-1 56.4%

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

   6. Simplified 56.4%

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

   if -1 < (/.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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 42.7%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 87.9%

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

Alternative 2: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.6 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.6 \cdot 10^{-80}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 8.2 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.6e-175)
   1.0
   (if (<= p_m 3.6e-80) (/ p_m (- x)) (if (<= p_m 8.2e-38) 1.0 (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.6e-175) {
		tmp = 1.0;
	} else if (p_m <= 3.6e-80) {
		tmp = p_m / -x;
	} else if (p_m <= 8.2e-38) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 2.6d-175) then
        tmp = 1.0d0
    else if (p_m <= 3.6d-80) then
        tmp = p_m / -x
    else if (p_m <= 8.2d-38) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.6e-175) {
		tmp = 1.0;
	} else if (p_m <= 3.6e-80) {
		tmp = p_m / -x;
	} else if (p_m <= 8.2e-38) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.6e-175:
		tmp = 1.0
	elif p_m <= 3.6e-80:
		tmp = p_m / -x
	elif p_m <= 8.2e-38:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.6e-175)
		tmp = 1.0;
	elseif (p_m <= 3.6e-80)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 8.2e-38)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 2.6e-175)
		tmp = 1.0;
	elseif (p_m <= 3.6e-80)
		tmp = p_m / -x;
	elseif (p_m <= 8.2e-38)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 2.6e-175], 1.0, If[LessEqual[p$95$m, 3.6e-80], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 8.2e-38], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.6e-175 or 3.6e-80 < p < 8.1999999999999996e-38

   1. Initial program 77.1%

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

      1. +-commutative 77.1%

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

      2. distribute-rgt-in 77.1%

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

      3. +-commutative 77.1%

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

      4. add-sqr-sqrt 77.1%

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

      5. hypot-define 77.1%

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

      6. associate-*l* 77.1%

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

      7. sqrt-prod 77.1%

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

      8. metadata-eval 77.1%

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

      9. sqrt-unprod 11.1%

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

      10. add-sqr-sqrt 77.1%

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

      11. metadata-eval 77.1%

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

   4. Applied egg-rr 77.1%

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

   5. Taylor expanded in x around inf 40.5%

      \[\leadsto \sqrt{\color{blue}{0.5} + 0.5} \]
   6. Step-by-step derivation

      1. metadata-eval 40.5%

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

      2. metadata-eval 40.5%

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

   7. Applied egg-rr 40.5%

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

   if 2.6e-175 < p < 3.6e-80

   1. Initial program 25.4%

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

   3. Taylor expanded in x around -inf 31.6%

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

   4. Taylor expanded in p around -inf 79.8%

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

      1. associate-*r/ 79.8%

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

      2. neg-mul-1 79.8%

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

   6. Simplified 79.8%

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

   if 8.1999999999999996e-38 < p

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0 85.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.6 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.6 \cdot 10^{-80}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 8.2 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.8e-124) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.8e-124) {
		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 <= (-3.8d-124)) 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 <= -3.8e-124) {
		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 <= -3.8e-124:
		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 <= -3.8e-124)
		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 <= -3.8e-124)
		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, -3.8e-124], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.80000000000000012e-124

   1. Initial program 53.2%

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

   3. Taylor expanded in x around -inf 32.1%

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

   4. Taylor expanded in p around -inf 32.4%

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

      1. associate-*r/ 32.4%

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

      2. neg-mul-1 32.4%

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

   6. Simplified 32.4%

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

   if -3.80000000000000012e-124 < 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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 38.9%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 55.7%

      \[\leadsto \sqrt{\color{blue}{0.5} + 0.5} \]
   6. Step-by-step derivation

      1. metadata-eval 55.7%

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

      2. metadata-eval 55.7%

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

   7. Applied egg-rr 55.7%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-124}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.0% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.6e+92) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.6e+92) {
		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 <= (-1.6d+92)) 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 <= -1.6e+92) {
		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 <= -1.6e+92:
		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 <= -1.6e+92)
		tmp = 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 <= -1.6e+92)
		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, -1.6e+92], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000013e92

   1. Initial program 45.5%

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

   3. Taylor expanded in x around -inf 48.1%

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

   4. Taylor expanded in p around 0 48.1%

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

   if -1.60000000000000013e92 < x

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. distribute-rgt-in 79.4%

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

      3. +-commutative 79.4%

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

      4. add-sqr-sqrt 79.4%

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

      5. hypot-define 79.4%

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

      6. associate-*l* 79.4%

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

      7. sqrt-prod 79.4%

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

      8. metadata-eval 79.4%

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

      9. sqrt-unprod 33.6%

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

      10. add-sqr-sqrt 79.4%

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

      11. metadata-eval 79.4%

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

   4. Applied egg-rr 79.4%

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

   5. Taylor expanded in x around inf 36.3%

      \[\leadsto \sqrt{\color{blue}{0.5} + 0.5} \]
   6. Step-by-step derivation

      1. metadata-eval 36.3%

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

      2. metadata-eval 36.3%

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

   7. Applied egg-rr 36.3%

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

Alternative 5: 35.9% 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 76.2%

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

   1. +-commutative 76.2%

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

   2. distribute-rgt-in 76.2%

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

   3. +-commutative 76.2%

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

   4. add-sqr-sqrt 76.2%

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

   5. hypot-define 76.3%

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

   6. associate-*l* 76.3%

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

   7. sqrt-prod 76.3%

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

   8. metadata-eval 76.3%

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

   9. sqrt-unprod 33.5%

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

   10. add-sqr-sqrt 76.3%

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

   11. metadata-eval 76.3%

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

4. Applied egg-rr 76.3%

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

5. Taylor expanded in x around inf 33.5%

   \[\leadsto \sqrt{\color{blue}{0.5} + 0.5} \]
6. Step-by-step derivation

   1. metadata-eval 33.5%

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

   2. metadata-eval 33.5%

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

7. Applied egg-rr 33.5%

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

Developer Target 1: 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 2024150 
(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))))))))