Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.7%

Time: 10.2s

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: 79.3% 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

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

FPCore

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

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 16.1%

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

   3. Taylor expanded in x around -inf 38.5%

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

      1. associate-*r/ 38.5%

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

   5. Simplified 38.5%

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

   7. Applied egg-rr 60.2%

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

   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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 48.4%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 90.1%

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

Alternative 2: 68.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.25 \cdot 10^{-279}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2 \cdot 10^{-212}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 9.5 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.25e-279)
   1.0
   (if (<= p_m 2e-212) (/ p_m (- x)) (if (<= p_m 9.5e-14) 1.0 (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.25e-279) {
		tmp = 1.0;
	} else if (p_m <= 2e-212) {
		tmp = p_m / -x;
	} else if (p_m <= 9.5e-14) {
		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 <= 2.25d-279) then
        tmp = 1.0d0
    else if (p_m <= 2d-212) then
        tmp = p_m / -x
    else if (p_m <= 9.5d-14) 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 <= 2.25e-279) {
		tmp = 1.0;
	} else if (p_m <= 2e-212) {
		tmp = p_m / -x;
	} else if (p_m <= 9.5e-14) {
		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 <= 2.25e-279:
		tmp = 1.0
	elif p_m <= 2e-212:
		tmp = p_m / -x
	elif p_m <= 9.5e-14:
		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 <= 2.25e-279)
		tmp = 1.0;
	elseif (p_m <= 2e-212)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 9.5e-14)
		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 <= 2.25e-279)
		tmp = 1.0;
	elseif (p_m <= 2e-212)
		tmp = p_m / -x;
	elseif (p_m <= 9.5e-14)
		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, 2.25e-279], 1.0, If[LessEqual[p$95$m, 2e-212], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 9.5e-14], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.24999999999999998e-279 or 1.99999999999999991e-212 < p < 9.4999999999999999e-14

   1. Initial program 76.7%

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

      1. add-sqr-sqrt 76.7%

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

      2. hypot-define 76.7%

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

      3. associate-*l* 76.7%

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

      4. sqrt-prod 76.7%

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

      5. metadata-eval 76.7%

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

      6. sqrt-unprod 21.1%

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

      7. add-sqr-sqrt 76.7%

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

   4. Applied egg-rr 76.7%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. mul-1-neg 38.5%

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

      2. unpow2 38.5%

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

      3. associate-/l/ 38.5%

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

      4. unpow2 38.5%

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

      5. associate-*l/ 38.5%

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

      6. associate-/l* 38.5%

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

      7. unpow2 38.5%

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

      8. unsub-neg 38.5%

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

      9. unpow2 38.5%

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

      10. *-lft-identity 38.5%

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

      11. associate-*l/ 38.5%

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

      12. associate-/r/ 38.5%

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

      13. unpow-1 38.5%

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

      14. *-lft-identity 38.5%

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

      15. associate-*l/ 38.5%

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

      16. associate-/r/ 38.5%

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

      17. unpow-1 38.5%

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

      18. pow-sqr 38.5%

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

      19. metadata-eval 38.5%

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

   7. Simplified 38.5%

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

   8. Taylor expanded in x around inf 45.6%

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

   if 2.24999999999999998e-279 < p < 1.99999999999999991e-212

   1. Initial program 53.1%

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

   3. Taylor expanded in x around -inf 27.6%

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

      1. associate-*r/ 27.6%

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

   5. Simplified 27.6%

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

   7. Applied egg-rr 74.3%

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

   if 9.4999999999999999e-14 < p

   1. Initial program 96.4%

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

   3. Taylor expanded in x around 0 93.5%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.25 \cdot 10^{-279}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-212}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.3% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-146}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.4e-146) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.4e-146) {
		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 <= (-3.4d-146)) 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 <= -3.4e-146) {
		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 <= -3.4e-146:
		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 <= -3.4e-146)
		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 <= -3.4e-146)
		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, -3.4e-146], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.4000000000000001e-146

   1. Initial program 53.7%

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

   3. Taylor expanded in x around -inf 23.4%

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

      1. associate-*r/ 23.4%

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

   5. Simplified 23.4%

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

   7. Applied egg-rr 34.6%

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

   if -3.4000000000000001e-146 < 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. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 50.7%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 53.6%

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

      1. mul-1-neg 53.6%

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

      2. unpow2 53.6%

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

      3. associate-/l/ 53.6%

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

      4. unpow2 53.6%

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

      5. associate-*l/ 53.6%

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

      6. associate-/l* 53.6%

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

      7. unpow2 53.6%

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

      8. unsub-neg 53.6%

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

      9. unpow2 53.6%

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

      10. *-lft-identity 53.6%

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

      11. associate-*l/ 53.6%

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

      12. associate-/r/ 53.6%

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

      13. unpow-1 53.6%

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

      14. *-lft-identity 53.6%

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

      15. associate-*l/ 53.6%

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

      16. associate-/r/ 53.6%

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

      17. unpow-1 53.6%

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

      18. pow-sqr 53.6%

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

      19. metadata-eval 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in x around inf 62.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-146}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.2% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+65}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2.5e+65) 0.0 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.5e+65) {
		tmp = 0.0;
	} 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 <= (-2.5d+65)) then
        tmp = 0.0d0
    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 <= -2.5e+65) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.5e+65:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.5e+65)
		tmp = 0.0;
	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 <= -2.5e+65)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.5e+65], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999986e65

   1. Initial program 58.2%

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

   3. Taylor expanded in x around -inf 27.1%

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

      1. metadata-eval 27.1%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{0}} \]

      2. metadata-eval 27.1%

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

      3. metadata-eval 27.1%

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

      4. pow1/2 27.1%

         \[\leadsto \color{blue}{{\left(1 + -1\right)}^{0.5}} \]

      5. metadata-eval 27.1%

         \[\leadsto {\color{blue}{0}}^{0.5} \]

      6. metadata-eval 27.1%

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

   5. Applied egg-rr 27.1%

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

   if -2.49999999999999986e65 < x

   1. Initial program 81.6%

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

      1. add-sqr-sqrt 81.6%

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

      2. hypot-define 81.6%

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

      3. associate-*l* 81.6%

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

      4. sqrt-prod 81.6%

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

      5. metadata-eval 81.6%

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

      6. sqrt-unprod 39.4%

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

      7. add-sqr-sqrt 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in x around inf 34.1%

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

      1. mul-1-neg 34.1%

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

      2. unpow2 34.1%

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

      3. associate-/l/ 34.1%

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

      4. unpow2 34.1%

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

      5. associate-*l/ 34.1%

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

      6. associate-/l* 34.1%

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

      7. unpow2 34.1%

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

      8. unsub-neg 34.1%

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

      9. unpow2 34.1%

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

      10. *-lft-identity 34.1%

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

      11. associate-*l/ 34.1%

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

      12. associate-/r/ 34.1%

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

      13. unpow-1 34.1%

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

      14. *-lft-identity 34.1%

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

      15. associate-*l/ 34.1%

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

      16. associate-/r/ 34.1%

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

      17. unpow-1 34.1%

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

      18. pow-sqr 34.1%

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

      19. metadata-eval 34.1%

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

   7. Simplified 34.1%

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

   8. Taylor expanded in x around inf 43.3%

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

Alternative 5: 6.1% 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 79.0%

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

3. Taylor expanded in x around -inf 6.1%

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

   1. metadata-eval 6.1%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{0}} \]

   2. metadata-eval 6.1%

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

   3. metadata-eval 6.1%

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

   4. pow1/2 6.1%

      \[\leadsto \color{blue}{{\left(1 + -1\right)}^{0.5}} \]

   5. metadata-eval 6.1%

      \[\leadsto {\color{blue}{0}}^{0.5} \]

   6. metadata-eval 6.1%

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

5. Applied egg-rr 6.1%

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

Developer Target 1: 79.3% 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 2024152 
(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))))))))