Given's Rotation SVD example

Percentage Accurate: 79.9% → 99.7%

Time: 6.2s

Alternatives: 6

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: 79.9% 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(Float64(-p_m) / 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 4 p) p) (*.f64 x x)))) < -0.5

   1. Initial program 16.3%

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

   2. Taylor expanded in x around -inf 52.4%

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

   3. Taylor expanded in p around -inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. neg-mul-1 56.4%

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

   5. Simplified 56.4%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. 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-def 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 51.0%

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

      7. add-sqr-sqrt 100.0%

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

   3. 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 90.5%

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

Alternative 2: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 9.2 \cdot 10^{-231}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.36 \cdot 10^{-199}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{elif}\;p_m \leq 3.85 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;p_m \cdot \sqrt{{x}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 9.2e-231)
   1.0
   (if (<= p_m 1.36e-199)
     (/ (- p_m) x)
     (if (<= p_m 3.85e-115)
       1.0
       (if (<= p_m 3.5e-44) (* p_m (sqrt (pow x -2.0))) (sqrt 0.5))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 9.2e-231) {
		tmp = 1.0;
	} else if (p_m <= 1.36e-199) {
		tmp = -p_m / x;
	} else if (p_m <= 3.85e-115) {
		tmp = 1.0;
	} else if (p_m <= 3.5e-44) {
		tmp = p_m * sqrt(pow(x, -2.0));
	} 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 <= 9.2d-231) then
        tmp = 1.0d0
    else if (p_m <= 1.36d-199) then
        tmp = -p_m / x
    else if (p_m <= 3.85d-115) then
        tmp = 1.0d0
    else if (p_m <= 3.5d-44) then
        tmp = p_m * sqrt((x ** (-2.0d0)))
    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 <= 9.2e-231) {
		tmp = 1.0;
	} else if (p_m <= 1.36e-199) {
		tmp = -p_m / x;
	} else if (p_m <= 3.85e-115) {
		tmp = 1.0;
	} else if (p_m <= 3.5e-44) {
		tmp = p_m * Math.sqrt(Math.pow(x, -2.0));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 9.2e-231:
		tmp = 1.0
	elif p_m <= 1.36e-199:
		tmp = -p_m / x
	elif p_m <= 3.85e-115:
		tmp = 1.0
	elif p_m <= 3.5e-44:
		tmp = p_m * math.sqrt(math.pow(x, -2.0))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 9.2e-231)
		tmp = 1.0;
	elseif (p_m <= 1.36e-199)
		tmp = Float64(Float64(-p_m) / x);
	elseif (p_m <= 3.85e-115)
		tmp = 1.0;
	elseif (p_m <= 3.5e-44)
		tmp = Float64(p_m * sqrt((x ^ -2.0)));
	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 <= 9.2e-231)
		tmp = 1.0;
	elseif (p_m <= 1.36e-199)
		tmp = -p_m / x;
	elseif (p_m <= 3.85e-115)
		tmp = 1.0;
	elseif (p_m <= 3.5e-44)
		tmp = p_m * sqrt((x ^ -2.0));
	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, 9.2e-231], 1.0, If[LessEqual[p$95$m, 1.36e-199], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[p$95$m, 3.85e-115], 1.0, If[LessEqual[p$95$m, 3.5e-44], N[(p$95$m * N[Sqrt[N[Power[x, -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 9.2e-231 or 1.3600000000000001e-199 < p < 3.8500000000000001e-115

   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. Taylor expanded in x around inf 42.3%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 9.2e-231 < p < 1.3600000000000001e-199

   1. Initial program 39.7%

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

   2. Taylor expanded in x around -inf 20.4%

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

   3. Taylor expanded in p around -inf 80.6%

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

      1. associate-*r/ 80.6%

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

      2. neg-mul-1 80.6%

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

   5. Simplified 80.6%

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

   if 3.8500000000000001e-115 < p < 3.4999999999999998e-44

   1. Initial program 39.3%

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

   2. Taylor expanded in x around -inf 40.3%

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

      1. associate-*r* 40.3%

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

      2. metadata-eval 40.3%

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

      3. *-un-lft-identity 40.3%

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

      4. div-inv 40.3%

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

      5. sqrt-prod 65.6%

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

      6. unpow2 65.6%

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

      7. sqrt-prod 65.6%

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

      8. add-sqr-sqrt 65.6%

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

      9. pow-flip 65.6%

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

      10. metadata-eval 65.6%

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

   4. Applied egg-rr 65.6%

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

   if 3.4999999999999998e-44 < p

   1. Initial program 90.6%

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

   2. Taylor expanded in x around 0 78.6%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 9.2 \cdot 10^{-231}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.36 \cdot 10^{-199}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.85 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;p \cdot \sqrt{{x}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 67.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 4.6 \cdot 10^{-230}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.35 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 4.6e-230)
     1.0
     (if (<= p_m 1.35e-199)
       t_0
       (if (<= p_m 3.5e-115) 1.0 (if (<= p_m 1.05e-42) t_0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 4.6e-230) {
		tmp = 1.0;
	} else if (p_m <= 1.35e-199) {
		tmp = t_0;
	} else if (p_m <= 3.5e-115) {
		tmp = 1.0;
	} else if (p_m <= 1.05e-42) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 4.6d-230) then
        tmp = 1.0d0
    else if (p_m <= 1.35d-199) then
        tmp = t_0
    else if (p_m <= 3.5d-115) then
        tmp = 1.0d0
    else if (p_m <= 1.05d-42) then
        tmp = t_0
    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 t_0 = -p_m / x;
	double tmp;
	if (p_m <= 4.6e-230) {
		tmp = 1.0;
	} else if (p_m <= 1.35e-199) {
		tmp = t_0;
	} else if (p_m <= 3.5e-115) {
		tmp = 1.0;
	} else if (p_m <= 1.05e-42) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 4.6e-230:
		tmp = 1.0
	elif p_m <= 1.35e-199:
		tmp = t_0
	elif p_m <= 3.5e-115:
		tmp = 1.0
	elif p_m <= 1.05e-42:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 4.6e-230)
		tmp = 1.0;
	elseif (p_m <= 1.35e-199)
		tmp = t_0;
	elseif (p_m <= 3.5e-115)
		tmp = 1.0;
	elseif (p_m <= 1.05e-42)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 4.6e-230)
		tmp = 1.0;
	elseif (p_m <= 1.35e-199)
		tmp = t_0;
	elseif (p_m <= 3.5e-115)
		tmp = 1.0;
	elseif (p_m <= 1.05e-42)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 4.6e-230], 1.0, If[LessEqual[p$95$m, 1.35e-199], t$95$0, If[LessEqual[p$95$m, 3.5e-115], 1.0, If[LessEqual[p$95$m, 1.05e-42], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.5999999999999995e-230 or 1.34999999999999995e-199 < p < 3.5000000000000002e-115

   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. Taylor expanded in x around inf 42.3%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 4.5999999999999995e-230 < p < 1.34999999999999995e-199 or 3.5000000000000002e-115 < p < 1.05000000000000003e-42

   1. Initial program 39.4%

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

   2. Taylor expanded in x around -inf 34.1%

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

   3. Taylor expanded in p around -inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. neg-mul-1 69.7%

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

   5. Simplified 69.7%

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

   if 1.05000000000000003e-42 < p

   1. Initial program 90.6%

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

   2. Taylor expanded in x around 0 78.6%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.6 \cdot 10^{-230}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.35 \cdot 10^{-199}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 67.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 3.8e-44) (/ (- p_m) x) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.8e-44) {
		tmp = -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 <= 3.8d-44) then
        tmp = -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 <= 3.8e-44) {
		tmp = -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 <= 3.8e-44:
		tmp = -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 <= 3.8e-44)
		tmp = Float64(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 <= 3.8e-44)
		tmp = -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, 3.8e-44], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 3.8000000000000001e-44

   1. Initial program 77.8%

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

   2. Taylor expanded in x around -inf 15.3%

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

   3. Taylor expanded in p around -inf 15.8%

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

      1. associate-*r/ 15.8%

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

      2. neg-mul-1 15.8%

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

   5. Simplified 15.8%

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

   if 3.8000000000000001e-44 < p

   1. Initial program 90.6%

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

   2. Taylor expanded in x around 0 78.6%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 28.3% accurate, 35.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5e-310) (/ (- p_m) x) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p_m / x;
	} else {
		tmp = 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 <= (-5d-310)) then
        tmp = -p_m / x
    else
        tmp = 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 <= -5e-310) {
		tmp = -p_m / x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e-310:
		tmp = -p_m / x
	else:
		tmp = p_m / x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = Float64(p_m / x);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -p_m / x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e-310], N[((-p$95$m) / x), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 60.3%

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

   2. Taylor expanded in x around -inf 27.4%

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

   3. Taylor expanded in p around -inf 28.2%

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

      1. associate-*r/ 28.2%

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

      2. neg-mul-1 28.2%

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

   5. Simplified 28.2%

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

   if -4.999999999999985e-310 < 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. Taylor expanded in x around -inf 4.6%

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

   3. Taylor expanded in p around 0 3.6%

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

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]

Alternative 6: 6.6% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

p_m = abs(p)
function code(p_m, x)
	return Float64(p_m / x)
end

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[(p$95$m / x), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

2. Taylor expanded in x around -inf 15.1%

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

3. Taylor expanded in p around 0 15.1%

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

4. Final simplification 15.1%

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

Developer target: 79.9% 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 2023333 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

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