Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.7%

Time: 12.7s

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: 79.1% 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

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}} \leq -0.9:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))) -0.9)
   (- (/ p_m x))
   (sqrt (+ 0.5 (/ (* x 0.5) (hypot x (* p_m 2.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt((((4.0 * p_m) * p_m) + (x * x)))) <= -0.9) {
		tmp = -(p_m / x);
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt((((4.0 * p_m) * p_m) + (x * x)))) <= -0.9) {
		tmp = -(p_m / x);
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt((((4.0 * p_m) * p_m) + (x * x)))) <= -0.9:
		tmp = -(p_m / x)
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / math.hypot(x, (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))) <= -0.9)
		tmp = Float64(-Float64(p_m / x));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / hypot(x, Float64(p_m * 2.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt((((4.0 * p_m) * p_m) + (x * x)))) <= -0.9)
		tmp = -(p_m / x);
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9], (-N[(p$95$m / x), $MachinePrecision]), N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $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)))) < -0.900000000000000022

   1. Initial program 18.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -0.900000000000000022 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   8. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 68.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 10^{-118}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.6 \cdot 10^{-108}:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;p\_m \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5 \cdot x}{2 \cdot p\_m + x \cdot \left(0.25 \cdot \frac{x}{p\_m}\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1e-118)
   1.0
   (if (<= p_m 1.6e-108)
     (- (/ p_m x))
     (if (<= p_m 3.8e-51)
       1.0
       (sqrt
        (+ 0.5 (/ (* 0.5 x) (+ (* 2.0 p_m) (* x (* 0.25 (/ x p_m)))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1e-118) {
		tmp = 1.0;
	} else if (p_m <= 1.6e-108) {
		tmp = -(p_m / x);
	} else if (p_m <= 3.8e-51) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((0.5 * x) / ((2.0 * p_m) + (x * (0.25 * (x / p_m)))))));
	}
	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 <= 1d-118) then
        tmp = 1.0d0
    else if (p_m <= 1.6d-108) then
        tmp = -(p_m / x)
    else if (p_m <= 3.8d-51) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((0.5d0 * x) / ((2.0d0 * p_m) + (x * (0.25d0 * (x / p_m)))))))
    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 <= 1e-118) {
		tmp = 1.0;
	} else if (p_m <= 1.6e-108) {
		tmp = -(p_m / x);
	} else if (p_m <= 3.8e-51) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((0.5 * x) / ((2.0 * p_m) + (x * (0.25 * (x / p_m)))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1e-118:
		tmp = 1.0
	elif p_m <= 1.6e-108:
		tmp = -(p_m / x)
	elif p_m <= 3.8e-51:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((0.5 * x) / ((2.0 * p_m) + (x * (0.25 * (x / p_m)))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1e-118)
		tmp = 1.0;
	elseif (p_m <= 1.6e-108)
		tmp = Float64(-Float64(p_m / x));
	elseif (p_m <= 3.8e-51)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(0.5 * x) / Float64(Float64(2.0 * p_m) + Float64(x * Float64(0.25 * Float64(x / p_m)))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1e-118)
		tmp = 1.0;
	elseif (p_m <= 1.6e-108)
		tmp = -(p_m / x);
	elseif (p_m <= 3.8e-51)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((0.5 * x) / ((2.0 * p_m) + (x * (0.25 * (x / p_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1e-118], 1.0, If[LessEqual[p$95$m, 1.6e-108], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[p$95$m, 3.8e-51], 1.0, N[Sqrt[N[(0.5 + N[(N[(0.5 * x), $MachinePrecision] / N[(N[(2.0 * p$95$m), $MachinePrecision] + N[(x * N[(0.25 * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 9.99999999999999985e-119 or 1.6e-108 < p < 3.80000000000000003e-51

   1. Initial program 83.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 9.99999999999999985e-119 < p < 1.6e-108

   1. Initial program 4.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 3.80000000000000003e-51 < p

   1. Initial program 88.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 3.8 \cdot 10^{-118}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.8 \cdot 10^{-109}:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;p\_m \leq 3.4 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 3.8e-118)
   1.0
   (if (<= p_m 3.8e-109) (- (/ p_m x)) (if (<= p_m 3.4e-50) 1.0 (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.8e-118) {
		tmp = 1.0;
	} else if (p_m <= 3.8e-109) {
		tmp = -(p_m / x);
	} else if (p_m <= 3.4e-50) {
		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 <= 3.8d-118) then
        tmp = 1.0d0
    else if (p_m <= 3.8d-109) then
        tmp = -(p_m / x)
    else if (p_m <= 3.4d-50) 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 <= 3.8e-118) {
		tmp = 1.0;
	} else if (p_m <= 3.8e-109) {
		tmp = -(p_m / x);
	} else if (p_m <= 3.4e-50) {
		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 <= 3.8e-118:
		tmp = 1.0
	elif p_m <= 3.8e-109:
		tmp = -(p_m / x)
	elif p_m <= 3.4e-50:
		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 <= 3.8e-118)
		tmp = 1.0;
	elseif (p_m <= 3.8e-109)
		tmp = Float64(-Float64(p_m / x));
	elseif (p_m <= 3.4e-50)
		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 <= 3.8e-118)
		tmp = 1.0;
	elseif (p_m <= 3.8e-109)
		tmp = -(p_m / x);
	elseif (p_m <= 3.4e-50)
		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, 3.8e-118], 1.0, If[LessEqual[p$95$m, 3.8e-109], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[p$95$m, 3.4e-50], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.8000000000000001e-118 or 3.80000000000000002e-109 < p < 3.40000000000000014e-50

   1. Initial program 83.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 3.8000000000000001e-118 < p < 3.80000000000000002e-109

   1. Initial program 4.3%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 3.40000000000000014e-50 < p

   1. Initial program 88.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 54.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-103}:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3e-103) (- (/ p_m x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3e-103) {
		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 <= (-3d-103)) 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 <= -3e-103) {
		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 <= -3e-103:
		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 <= -3e-103)
		tmp = Float64(-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 <= -3e-103)
		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, -3e-103], (-N[(p$95$m / x), $MachinePrecision]), 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3e-103

   1. Initial program 58.2%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around -inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -3e-103 < x

   1. Initial program 99.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 35.8% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Developer target: 79.1% 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 2024110 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

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

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