Given's Rotation SVD example

Percentage Accurate: 78.9% → 78.9%

Time: 3.5s

Alternatives: 8

Speedup: 1.0×

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: 78.9% 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: 78.9% 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]

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. Add Preprocessing

Alternative 2: 39.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq 0.004:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))
   (if (<= t_0 0.004) (+ 1.0 t_0) t_0)))

C

double code(double p, double x) {
	double t_0 = x / sqrt((((4.0 * p) * p) + (x * x)));
	double tmp;
	if (t_0 <= 0.004) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / sqrt((((4.0d0 * p) * p) + (x * x)))
    if (t_0 <= 0.004d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = x / Math.sqrt((((4.0 * p) * p) + (x * x)));
	double tmp;
	if (t_0 <= 0.004) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = x / math.sqrt((((4.0 * p) * p) + (x * x)))
	tmp = 0
	if t_0 <= 0.004:
		tmp = 1.0 + t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(p, x)
	t_0 = Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= 0.004)
		tmp = Float64(1.0 + t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = x / sqrt((((4.0 * p) * p) + (x * x)));
	tmp = 0.0;
	if (t_0 <= 0.004)
		tmp = 1.0 + t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.004], N[(1.0 + t$95$0), $MachinePrecision], t$95$0]]

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)))) < 0.0040000000000000001

   1. Initial program 72.5%

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

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2} + {p}^{2} \cdot \left(-1 \cdot \frac{\sqrt{\frac{1}{2}}}{{x}^{2} \cdot \sqrt{2}} + \frac{1}{2} \cdot \frac{{p}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2}}\right)} \]

   5. Applied rewrites 19.4%

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

   if 0.0040000000000000001 < (/.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. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2} + {p}^{2} \cdot \left(-1 \cdot \frac{\sqrt{\frac{1}{2}}}{{x}^{2} \cdot \sqrt{2}} + {p}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{p}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(-1 \cdot \frac{6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{2}} + 20 \cdot \frac{1}{{x}^{6}}\right)\right)}{\sqrt{2}} + \frac{1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}}\right)\right)} \]

   5. Applied rewrites 99.2%

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

Alternative 3: 31.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot p\right) \cdot p\\ t_1 := \frac{x}{\sqrt{t\_0 + x \cdot x}}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (* (* 4.0 p) p)) (t_1 (/ x (sqrt (+ t_0 (* x x))))))
   (if (<= t_1 5e-143) (* 0.5 t_0) t_1)))

C

double code(double p, double x) {
	double t_0 = (4.0 * p) * p;
	double t_1 = x / sqrt((t_0 + (x * x)));
	double tmp;
	if (t_1 <= 5e-143) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * p) * p
    t_1 = x / sqrt((t_0 + (x * x)))
    if (t_1 <= 5d-143) then
        tmp = 0.5d0 * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = (4.0 * p) * p;
	double t_1 = x / Math.sqrt((t_0 + (x * x)));
	double tmp;
	if (t_1 <= 5e-143) {
		tmp = 0.5 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = (4.0 * p) * p
	t_1 = x / math.sqrt((t_0 + (x * x)))
	tmp = 0
	if t_1 <= 5e-143:
		tmp = 0.5 * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(p, x)
	t_0 = Float64(Float64(4.0 * p) * p)
	t_1 = Float64(x / sqrt(Float64(t_0 + Float64(x * x))))
	tmp = 0.0
	if (t_1 <= 5e-143)
		tmp = Float64(0.5 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = (4.0 * p) * p;
	t_1 = x / sqrt((t_0 + (x * x)));
	tmp = 0.0;
	if (t_1 <= 5e-143)
		tmp = 0.5 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-143], N[(0.5 * t$95$0), $MachinePrecision], t$95$1]]]

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)))) < 5.0000000000000002e-143

   1. Initial program 69.4%

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

   3. Taylor expanded in p around 0

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(2 + \color{blue}{{p}^{2} \cdot \left(6 \cdot \frac{{p}^{2}}{{x}^{4}} - 2 \cdot \frac{1}{{x}^{2}}\right)}\right) \]

   8. Applied rewrites 4.0%

      \[\leadsto 0.5 \cdot \left(4 \cdot \color{blue}{p}\right) \]

   9. Taylor expanded in p around 0

      \[\leadsto \frac{1}{2} \cdot \left(4 \cdot p\right) \]

   11. Applied rewrites 9.9%

       \[\leadsto 0.5 \cdot \left(\left(4 \cdot p\right) \cdot p\right) \]

   if 5.0000000000000002e-143 < (/.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. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2} + {p}^{2} \cdot \left(-1 \cdot \frac{\sqrt{\frac{1}{2}}}{{x}^{2} \cdot \sqrt{2}} + {p}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{p}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(-1 \cdot \frac{6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{2}} + 20 \cdot \frac{1}{{x}^{6}}\right)\right)}{\sqrt{2}} + \frac{1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}}\right)\right)} \]

   5. Applied rewrites 76.2%

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

Alternative 4: 39.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double p, double x) {
	return 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 = 0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))
end function

Java

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

Python

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

Julia

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

Wolfram

code[p_, x_] := 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]

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 p around 0

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

5. Applied rewrites 38.5%

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

Alternative 5: 9.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot p\right) \cdot p\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 + x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (* (* 4.0 p) p)))
   (if (<= x -2.6e+25) t_0 (sqrt (sqrt (+ t_0 (* x x)))))))

C

double code(double p, double x) {
	double t_0 = (4.0 * p) * p;
	double tmp;
	if (x <= -2.6e+25) {
		tmp = t_0;
	} else {
		tmp = sqrt(sqrt((t_0 + (x * x))));
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * p) * p
    if (x <= (-2.6d+25)) then
        tmp = t_0
    else
        tmp = sqrt(sqrt((t_0 + (x * x))))
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = (4.0 * p) * p;
	double tmp;
	if (x <= -2.6e+25) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(Math.sqrt((t_0 + (x * x))));
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = (4.0 * p) * p
	tmp = 0
	if x <= -2.6e+25:
		tmp = t_0
	else:
		tmp = math.sqrt(math.sqrt((t_0 + (x * x))))
	return tmp

Julia

function code(p, x)
	t_0 = Float64(Float64(4.0 * p) * p)
	tmp = 0.0
	if (x <= -2.6e+25)
		tmp = t_0;
	else
		tmp = sqrt(sqrt(Float64(t_0 + Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = (4.0 * p) * p;
	tmp = 0.0;
	if (x <= -2.6e+25)
		tmp = t_0;
	else
		tmp = sqrt(sqrt((t_0 + (x * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision]}, If[LessEqual[x, -2.6e+25], t$95$0, N[Sqrt[N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5999999999999998e25

   1. Initial program 47.6%

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

   3. Taylor expanded in p around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2} + {p}^{2} \cdot \left(-1 \cdot \frac{\sqrt{\frac{1}{2}}}{{x}^{2} \cdot \sqrt{2}} + \frac{1}{2} \cdot \frac{{p}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2}}\right)} \]

   5. Applied rewrites 26.8%

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

   6. Taylor expanded in p around 0

      \[\leadsto 2 + \color{blue}{{p}^{2} \cdot \left(6 \cdot \frac{{p}^{2}}{{x}^{4}} - 2 \cdot \frac{1}{{x}^{2}}\right)} \]

   8. Applied rewrites 4.7%

      \[\leadsto 4 \cdot \color{blue}{p} \]

   9. Taylor expanded in p around 0

      \[\leadsto 4 \cdot p \]

   11. Applied rewrites 23.5%

       \[\leadsto \left(4 \cdot p\right) \cdot p \]

   if -2.5999999999999998e25 < x

   1. Initial program 86.6%

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

   3. Taylor expanded in p around 0

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

   5. Applied rewrites 6.9%

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

Alternative 6: 7.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(4 \cdot p\right) \cdot p\right) \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (* 0.5 (* (* 4.0 p) p)))

C

double code(double p, double x) {
	return 0.5 * ((4.0 * p) * p);
}

Fortran

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

Java

public static double code(double p, double x) {
	return 0.5 * ((4.0 * p) * p);
}

Python

def code(p, x):
	return 0.5 * ((4.0 * p) * p)

Julia

function code(p, x)
	return Float64(0.5 * Float64(Float64(4.0 * p) * p))
end

MATLAB

function tmp = code(p, x)
	tmp = 0.5 * ((4.0 * p) * p);
end

Wolfram

code[p_, x_] := N[(0.5 * N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision]), $MachinePrecision]

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 p around 0

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

5. Applied rewrites 38.5%

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

6. Taylor expanded in p around 0

   \[\leadsto \frac{1}{2} \cdot \left(2 + \color{blue}{{p}^{2} \cdot \left(6 \cdot \frac{{p}^{2}}{{x}^{4}} - 2 \cdot \frac{1}{{x}^{2}}\right)}\right) \]

8. Applied rewrites 3.9%

   \[\leadsto 0.5 \cdot \left(4 \cdot \color{blue}{p}\right) \]

9. Taylor expanded in p around 0

   \[\leadsto \frac{1}{2} \cdot \left(4 \cdot p\right) \]

11. Applied rewrites 8.2%

    \[\leadsto 0.5 \cdot \left(\left(4 \cdot p\right) \cdot p\right) \]
12. Add Preprocessing

Alternative 7: 7.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \left(4 \cdot p\right) \cdot p \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (* (* 4.0 p) p))

C

double code(double p, double x) {
	return (4.0 * p) * p;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = (4.0d0 * p) * p
end function

Java

public static double code(double p, double x) {
	return (4.0 * p) * p;
}

Python

def code(p, x):
	return (4.0 * p) * p

Julia

function code(p, x)
	return Float64(Float64(4.0 * p) * p)
end

MATLAB

function tmp = code(p, x)
	tmp = (4.0 * p) * p;
end

Wolfram

code[p_, x_] := N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision]

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 p around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2} + {p}^{2} \cdot \left(-1 \cdot \frac{\sqrt{\frac{1}{2}}}{{x}^{2} \cdot \sqrt{2}} + \frac{1}{2} \cdot \frac{{p}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2}}\right)} \]

5. Applied rewrites 19.3%

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

6. Taylor expanded in p around 0

   \[\leadsto 2 + \color{blue}{{p}^{2} \cdot \left(6 \cdot \frac{{p}^{2}}{{x}^{4}} - 2 \cdot \frac{1}{{x}^{2}}\right)} \]

8. Applied rewrites 3.9%

   \[\leadsto 4 \cdot \color{blue}{p} \]

9. Taylor expanded in p around 0

   \[\leadsto 4 \cdot p \]

11. Applied rewrites 8.2%

    \[\leadsto \left(4 \cdot p\right) \cdot p \]
12. Add Preprocessing

Alternative 8: 4.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot p \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (* 4.0 p))

C

double code(double p, double x) {
	return 4.0 * p;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = 4.0d0 * p
end function

Java

public static double code(double p, double x) {
	return 4.0 * p;
}

Python

def code(p, x):
	return 4.0 * p

Julia

function code(p, x)
	return Float64(4.0 * p)
end

MATLAB

function tmp = code(p, x)
	tmp = 4.0 * p;
end

Wolfram

code[p_, x_] := N[(4.0 * p), $MachinePrecision]

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 p around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2} + {p}^{2} \cdot \left(-1 \cdot \frac{\sqrt{\frac{1}{2}}}{{x}^{2} \cdot \sqrt{2}} + \frac{1}{2} \cdot \frac{{p}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(6 \cdot \frac{1}{{x}^{4}} - \frac{1}{{x}^{4} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2}}\right)} \]

5. Applied rewrites 19.3%

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

6. Taylor expanded in p around 0

   \[\leadsto 2 + \color{blue}{{p}^{2} \cdot \left(6 \cdot \frac{{p}^{2}}{{x}^{4}} - 2 \cdot \frac{1}{{x}^{2}}\right)} \]

8. Applied rewrites 3.9%

   \[\leadsto 4 \cdot \color{blue}{p} \]
9. Add Preprocessing

Developer Target 1: 78.9% accurate, 0.2× 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 2024321 
(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))))))))