Given's Rotation SVD example

Percentage Accurate: 98.4% → 98.4%

Time: 12.3s

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: 98.4% 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: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (log (exp (+ 1.0 (/ x (hypot x (* 2.0 p)))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * log(exp((1.0 + (x / hypot(x, (2.0 * p))))))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * Math.log(Math.exp((1.0 + (x / Math.hypot(x, (2.0 * p))))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * math.log(math.exp((1.0 + (x / math.hypot(x, (2.0 * p))))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * log(exp(Float64(1.0 + Float64(x / hypot(x, Float64(2.0 * p))))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * log(exp((1.0 + (x / hypot(x, (2.0 * p))))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(2.0 * p), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. add-log-exp 97.9%

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

   2. +-commutative 97.9%

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

   3. add-sqr-sqrt 97.9%

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

   4. hypot-define 97.9%

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

   5. associate-*l* 97.9%

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

   6. sqrt-prod 97.9%

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

   7. metadata-eval 97.9%

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

   8. sqrt-unprod 45.3%

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

   9. add-sqr-sqrt 97.9%

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

   \[\leadsto \sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} \]
6. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / hypot((2.0 * p), x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.hypot((2.0 * p), x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.hypot((2.0 * p), x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(2.0 * p), x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / hypot((2.0 * p), x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(2.0 * p), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. add-sqr-sqrt 97.9%

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

   2. hypot-define 97.9%

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

   3. associate-*l* 97.9%

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

   4. sqrt-prod 97.9%

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

   5. metadata-eval 97.9%

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

   6. sqrt-unprod 45.2%

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

   7. add-sqr-sqrt 97.9%

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \]
6. Add Preprocessing

Alternative 3: 51.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq 2.65 \cdot 10^{-138}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{elif}\;p \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{2 \cdot p}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p 2.65e-138)
   1.0
   (if (<= p 7.5e-93)
     (sqrt 0.0)
     (if (<= p 1.35e-5) 1.0 (sqrt (* 0.5 (+ 1.0 (/ x (* 2.0 p)))))))))

C

double code(double p, double x) {
	double tmp;
	if (p <= 2.65e-138) {
		tmp = 1.0;
	} else if (p <= 7.5e-93) {
		tmp = sqrt(0.0);
	} else if (p <= 1.35e-5) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (2.0 * p)))));
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 2.65d-138) then
        tmp = 1.0d0
    else if (p <= 7.5d-93) then
        tmp = sqrt(0.0d0)
    else if (p <= 1.35d-5) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (2.0d0 * p)))))
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= 2.65e-138) {
		tmp = 1.0;
	} else if (p <= 7.5e-93) {
		tmp = Math.sqrt(0.0);
	} else if (p <= 1.35e-5) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (2.0 * p)))));
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= 2.65e-138:
		tmp = 1.0
	elif p <= 7.5e-93:
		tmp = math.sqrt(0.0)
	elif p <= 1.35e-5:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (2.0 * p)))))
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= 2.65e-138)
		tmp = 1.0;
	elseif (p <= 7.5e-93)
		tmp = sqrt(0.0);
	elseif (p <= 1.35e-5)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(2.0 * p)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 2.65e-138)
		tmp = 1.0;
	elseif (p <= 7.5e-93)
		tmp = sqrt(0.0);
	elseif (p <= 1.35e-5)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (2.0 * p)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, 2.65e-138], 1.0, If[LessEqual[p, 7.5e-93], N[Sqrt[0.0], $MachinePrecision], If[LessEqual[p, 1.35e-5], 1.0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(2.0 * p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.65000000000000013e-138 or 7.50000000000000034e-93 < p < 1.3499999999999999e-5

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf 42.3%

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

   if 2.65000000000000013e-138 < p < 7.50000000000000034e-93

   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 x around -inf 63.7%

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

   if 1.3499999999999999e-5 < p

   1. Initial program 96.2%

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

   3. Taylor expanded in p around inf 85.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.65 \cdot 10^{-138}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7.5 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{elif}\;p \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{2 \cdot p}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{elif}\;p \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p 1.15e-137)
   1.0
   (if (<= p 1.15e-92) (sqrt 0.0) (if (<= p 2.1e-6) 1.0 (sqrt 0.5)))))

C

double code(double p, double x) {
	double tmp;
	if (p <= 1.15e-137) {
		tmp = 1.0;
	} else if (p <= 1.15e-92) {
		tmp = sqrt(0.0);
	} else if (p <= 2.1e-6) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 1.15d-137) then
        tmp = 1.0d0
    else if (p <= 1.15d-92) then
        tmp = sqrt(0.0d0)
    else if (p <= 2.1d-6) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= 1.15e-137) {
		tmp = 1.0;
	} else if (p <= 1.15e-92) {
		tmp = Math.sqrt(0.0);
	} else if (p <= 2.1e-6) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= 1.15e-137:
		tmp = 1.0
	elif p <= 1.15e-92:
		tmp = math.sqrt(0.0)
	elif p <= 2.1e-6:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= 1.15e-137)
		tmp = 1.0;
	elseif (p <= 1.15e-92)
		tmp = sqrt(0.0);
	elseif (p <= 2.1e-6)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 1.15e-137)
		tmp = 1.0;
	elseif (p <= 1.15e-92)
		tmp = sqrt(0.0);
	elseif (p <= 2.1e-6)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, 1.15e-137], 1.0, If[LessEqual[p, 1.15e-92], N[Sqrt[0.0], $MachinePrecision], If[LessEqual[p, 2.1e-6], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.15000000000000004e-137 or 1.15000000000000008e-92 < p < 2.0999999999999998e-6

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf 42.3%

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

   if 1.15000000000000004e-137 < p < 1.15000000000000008e-92

   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 x around -inf 63.7%

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

   if 2.0999999999999998e-6 < p

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 0 84.6%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{elif}\;p \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 30.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq 7.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= p 7.5e-161) (/ p (- x)) (sqrt 0.5)))

C

double code(double p, double x) {
	double tmp;
	if (p <= 7.5e-161) {
		tmp = p / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 7.5d-161) then
        tmp = p / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= 7.5e-161) {
		tmp = p / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= 7.5e-161:
		tmp = p / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= 7.5e-161)
		tmp = Float64(p / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 7.5e-161)
		tmp = p / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, 7.5e-161], N[(p / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 7.49999999999999991e-161

   1. Initial program 98.0%

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

   3. Taylor expanded in x around -inf 9.8%

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

      1. mul-1-neg 9.8%

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

      2. associate-/l* 9.8%

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

      3. distribute-lft-neg-in 9.8%

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

      4. associate-/l* 9.8%

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

   5. Simplified 9.8%

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

      1. distribute-lft-neg-out 9.8%

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

      2. neg-sub0 9.8%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 9.8%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 9.8%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 9.8%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 9.8%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 9.8%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 9.8%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 9.8%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 9.8%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 9.8%

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

      2. distribute-frac-neg 9.8%

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

   9. Simplified 9.8%

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

   if 7.49999999999999991e-161 < p

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0 60.2%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= p 2.1e-6) 1.0 (sqrt 0.5)))

C

double code(double p, double x) {
	double tmp;
	if (p <= 2.1e-6) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 2.1d-6) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= 2.1e-6) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= 2.1e-6:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= 2.1e-6)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 2.1e-6)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, 2.1e-6], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 2.0999999999999998e-6

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf 40.8%

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

   if 2.0999999999999998e-6 < p

   1. Initial program 96.2%

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

   3. Taylor expanded in x around 0 84.6%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 7.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-147}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= x -7e-147) (/ p (- x)) (/ p x)))

C

double code(double p, double x) {
	double tmp;
	if (x <= -7e-147) {
		tmp = p / -x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-7d-147)) then
        tmp = p / -x
    else
        tmp = p / x
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (x <= -7e-147) {
		tmp = p / -x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if x <= -7e-147:
		tmp = p / -x
	else:
		tmp = p / x
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (x <= -7e-147)
		tmp = Float64(p / Float64(-x));
	else
		tmp = Float64(p / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -7e-147)
		tmp = p / -x;
	else
		tmp = p / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[x, -7e-147], N[(p / (-x)), $MachinePrecision], N[(p / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000007e-147

   1. Initial program 95.6%

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

   3. Taylor expanded in x around -inf 13.6%

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

      1. mul-1-neg 13.6%

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

      2. associate-/l* 13.6%

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

      3. distribute-lft-neg-in 13.6%

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

      4. associate-/l* 13.6%

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

   5. Simplified 13.6%

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

      1. distribute-lft-neg-out 13.6%

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

      2. neg-sub0 13.6%

         \[\leadsto \color{blue}{0 - p \cdot \left(\sqrt{0.5} \cdot \frac{\sqrt{2}}{x}\right)} \]

      3. associate-*r/ 13.6%

         \[\leadsto 0 - p \cdot \color{blue}{\frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      4. sqrt-unprod 13.6%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      5. metadata-eval 13.6%

         \[\leadsto 0 - p \cdot \frac{\sqrt{\color{blue}{1}}}{x} \]

      6. metadata-eval 13.6%

         \[\leadsto 0 - p \cdot \frac{\color{blue}{1}}{x} \]

      7. associate-*r/ 13.7%

         \[\leadsto 0 - \color{blue}{\frac{p \cdot 1}{x}} \]

      8. *-commutative 13.7%

         \[\leadsto 0 - \frac{\color{blue}{1 \cdot p}}{x} \]

      9. *-un-lft-identity 13.7%

         \[\leadsto 0 - \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 13.7%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 13.7%

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

      2. distribute-frac-neg 13.7%

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

   9. Simplified 13.7%

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

   if -7.00000000000000007e-147 < 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 x around -inf 3.6%

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

      1. mul-1-neg 3.6%

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

      2. associate-/l* 3.6%

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

      3. distribute-lft-neg-in 3.6%

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

      4. associate-/l* 3.6%

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

   5. Simplified 3.6%

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

      1. *-commutative 3.6%

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

      2. associate-*r/ 3.6%

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

      3. sqrt-unprod 3.6%

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

      4. metadata-eval 3.6%

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

      5. metadata-eval 3.6%

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

      6. associate-*l/ 3.6%

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

      7. *-un-lft-identity 3.6%

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

      8. add-sqr-sqrt 2.9%

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

      9. sqrt-unprod 5.3%

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

      10. sqr-neg 5.3%

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

      11. sqrt-unprod 2.1%

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

      12. add-sqr-sqrt 3.0%

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

   7. Applied egg-rr 3.0%

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

4. Final simplification 8.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-147}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 7.1% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ \frac{p}{x} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (/ p x))

C

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

Fortran

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

Java

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

Python

def code(p, x):
	return p / x

Julia

function code(p, x)
	return Float64(p / x)
end

MATLAB

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

Wolfram

code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around -inf 8.4%

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

   1. mul-1-neg 8.4%

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

   2. associate-/l* 8.4%

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

   3. distribute-lft-neg-in 8.4%

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

   4. associate-/l* 8.4%

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

5. Simplified 8.4%

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

   1. *-commutative 8.4%

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

   2. associate-*r/ 8.4%

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

   3. sqrt-unprod 8.4%

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

   4. metadata-eval 8.4%

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

   5. metadata-eval 8.4%

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

   6. associate-*l/ 8.4%

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

   7. *-un-lft-identity 8.4%

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

   8. add-sqr-sqrt 4.3%

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

   9. sqrt-unprod 13.0%

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

   10. sqr-neg 13.0%

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

   11. sqrt-unprod 2.9%

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

   12. add-sqr-sqrt 7.6%

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

7. Applied egg-rr 7.6%

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

8. Final simplification 7.6%

   \[\leadsto \frac{p}{x} \]
9. Add Preprocessing

Developer target: 98.4% 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 2024052 
(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))))))))