Given's Rotation SVD example, simplified

Percentage Accurate: 76.6% → 99.8%

Time: 10.6s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;\frac{-0.1875 \cdot {x}^{4} + 0.25 \cdot {x}^{2}}{1 + \left(1 + {x}^{2} \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\mathsf{log1p}\left(-{t_1}^{2}\right)}}{t_0 + 1.5}}{1 + \sqrt{t_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.005)
     (/
      (+ (* -0.1875 (pow x 4.0)) (* 0.25 (pow x 2.0)))
      (+ 1.0 (+ 1.0 (* (pow x 2.0) -0.125))))
     (/ (/ (exp (log1p (- (pow t_1 2.0)))) (+ t_0 1.5)) (+ 1.0 (sqrt t_1))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = ((-0.1875 * pow(x, 4.0)) + (0.25 * pow(x, 2.0))) / (1.0 + (1.0 + (pow(x, 2.0) * -0.125)));
	} else {
		tmp = (exp(log1p(-pow(t_1, 2.0))) / (t_0 + 1.5)) / (1.0 + sqrt(t_1));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (Math.hypot(1.0, x) <= 1.005) {
		tmp = ((-0.1875 * Math.pow(x, 4.0)) + (0.25 * Math.pow(x, 2.0))) / (1.0 + (1.0 + (Math.pow(x, 2.0) * -0.125)));
	} else {
		tmp = (Math.exp(Math.log1p(-Math.pow(t_1, 2.0))) / (t_0 + 1.5)) / (1.0 + Math.sqrt(t_1));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	tmp = 0
	if math.hypot(1.0, x) <= 1.005:
		tmp = ((-0.1875 * math.pow(x, 4.0)) + (0.25 * math.pow(x, 2.0))) / (1.0 + (1.0 + (math.pow(x, 2.0) * -0.125)))
	else:
		tmp = (math.exp(math.log1p(-math.pow(t_1, 2.0))) / (t_0 + 1.5)) / (1.0 + math.sqrt(t_1))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64(Float64(-0.1875 * (x ^ 4.0)) + Float64(0.25 * (x ^ 2.0))) / Float64(1.0 + Float64(1.0 + Float64((x ^ 2.0) * -0.125))));
	else
		tmp = Float64(Float64(exp(log1p(Float64(-(t_1 ^ 2.0)))) / Float64(t_0 + 1.5)) / Float64(1.0 + sqrt(t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(N[(-0.1875 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[Log[1 + (-N[Power[t$95$1, 2.0], $MachinePrecision])], $MachinePrecision]], $MachinePrecision] / N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;\frac{-0.1875 \cdot {x}^{4} + 0.25 \cdot {x}^{2}}{1 + \left(1 + {x}^{2} \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {t_1}^{2}}{t_0 + 1.5}}{1 + \sqrt{t_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.005)
     (/
      (+ (* -0.1875 (pow x 4.0)) (* 0.25 (pow x 2.0)))
      (+ 1.0 (+ 1.0 (* (pow x 2.0) -0.125))))
     (/ (/ (- 1.0 (pow t_1 2.0)) (+ t_0 1.5)) (+ 1.0 (sqrt t_1))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = ((-0.1875 * pow(x, 4.0)) + (0.25 * pow(x, 2.0))) / (1.0 + (1.0 + (pow(x, 2.0) * -0.125)));
	} else {
		tmp = ((1.0 - pow(t_1, 2.0)) / (t_0 + 1.5)) / (1.0 + sqrt(t_1));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (Math.hypot(1.0, x) <= 1.005) {
		tmp = ((-0.1875 * Math.pow(x, 4.0)) + (0.25 * Math.pow(x, 2.0))) / (1.0 + (1.0 + (Math.pow(x, 2.0) * -0.125)));
	} else {
		tmp = ((1.0 - Math.pow(t_1, 2.0)) / (t_0 + 1.5)) / (1.0 + Math.sqrt(t_1));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	tmp = 0
	if math.hypot(1.0, x) <= 1.005:
		tmp = ((-0.1875 * math.pow(x, 4.0)) + (0.25 * math.pow(x, 2.0))) / (1.0 + (1.0 + (math.pow(x, 2.0) * -0.125)))
	else:
		tmp = ((1.0 - math.pow(t_1, 2.0)) / (t_0 + 1.5)) / (1.0 + math.sqrt(t_1))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64(Float64(-0.1875 * (x ^ 4.0)) + Float64(0.25 * (x ^ 2.0))) / Float64(1.0 + Float64(1.0 + Float64((x ^ 2.0) * -0.125))));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_1 ^ 2.0)) / Float64(t_0 + 1.5)) / Float64(1.0 + sqrt(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = 0.5 + t_0;
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.005)
		tmp = ((-0.1875 * (x ^ 4.0)) + (0.25 * (x ^ 2.0))) / (1.0 + (1.0 + ((x ^ 2.0) * -0.125)));
	else
		tmp = ((1.0 - (t_1 ^ 2.0)) / (t_0 + 1.5)) / (1.0 + sqrt(t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(N[(-0.1875 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;\frac{-0.1875 \cdot {x}^{4} + 0.25 \cdot {x}^{2}}{1 + \left(1 + {x}^{2} \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.005)
     (/
      (+ (* -0.1875 (pow x 4.0)) (* 0.25 (pow x 2.0)))
      (+ 1.0 (+ 1.0 (* (pow x 2.0) -0.125))))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = ((-0.1875 * pow(x, 4.0)) + (0.25 * pow(x, 2.0))) / (1.0 + (1.0 + (pow(x, 2.0) * -0.125)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.005) {
		tmp = ((-0.1875 * Math.pow(x, 4.0)) + (0.25 * Math.pow(x, 2.0))) / (1.0 + (1.0 + (Math.pow(x, 2.0) * -0.125)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.005:
		tmp = ((-0.1875 * math.pow(x, 4.0)) + (0.25 * math.pow(x, 2.0))) / (1.0 + (1.0 + (math.pow(x, 2.0) * -0.125)))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64(Float64(-0.1875 * (x ^ 4.0)) + Float64(0.25 * (x ^ 2.0))) / Float64(1.0 + Float64(1.0 + Float64((x ^ 2.0) * -0.125))));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.005)
		tmp = ((-0.1875 * (x ^ 4.0)) + (0.25 * (x ^ 2.0))) / (1.0 + (1.0 + ((x ^ 2.0) * -0.125)));
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(N[(-0.1875 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.005:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + {x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.005)
     (+ (* (pow x 4.0) -0.0859375) (* (pow x 2.0) 0.125))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.005) {
		tmp = (pow(x, 4.0) * -0.0859375) + (pow(x, 2.0) * 0.125);
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.005) {
		tmp = (Math.pow(x, 4.0) * -0.0859375) + (Math.pow(x, 2.0) * 0.125);
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.005:
		tmp = (math.pow(x, 4.0) * -0.0859375) + (math.pow(x, 2.0) * 0.125)
	else:
		tmp = (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.005)
		tmp = Float64(Float64((x ^ 4.0) * -0.0859375) + Float64((x ^ 2.0) * 0.125));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.005)
		tmp = ((x ^ 4.0) * -0.0859375) + ((x ^ 2.0) * 0.125);
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.005], N[(N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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1. Add Preprocessing

Alternative 5: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + {x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+ (* (pow x 4.0) -0.0859375) (* (pow x 2.0) 0.125))
   (/ (- 0.5 (/ 0.5 (hypot 1.0 x))) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (pow(x, 4.0) * -0.0859375) + (pow(x, 2.0) * 0.125);
	} else {
		tmp = (0.5 - (0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (Math.pow(x, 4.0) * -0.0859375) + (Math.pow(x, 2.0) * 0.125);
	} else {
		tmp = (0.5 - (0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (math.pow(x, 4.0) * -0.0859375) + (math.pow(x, 2.0) * 0.125)
	else:
		tmp = (0.5 - (0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64((x ^ 4.0) * -0.0859375) + Float64((x ^ 2.0) * 0.125));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / hypot(1.0, x))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = ((x ^ 4.0) * -0.0859375) + ((x ^ 2.0) * 0.125);
	else
		tmp = (0.5 - (0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{4} \cdot -0.0859375 + {x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+ (* (pow x 4.0) -0.0859375) (* (pow x 2.0) 0.125))
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (pow(x, 4.0) * -0.0859375) + (pow(x, 2.0) * 0.125);
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (Math.pow(x, 4.0) * -0.0859375) + (Math.pow(x, 2.0) * 0.125);
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (math.pow(x, 4.0) * -0.0859375) + (math.pow(x, 2.0) * 0.125)
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64((x ^ 4.0) * -0.0859375) + Float64((x ^ 2.0) * 0.125));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = ((x ^ 4.0) * -0.0859375) + ((x ^ 2.0) * 0.125);
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (pow x 2.0) 0.125)
   (/ (+ 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ -0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * 0.125;
	} else {
		tmp = (0.5 + (0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = Math.pow(x, 2.0) * 0.125;
	} else {
		tmp = (0.5 + (0.5 / x)) / (1.0 + Math.sqrt((0.5 + (-0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = math.pow(x, 2.0) * 0.125
	else:
		tmp = (0.5 + (0.5 / x)) / (1.0 + math.sqrt((0.5 + (-0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64((x ^ 2.0) * 0.125);
	else
		tmp = Float64(Float64(0.5 + Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(-0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * 0.125;
	else
		tmp = (0.5 + (0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (pow x 2.0) 0.125)
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * 0.125;
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = Math.pow(x, 2.0) * 0.125;
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = math.pow(x, 2.0) * 0.125
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64((x ^ 2.0) * 0.125);
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * 0.125;
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* (pow x 2.0) 0.125) (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * 0.125;
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = Math.pow(x, 2.0) * 0.125;
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = math.pow(x, 2.0) * 0.125
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64((x ^ 2.0) * 0.125);
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * 0.125;
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.52) (not (<= x 1.55)))
   (- 1.0 (sqrt 0.5))
   (* (pow x 2.0) 0.125)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.52) || !(x <= 1.55)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = pow(x, 2.0) * 0.125;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.52d0)) .or. (.not. (x <= 1.55d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = (x ** 2.0d0) * 0.125d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.52) || !(x <= 1.55)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = Math.pow(x, 2.0) * 0.125;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.52) or not (x <= 1.55):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = math.pow(x, 2.0) * 0.125
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.52) || !(x <= 1.55))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64((x ^ 2.0) * 0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.52) || ~((x <= 1.55)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = (x ^ 2.0) * 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.52], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 75.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-77} \lor \neg \left(x \leq 2.15 \cdot 10^{-77}\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.2e-77) (not (<= x 2.15e-77))) (- 1.0 (sqrt 0.5)) 0.0))

C

double code(double x) {
	double tmp;
	if ((x <= -2.2e-77) || !(x <= 2.15e-77)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.2d-77)) .or. (.not. (x <= 2.15d-77))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.2e-77) || !(x <= 2.15e-77)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.2e-77) or not (x <= 2.15e-77):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.2e-77) || !(x <= 2.15e-77))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.2e-77) || ~((x <= 2.15e-77)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.2e-77], N[Not[LessEqual[x, 2.15e-77]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 37.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.15e-77) 0.25 (if (<= x 2.15e-77) 0.0 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -2.15e-77) {
		tmp = 0.25;
	} else if (x <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.15d-77)) then
        tmp = 0.25d0
    else if (x <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.15e-77) {
		tmp = 0.25;
	} else if (x <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.15e-77:
		tmp = 0.25
	elif x <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.15e-77)
		tmp = 0.25;
	elseif (x <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.15e-77)
		tmp = 0.25;
	elseif (x <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.15e-77], 0.25, If[LessEqual[x, 2.15e-77], 0.0, 0.25]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 27.2% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023350 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))