Given's Rotation SVD example, simplified

Percentage Accurate: 75.8% → 99.4%

Time: 7.3s

Alternatives: 6

Speedup: 6.7×

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: 75.8% 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: 99.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ \mathbf{if}\;x\_m \leq 1.12:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0859375, \left(-x\_m\right) \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)))
   (if (<= x_m 1.12)
     (* (* (fma 0.0859375 (* (- x_m) x_m) 0.125) x_m) x_m)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.12) {
		tmp = (fma(0.0859375, (-x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	tmp = 0.0
	if (x_m <= 1.12)
		tmp = Float64(Float64(fma(0.0859375, Float64(Float64(-x_m) * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.12], N[(N[(N[(0.0859375 * N[((-x$95$m) * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001

   1. Initial program 69.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}}} \]

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{1 - 0.5}{\sqrt{0.5} + 1}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

   10. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\frac{\sqrt{0.5}}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}, -0.0625, \frac{0.1875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right), \frac{0.25}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0859375, \left(-x\right) \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.1200000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

         \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5}{x}} + 0.5} \]

   5. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5}{x} + 0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 100.0%

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

Alternative 2: 98.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.12:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0859375, \left(-x\_m\right) \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.12)
   (* (* (fma 0.0859375 (* (- x_m) x_m) 0.125) x_m) x_m)
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.12) {
		tmp = (fma(0.0859375, (-x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.12)
		tmp = Float64(Float64(fma(0.0859375, Float64(Float64(-x_m) * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.12], N[(N[(N[(0.0859375 * N[((-x$95$m) * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001

   1. Initial program 69.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}}} \]

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{1 - 0.5}{\sqrt{0.5} + 1}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

   10. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\frac{\sqrt{0.5}}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}, -0.0625, \frac{0.1875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right), \frac{0.25}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0859375, \left(-x\right) \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.1200000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}}} \]

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{1 - 0.5}{\sqrt{0.5} + 1}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2}}}{\sqrt{\frac{1}{2}} + 1} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.8%

       \[\leadsto \frac{\color{blue}{0.5}}{\sqrt{0.5} + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.12:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0859375, \left(-x\_m\right) \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.12)
   (* (* (fma 0.0859375 (* (- x_m) x_m) 0.125) x_m) x_m)
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.12) {
		tmp = (fma(0.0859375, (-x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.12)
		tmp = Float64(Float64(fma(0.0859375, Float64(Float64(-x_m) * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.12], N[(N[(N[(0.0859375 * N[((-x$95$m) * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001

   1. Initial program 69.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 42.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}}} \]

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{1 - 0.5}{\sqrt{0.5} + 1}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

   10. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\frac{\sqrt{0.5}}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}, -0.0625, \frac{0.1875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right), \frac{0.25}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 59.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0859375, \left(-x\right) \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.1200000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.1015625, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (* (* (fma -0.1015625 (* x_m x_m) 0.125) x_m) x_m)
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = (fma(-0.1015625, (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(fma(-0.1015625, Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(N[(N[(-0.1015625 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 69.0%

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

   4. Applied rewrites 28.8%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\mathsf{fma}\left(\cosh \tanh^{-1} x, 0.5, 0.5\right)} + 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-13}{128} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-13}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-13}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-13}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-13}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-13}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-13}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-13}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-13}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 59.4

         \[\leadsto \left(\mathsf{fma}\left(-0.1015625, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 59.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.1015625, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 69.0%

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

   4. Applied rewrites 28.8%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\mathsf{fma}\left(\cosh \tanh^{-1} x, 0.5, 0.5\right)} + 1}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. lower-*.f64 60.8

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. Applied rewrites 60.8%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]

   if 1.55000000000000004 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.2%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 51.9% accurate, 12.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.125 \cdot \left(x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* 0.125 (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return 0.125 * (x_m * x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.125d0 * (x_m * x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.125 * (x_m * x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.125 * (x_m * x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(0.125 * Float64(x_m * x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.125 * (x_m * x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

4. Applied rewrites 22.1%

   \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\sqrt{\mathsf{fma}\left(\cosh \tanh^{-1} x, 0.5, 0.5\right)} + 1}} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. lower-*.f64 47.5

      \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

7. Applied rewrites 47.5%

   \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024342 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))