Given's Rotation SVD example, simplified

Percentage Accurate: 76.0% → 99.8%

Time: 9.2s

Alternatives: 10

Speedup: 1.1×

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: 76.0% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0859375, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x x 1.0)))))
   (if (<= (hypot 1.0 x) 1.02)
     (* x (* x (fma x (* x -0.0859375) 0.125)))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(x, x, 1.0));
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = x * (x * fma(x, (x * -0.0859375), 0.125));
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(x, x, 1.0)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(x * Float64(x * fma(x, Float64(x * -0.0859375), 0.125)));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * N[(x * N[(x * -0.0859375), $MachinePrecision] + 0.125), $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

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 49.2%

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

   4. Applied rewrites 49.2%

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-11}{128}} + \frac{1}{8}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-11}{128} + \frac{1}{8}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-11}{128}\right)} + \frac{1}{8}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-11}{128}, \frac{1}{8}\right)}\right) \]

      10. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) 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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

Alternative 2: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \leq 0.0005:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), x \cdot \left(x \cdot x\right), x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 (hypot 1.0 x)) 0.0005)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (*
    x
    (fma (fma x (* x 0.0673828125) -0.0859375) (* x (* x x)) (* x 0.125)))))

C

double code(double x) {
	double tmp;
	if ((1.0 / hypot(1.0, x)) <= 0.0005) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else {
		tmp = x * fma(fma(x, (x * 0.0673828125), -0.0859375), (x * (x * x)), (x * 0.125));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / hypot(1.0, x)) <= 0.0005)
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(x * fma(fma(x, Float64(x * 0.0673828125), -0.0859375), Float64(x * Float64(x * x)), Float64(x * 0.125)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * N[(x * 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 5.0000000000000001e-4

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, \frac{1}{8}\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, \frac{1}{8}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, \frac{1}{8}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{69}{1024} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right)}, \frac{1}{8}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{69}{1024}} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right), \frac{1}{8}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{69}{1024} + \color{blue}{\frac{-11}{128}}, \frac{1}{8}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{69}{1024}, \frac{-11}{128}\right)}, \frac{1}{8}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{69}{1024}, \frac{-11}{128}\right), \frac{1}{8}\right)\right) \]

      14. lower-*.f64 99.5

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0673828125, -0.0859375\right), 0.125\right)\right) \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0673828125, -0.0859375\right), 0.125\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.0673828125, -0.0859375\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x \cdot 0.125\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0859375, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.02)
   (* x (* x (fma x (* x -0.0859375) 0.125)))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = x * (x * fma(x, (x * -0.0859375), 0.125));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / sqrt(fma(x, x, 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(x * Float64(x * fma(x, Float64(x * -0.0859375), 0.125)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * N[(x * N[(x * -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 49.2%

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

   4. Applied rewrites 49.2%

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-11}{128}} + \frac{1}{8}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-11}{128} + \frac{1}{8}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-11}{128}\right)} + \frac{1}{8}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-11}{128}, \frac{1}{8}\right)}\right) \]

      10. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.5

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

      11. lift-hypot.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-hypot.f64 N/A

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

      14. lift-hypot.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lift-hypot.f64 N/A

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

      17. lift-hypot.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. metadata-eval N/A

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

      20. +-commutative N/A

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

      21. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0859375, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \leq 0.0005:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0673828125, -0.0859375\right), 0.125\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 (hypot 1.0 x)) 0.0005)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (* x (* x (fma (* x x) (fma (* x x) 0.0673828125 -0.0859375) 0.125)))))

C

double code(double x) {
	double tmp;
	if ((1.0 / hypot(1.0, x)) <= 0.0005) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else {
		tmp = x * (x * fma((x * x), fma((x * x), 0.0673828125, -0.0859375), 0.125));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / hypot(1.0, x)) <= 0.0005)
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(x * Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.0673828125, -0.0859375), 0.125)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 0.0005], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0673828125 + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 5.0000000000000001e-4

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 98.3

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

   7. Applied rewrites 98.3%

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

   if 5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right)}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, \frac{1}{8}\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, \frac{1}{8}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, \frac{1}{8}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{69}{1024} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right)}, \frac{1}{8}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{69}{1024}} + \left(\mathsf{neg}\left(\frac{11}{128}\right)\right), \frac{1}{8}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{69}{1024} + \color{blue}{\frac{-11}{128}}, \frac{1}{8}\right)\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{69}{1024}, \frac{-11}{128}\right)}, \frac{1}{8}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{69}{1024}, \frac{-11}{128}\right), \frac{1}{8}\right)\right) \]

      14. lower-*.f64 99.5

          \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0673828125, -0.0859375\right), 0.125\right)\right) \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0673828125, -0.0859375\right), 0.125\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-11}{128}} + \frac{1}{8}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-11}{128} + \frac{1}{8}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-11}{128}\right)} + \frac{1}{8}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-11}{128}, \frac{1}{8}\right)}\right) \]

      10. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 98.3

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

   7. Applied rewrites 98.3%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* x (* x (fma x (* x -0.0859375) 0.125)))
   (- 1.0 (sqrt 0.5))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   6. Applied rewrites 49.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-11}{128}} + \frac{1}{8}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-11}{128} + \frac{1}{8}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{-11}{128}\right)} + \frac{1}{8}\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-11}{128}, \frac{1}{8}\right)}\right) \]

      10. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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 96.8%

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

Alternative 7: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* x x) (fma (* x x) -0.0859375 0.125))
   (- 1.0 (sqrt 0.5))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-11}{128}} + \frac{1}{8}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-11}{128}, \frac{1}{8}\right)} \]

      7. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-11}{128}, \frac{1}{8}\right) \]

      8. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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 96.8%

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

Alternative 8: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = 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 = 0.125 * (x * x);
	else
		tmp = 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[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   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 99.2

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

   7. Applied rewrites 99.2%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) 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 96.8%

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

Alternative 9: 52.0% accurate, 12.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.125 (* x x)))

C

double code(double x) {
	return 0.125 * (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.125 * (x * x);
}

Python

def code(x):
	return 0.125 * (x * x)

Julia

function code(x)
	return Float64(0.125 * Float64(x * x))
end

MATLAB

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

Wolfram

code[x_] := N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

4. Applied rewrites 73.6%

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

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 54.0

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

7. Applied rewrites 54.0%

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

Alternative 10: 27.9% accurate, 33.5× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 1.0))

C

double code(double x) {
	return 1.0 - 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - 1.0d0
end function

Java

public static double code(double x) {
	return 1.0 - 1.0;
}

Python

def code(x):
	return 1.0 - 1.0

Julia

function code(x)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 72.9%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 72.9

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

   11. lift-hypot.f64 N/A

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

   12. rem-square-sqrt N/A

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

   13. lift-hypot.f64 N/A

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

   14. lift-hypot.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lift-hypot.f64 N/A

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

   17. lift-hypot.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. metadata-eval N/A

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

   20. +-commutative N/A

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

   21. lower-fma.f64 72.9

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

4. Applied rewrites 72.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 26.9%

   \[\leadsto 1 - \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))