Given's Rotation SVD example, simplified

Percentage Accurate: 75.8% → 99.8%

Time: 7.9s

Alternatives: 8

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: 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 0.125, -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 2.0)
     (/
      (* (* (fma (fma -0.15625 (* x x) 0.1875) (* x x) -0.25) x) x)
      (fma (* x x) 0.125 -2.0))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = ((fma(fma(-0.15625, (x * x), 0.1875), (x * x), -0.25) * x) * x) / fma((x * x), 0.125, -2.0);
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(Float64(fma(fma(-0.15625, Float64(x * x), 0.1875), Float64(x * x), -0.25) * x) * x) / fma(Float64(x * x), 0.125, -2.0));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.15625 * N[(x * x), $MachinePrecision] + 0.1875), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * 0.125 + -2.0), $MachinePrecision]), $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 (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2}}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right)} \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\color{blue}{\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\frac{-5}{32} \cdot {x}^{2} + \frac{3}{16}}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5}{32}, {x}^{2}, \frac{3}{16}\right)}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), \color{blue}{x \cdot x}, -0.25\right) \cdot x\right) \cdot x}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1}{8} \cdot {x}^{2} - 2}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1}{8} \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{{x}^{2} \cdot \frac{1}{8}} + \left(\mathsf{neg}\left(2\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{8}, \mathsf{neg}\left(2\right)\right)}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{8}, \mathsf{neg}\left(2\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{8}, \mathsf{neg}\left(2\right)\right)} \]

      6. metadata-eval 100.0

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 0.125, \color{blue}{-2}\right)} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.125, -2\right)}} \]

   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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 0.125, -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 0.125, -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{x} + 0.5}{\sqrt{0.5 - \frac{-0.5}{x}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (/
    (* (* (fma (fma -0.15625 (* x x) 0.1875) (* x x) -0.25) x) x)
    (fma (* x x) 0.125 -2.0))
   (/ (+ (/ -0.5 x) 0.5) (+ (sqrt (- 0.5 (/ -0.5 x))) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = ((fma(fma(-0.15625, (x * x), 0.1875), (x * x), -0.25) * x) * x) / fma((x * x), 0.125, -2.0);
	} else {
		tmp = ((-0.5 / x) + 0.5) / (sqrt((0.5 - (-0.5 / x))) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(Float64(fma(fma(-0.15625, Float64(x * x), 0.1875), Float64(x * x), -0.25) * x) * x) / fma(Float64(x * x), 0.125, -2.0));
	else
		tmp = Float64(Float64(Float64(-0.5 / x) + 0.5) / Float64(sqrt(Float64(0.5 - Float64(-0.5 / x))) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.15625 * N[(x * x), $MachinePrecision] + 0.1875), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * 0.125 + -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision] / N[(N[Sqrt[N[(0.5 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2}}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right)} \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\color{blue}{\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\frac{-5}{32} \cdot {x}^{2} + \frac{3}{16}}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5}{32}, {x}^{2}, \frac{3}{16}\right)}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), \color{blue}{x \cdot x}, -0.25\right) \cdot x\right) \cdot x}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1}{8} \cdot {x}^{2} - 2}} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{\frac{1}{8} \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{{x}^{2} \cdot \frac{1}{8}} + \left(\mathsf{neg}\left(2\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{8}, \mathsf{neg}\left(2\right)\right)}} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{8}, \mathsf{neg}\left(2\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), x \cdot x, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{8}, \mathsf{neg}\left(2\right)\right)} \]

      6. metadata-eval 100.0

         \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(x \cdot x, 0.125, \color{blue}{-2}\right)} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\color{blue}{\mathsf{fma}\left(x \cdot x, 0.125, -2\right)}} \]

   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. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. inv-pow N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-pow N/A

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

      16. pow2 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. inv-pow N/A

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

      23. lift-/.f64 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 97.3

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

   7. Applied rewrites 97.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 - \frac{-0.5}{x}\right)}{\sqrt{0.5 - \frac{-0.5}{x}} + 1}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift--.f64 N/A

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

       3. associate--r- N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 98.9

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

   11. Applied rewrites 98.9%

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

Alternative 3: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.8%

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

   4. Applied rewrites 49.8%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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. 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. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. inv-pow N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-pow N/A

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

      16. pow2 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. inv-pow N/A

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

      23. lift-/.f64 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 97.3

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

   7. Applied rewrites 97.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{1 - \left(0.5 - \frac{-0.5}{x}\right)}{\sqrt{0.5 - \frac{-0.5}{x}} + 1}} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift--.f64 N/A

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

       3. associate--r- N/A

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 98.9

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

   11. Applied rewrites 98.9%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.8%

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

   4. Applied rewrites 49.8%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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%

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

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 98.4

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

   7. Applied rewrites 98.4%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} 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(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	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[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $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 49.8%

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

   4. Applied rewrites 49.8%

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

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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 96.9%

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

Alternative 6: 97.7% 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 49.8%

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

   4. Applied rewrites 49.8%

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

   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 99.3

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

   7. Applied rewrites 99.3%

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

   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 96.9%

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

Alternative 7: 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 72.4%

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

4. Applied rewrites 73.2%

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

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 55.1

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

7. Applied rewrites 55.1%

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

Alternative 8: 27.8% 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 72.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. inv-pow N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

   16. pow2 N/A

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

   17. sqr-neg N/A

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

   18. pow-prod-down N/A

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

   19. pow-sqr N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. inv-pow N/A

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

   23. lift-/.f64 N/A

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

4. Applied rewrites 72.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))