Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 100.0%

Time: 9.4s

Alternatives: 16

Speedup: 1.0×

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: 76.3% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(t\_1, 1.5 - t\_0, 1\right) \cdot \frac{\mathsf{fma}\left(-0.25, {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2}, 0.25\right)}{1 - {t\_1}^{3}}} + 1}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{t\_1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))) (t_1 (- 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/
      1.0
      (/
       (+
        (sqrt
         (*
          (fma t_1 (- 1.5 t_0) 1.0)
          (/ (fma -0.25 (pow (hypot 1.0 x) -2.0) 0.25) (- 1.0 (pow t_1 3.0)))))
        1.0)
       (/ (- 0.25 (/ 0.25 (fma x x 1.0))) t_1))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 / ((sqrt((fma(t_1, (1.5 - t_0), 1.0) * (fma(-0.25, pow(hypot(1.0, x), -2.0), 0.25) / (1.0 - pow(t_1, 3.0))))) + 1.0) / ((0.25 - (0.25 / fma(x, x, 1.0))) / t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 / Float64(Float64(sqrt(Float64(fma(t_1, Float64(1.5 - t_0), 1.0) * Float64(fma(-0.25, (hypot(1.0, x) ^ -2.0), 0.25) / Float64(1.0 - (t_1 ^ 3.0))))) + 1.0) / Float64(Float64(0.25 - Float64(0.25 / fma(x, x, 1.0))) / t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 / N[(N[(N[Sqrt[N[(N[(t$95$1 * N[(1.5 - t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-0.25 * N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -2.0], $MachinePrecision] + 0.25), $MachinePrecision] / N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(0.25 - N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 52.9%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{\mathsf{fma}\left(-0.25, {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2}, 0.25\right)}{1 - {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}} \cdot \mathsf{fma}\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}, 1.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}, 1\right)}} + 1}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\mathsf{fma}\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}, 1.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}, 1\right) \cdot \frac{\mathsf{fma}\left(-0.25, {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2}, 0.25\right)}{1 - {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}} + 1}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \mathsf{fma}\left(-1, \sqrt{t\_0}, -1\right)\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot t\_0 - t\_1}{{t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))) (t_1 (fma -1.0 (sqrt t_0) -1.0)))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/ (- (* t_1 t_0) t_1) (pow t_1 2.0)))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double t_1 = fma(-1.0, sqrt(t_0), -1.0);
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = ((t_1 * t_0) - t_1) / pow(t_1, 2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	t_1 = fma(-1.0, sqrt(t_0), -1.0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(Float64(t_1 * t_0) - t_1) / (t_1 ^ 2.0));
	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]}, Block[{t$95$1 = N[(-1.0 * N[Sqrt[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 52.9%

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right) \cdot \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - \mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}{{\left(\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{\left(t\_0 + -0.5\right) \cdot \left(\sqrt{0.5 - t\_0} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.05)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/
      (+ (/ 0.25 (fma x x 1.0)) -0.25)
      (* (+ t_0 -0.5) (+ (sqrt (- 0.5 t_0)) 1.0))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.05) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = ((0.25 / fma(x, x, 1.0)) + -0.25) / ((t_0 + -0.5) * (sqrt((0.5 - t_0)) + 1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.05)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(Float64(0.25 / fma(x, x, 1.0)) + -0.25) / Float64(Float64(t_0 + -0.5) * Float64(sqrt(Float64(0.5 - t_0)) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.05], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N[(t$95$0 + -0.5), $MachinePrecision] * N[(N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.05000000000000004

   1. Initial program 53.2%

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.05000000000000004 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\frac{-0.25 + \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{\left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1\right) \cdot \left(-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + -0.25}{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + -0.5\right) \cdot \left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + -0.5\right) \cdot \frac{-1}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.05)
   (*
    (*
     (fma
      (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
      (* x x)
      0.125)
     x)
    x)
   (*
    (+ (/ 0.5 (hypot 1.0 x)) -0.5)
    (/ -1.0 (+ (sqrt (- 0.5 (/ -0.5 (hypot 1.0 x)))) 1.0)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.05) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = ((0.5 / hypot(1.0, x)) + -0.5) * (-1.0 / (sqrt((0.5 - (-0.5 / hypot(1.0, x)))) + 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.05)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(Float64(0.5 / hypot(1.0, x)) + -0.5) * Float64(-1.0 / Float64(sqrt(Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.05], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.05000000000000004

   1. Initial program 53.2%

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.05000000000000004 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   10. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + -0.5\right) \cdot \frac{-1}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 0.5}{\sqrt{0.5 - t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.05)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/ (+ t_0 0.5) (+ (sqrt (- 0.5 t_0)) 1.0)))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.05) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = (t_0 + 0.5) / (sqrt((0.5 - t_0)) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.05)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(t_0 + 0.5) / Float64(sqrt(Float64(0.5 - t_0)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.05], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(t$95$0 + 0.5), $MachinePrecision] / N[(N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.05000000000000004

   1. Initial program 53.2%

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.05000000000000004 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 99.9%

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

Alternative 6: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{x} + 0.5\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x) 0.5)))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (*
       (fma
        (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
        (* x x)
        0.125)
       x)
      x)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

double code(double x) {
	double t_0 = (0.5 / x) + 0.5;
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 / x) + 0.5)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	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[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $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 53.9%

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 99.0%

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

Alternative 7: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{\frac{0.25}{x \cdot x} - 0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (*
     (fma
      (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
      (* x x)
      0.125)
     x)
    x)
   (- 1.0 (sqrt (- 0.5 (/ (- (/ 0.25 (* x x)) 0.5) x))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt((0.5 - (((0.25 / (x * x)) - 0.5) / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 - Float64(Float64(Float64(0.25 / Float64(x * x)) - 0.5) / x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 - N[(N[(N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.9%

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   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

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

      1. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}{x}}} \]

      2. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\color{blue}{\frac{1}{4} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}}{x}} \]

      3. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\color{blue}{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}} - \frac{1}{2}}{x}} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{\color{blue}{\frac{1}{4}}}{{x}^{2}} - \frac{1}{2}}{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{{x}^{2}}} - \frac{1}{2}}{x}} \]

      6. unpow2 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{\frac{1}{4}}{\color{blue}{x \cdot x}} - \frac{1}{2}}{x}} \]

      7. lower-*.f64 97.4

         \[\leadsto 1 - \sqrt{0.5 - \frac{\frac{0.25}{\color{blue}{x \cdot x}} - 0.5}{x}} \]

   7. Applied rewrites 97.4%

      \[\leadsto 1 - \sqrt{0.5 - \color{blue}{\frac{\frac{0.25}{x \cdot x} - 0.5}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (*
     (fma
      (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
      (* x x)
      0.125)
     x)
    x)
   (- 1.0 (sqrt (+ (/ 0.5 x) 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt(((0.5 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x) + 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x), $MachinePrecision] + 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 53.9%

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 9: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 (hypot 1.0 x)) 5e-5)
   (/ 0.5 (+ (sqrt 0.5) 1.0))
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)))

C

double code(double x) {
	double tmp;
	if ((1.0 / hypot(1.0, x)) <= 5e-5) {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	} else {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / hypot(1.0, x)) <= 5e-5)
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	else
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 5e-5], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 5.00000000000000024e-5

   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 98.3

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

   7. Applied rewrites 98.3%

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

   if 5.00000000000000024e-5 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 53.9%

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

   4. Applied rewrites 53.9%

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

   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 98.1

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

   7. Applied rewrites 98.1%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
   (- 1.0 (sqrt (+ (/ 0.5 x) 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt(((0.5 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x) + 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(0.0673828125 * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x), $MachinePrecision] + 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 53.9%

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 11: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 (hypot 1.0 x)) 5e-5)
   (- 1.0 (sqrt 0.5))
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)))

C

double code(double x) {
	double tmp;
	if ((1.0 / hypot(1.0, x)) <= 5e-5) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / hypot(1.0, x)) <= 5e-5)
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 5e-5], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 5.00000000000000024e-5

   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%

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

   if 5.00000000000000024e-5 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 53.9%

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

   4. Applied rewrites 53.9%

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

   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 98.1

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

   7. Applied rewrites 98.1%

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

Alternative 12: 98.2% accurate, 1.0× speedup

Math

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

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 x) 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 / x) + 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(Float64(Float64(0.5 / x) + 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[N[(N[(0.5 / x), $MachinePrecision] + 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 53.9%

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

   4. Applied rewrites 53.9%

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

   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 98.1

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

   7. Applied rewrites 98.1%

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

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 13: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\mathsf{hypot}\left(1, x\right)} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(0.125 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 1.0 (hypot 1.0 x)) 5e-5) (- 1.0 (sqrt 0.5)) (* (* 0.125 x) x)))

C

double code(double x) {
	double tmp;
	if ((1.0 / hypot(1.0, x)) <= 5e-5) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = (0.125 * x) * x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((1.0 / Math.hypot(1.0, x)) <= 5e-5) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = (0.125 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 / math.hypot(1.0, x)) <= 5e-5:
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = (0.125 * x) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 / hypot(1.0, x)) <= 5e-5)
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(Float64(0.125 * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 / hypot(1.0, x)) <= 5e-5)
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = (0.125 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 5e-5], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 5.00000000000000024e-5

   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%

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

   if 5.00000000000000024e-5 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 53.9%

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 97.6%

       \[\leadsto \left(0.125 \cdot x\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.3% accurate, 12.2× speedup

Math

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

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(Float64(0.125 * x) * x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

4. Applied rewrites 78.0%

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

5. Taylor expanded in x around 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. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 48.7

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

7. Applied rewrites 48.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 48.7%

    \[\leadsto \left(0.125 \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 15: 51.3% 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 77.2%

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

4. Applied rewrites 78.0%

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

5. Taylor expanded in x around 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 48.7

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

7. Applied rewrites 48.7%

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

Alternative 16: 27.6% 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 77.2%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 77.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 25.9%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))