Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 99.9%

Time: 5.0s

Alternatives: 11

Speedup: 1.5×

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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}\\ t_1 := \frac{0.5}{t\_0}\\ t_2 := \frac{{t\_1}^{3} + 0.125}{\frac{0.25}{t\_0 \cdot t\_0} + \left(0.25 - t\_1 \cdot 0.5\right)}\\ \mathbf{if}\;x\_m \leq 0.034:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.056243896484375, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375\right) \cdot x\_m, x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_2}{\sqrt{t\_2} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (fma x_m x_m 1.0)))
        (t_1 (/ 0.5 t_0))
        (t_2
         (/
          (+ (pow t_1 3.0) 0.125)
          (+ (/ 0.25 (* t_0 t_0)) (- 0.25 (* t_1 0.5))))))
   (if (<= x_m 0.034)
     (*
      (*
       (fma
        (*
         (-
          (* (* (fma (* x_m x_m) -0.056243896484375 0.0673828125) x_m) x_m)
          0.0859375)
         x_m)
        x_m
        0.125)
       x_m)
      x_m)
     (/ (- 1.0 t_2) (+ (sqrt t_2) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt(fma(x_m, x_m, 1.0));
	double t_1 = 0.5 / t_0;
	double t_2 = (pow(t_1, 3.0) + 0.125) / ((0.25 / (t_0 * t_0)) + (0.25 - (t_1 * 0.5)));
	double tmp;
	if (x_m <= 0.034) {
		tmp = (fma(((((fma((x_m * x_m), -0.056243896484375, 0.0673828125) * x_m) * x_m) - 0.0859375) * x_m), x_m, 0.125) * x_m) * x_m;
	} else {
		tmp = (1.0 - t_2) / (sqrt(t_2) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(fma(x_m, x_m, 1.0))
	t_1 = Float64(0.5 / t_0)
	t_2 = Float64(Float64((t_1 ^ 3.0) + 0.125) / Float64(Float64(0.25 / Float64(t_0 * t_0)) + Float64(0.25 - Float64(t_1 * 0.5))))
	tmp = 0.0
	if (x_m <= 0.034)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(fma(Float64(x_m * x_m), -0.056243896484375, 0.0673828125) * x_m) * x_m) - 0.0859375) * x_m), x_m, 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(1.0 - t_2) / Float64(sqrt(t_2) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + 0.125), $MachinePrecision] / N[(N[(0.25 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.25 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.034], N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 - t$95$2), $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.034000000000000002

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 0.034000000000000002 < x

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

   5. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1 - \color{blue}{\frac{{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}^{3} + 0.125}{\frac{0.25}{\sqrt{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}} + \left(0.25 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot 0.5\right)}}}{\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + 0.5} + 1} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. flip3-+ N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-/.f64 N/A

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

   9. Applied rewrites 99.9%

      \[\leadsto \frac{1 - \frac{{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}^{3} + 0.125}{\frac{0.25}{\sqrt{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}} + \left(0.25 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot 0.5\right)}}{\sqrt{\color{blue}{\frac{{\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}^{3} + 0.125}{\frac{0.25}{\sqrt{\mathsf{fma}\left(x, x, 1\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}} + \left(0.25 - \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} \cdot 0.5\right)}}} + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{if}\;x\_m \leq 0.0305:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.056243896484375, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375\right) \cdot x\_m, x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - t\_0\right) - 0.5}{\sqrt{t\_0 + 0.5} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x_m x_m 1.0)))))
   (if (<= x_m 0.0305)
     (*
      (*
       (fma
        (*
         (-
          (* (* (fma (* x_m x_m) -0.056243896484375 0.0673828125) x_m) x_m)
          0.0859375)
         x_m)
        x_m
        0.125)
       x_m)
      x_m)
     (/ (- (- 1.0 t_0) 0.5) (+ (sqrt (+ t_0 0.5)) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 / sqrt(fma(x_m, x_m, 1.0));
	double tmp;
	if (x_m <= 0.0305) {
		tmp = (fma(((((fma((x_m * x_m), -0.056243896484375, 0.0673828125) * x_m) * x_m) - 0.0859375) * x_m), x_m, 0.125) * x_m) * x_m;
	} else {
		tmp = ((1.0 - t_0) - 0.5) / (sqrt((t_0 + 0.5)) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 / sqrt(fma(x_m, x_m, 1.0)))
	tmp = 0.0
	if (x_m <= 0.0305)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(fma(Float64(x_m * x_m), -0.056243896484375, 0.0673828125) * x_m) * x_m) - 0.0859375) * x_m), x_m, 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(Float64(1.0 - t_0) - 0.5) / Float64(sqrt(Float64(t_0 + 0.5)) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0305], N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(1.0 - t$95$0), $MachinePrecision] - 0.5), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.030499999999999999

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 0.030499999999999999 < x

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

   5. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate--r+ N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 3: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.03:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.056243896484375, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375\right) \cdot x\_m, x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (*
    (*
     (fma
      (*
       (-
        (* (* (fma (* x_m x_m) -0.056243896484375 0.0673828125) x_m) x_m)
        0.0859375)
       x_m)
      x_m
      0.125)
     x_m)
    x_m)
   (- 1.0 (sqrt (+ (/ 0.5 (sqrt (fma x_m x_m 1.0))) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = (fma(((((fma((x_m * x_m), -0.056243896484375, 0.0673828125) * x_m) * x_m) - 0.0859375) * x_m), x_m, 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.03)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(fma(Float64(x_m * x_m), -0.056243896484375, 0.0673828125) * x_m) * x_m) - 0.0859375) * x_m), x_m, 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.03], N[(N[(N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 0.029999999999999999 < x

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.0673828125 - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.012)
   (*
    (* (fma (- (* (* x_m x_m) 0.0673828125) 0.0859375) (* x_m x_m) 0.125) x_m)
    x_m)
   (- 1.0 (sqrt (+ (/ 0.5 (sqrt (fma x_m x_m 1.0))) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.012) {
		tmp = (fma((((x_m * x_m) * 0.0673828125) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(x_m * x_m) * 0.0673828125) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.012], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

   if 0.012 < x

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 5: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.012:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.0673828125 - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.012)
   (*
    (* (fma (- (* (* x_m x_m) 0.0673828125) 0.0859375) (* x_m x_m) 0.125) x_m)
    x_m)
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.012) {
		tmp = (fma((((x_m * x_m) * 0.0673828125) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.012)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(x_m * x_m) * 0.0673828125) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.012], N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.012

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 100.0%

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

   if 0.012 < x

   1. Initial program 98.4%

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

Alternative 6: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0025:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0025)
   (* (* (fma (* x_m x_m) -0.0859375 0.125) x_m) x_m)
   (- 1.0 (sqrt (+ (/ 0.5 (sqrt (fma x_m x_m 1.0))) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0025) {
		tmp = (fma((x_m * x_m), -0.0859375, 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / sqrt(fma(x_m, x_m, 1.0))) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0025)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / sqrt(fma(x_m, x_m, 1.0))) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0025], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00250000000000000005

   1. Initial program 54.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 0.00250000000000000005 < x

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. 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. lift-/.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-in N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. pow2 N/A

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

      15. lower-fma.f64 98.4

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

   3. Applied rewrites 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 7: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (* (fma (* x_m x_m) -0.0859375 0.125) x_m) x_m)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = (fma((x_m * x_m), -0.0859375, 0.125) * x_m) * x_m;
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * x_m) * x_m);
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 54.7%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \frac{1 - \color{blue}{\frac{1}{2}}}{1 + \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + 1\right) \cdot \frac{1}{2}}} \]
   5. Step-by-step derivation

      1. pow2 97.8

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

      2. +-commutative 97.8

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

      3. pow2 97.8

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

      4. +-commutative 97.8

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

      5. *-commutative 97.8

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

      6. metadata-eval 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in x around inf

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

      1. pow2 97.8

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

      2. +-commutative 97.8

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

      3. pow2 97.8

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

      4. +-commutative 97.8

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

      5. *-commutative 97.8

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

      6. metadata-eval 97.8

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

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 97.8%

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

Alternative 8: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (* (* x_m x_m) 0.125)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else {
		tmp = (x_m * x_m) * 0.125;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x_m))))) <= 0.8) {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	} else {
		tmp = (x_m * x_m) * 0.125;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x_m))))) <= 0.8:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	else:
		tmp = (x_m * x_m) * 0.125
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = 0.5 / (1.0 + sqrt(0.5));
	else
		tmp = (x_m * x_m) * 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

      \[\leadsto \frac{1 - \color{blue}{\frac{1}{2}}}{1 + \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + 1\right) \cdot \frac{1}{2}}} \]
   5. Step-by-step derivation

      1. pow2 98.0

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

      2. +-commutative 98.0

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

      3. pow2 98.0

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

      4. +-commutative 98.0

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

      5. *-commutative 98.0

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

      6. metadata-eval 98.0

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

   6. Applied rewrites 98.0%

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

   7. Taylor expanded in x around inf

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

      1. pow2 98.0

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

      2. +-commutative 98.0

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

      3. pow2 98.0

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

      4. +-commutative 98.0

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

      5. *-commutative 98.0

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

      6. metadata-eval 98.0

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

   9. Applied rewrites 98.0%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 98.0%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 54.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 98.7

         \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   4. Applied rewrites 98.7%

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

Alternative 9: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
   (- 1.0 (sqrt 0.5))
   (* (* x_m x_m) 0.125)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = (x_m * x_m) * 0.125;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x_m))))) <= 0.8) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = (x_m * x_m) * 0.125;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x_m))))) <= 0.8:
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = (x_m * x_m) * 0.125
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = (x_m * x_m) * 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 96.5%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 54.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 98.7

         \[\leadsto \left(x \cdot x\right) \cdot 0.125 \]

   4. Applied rewrites 98.7%

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

Alternative 10: 74.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15e-77) (- 1.0 1.0) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.15d-77) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15e-77:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15e-77], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 69.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.0%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 80.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 78.5%

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

Alternative 11: 28.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (- 1.0 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - 1.0;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1.0d0 - 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 - 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 - 1.0

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - 1.0)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 28.4%

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

Reproduce

herbie shell --seed 2025134 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))