Given's Rotation SVD example, simplified

Percentage Accurate: 76.1% → 99.7%

Time: 5.9s

Alternatives: 12

Speedup: 6.7×

Specification

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x)))));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

Wolfram

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

Initial Program: 76.1% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := \mathsf{fma}\left(-0.5, t\_0, -0.5\right)\\ \mathbf{if}\;x\_m \leq 0.00013:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + {t\_1}^{3}}{1 + \left({\left(\mathsf{fma}\left(t\_0, 0.5, 0.5\right)\right)}^{2} - 1 \cdot t\_1\right)}}{1 + \sqrt{\left(t\_0 + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m))) (t_1 (fma -0.5 t_0 -0.5)))
   (if (<= x_m 0.00013)
     (* (* x_m x_m) 0.125)
     (/
      (/
       (+ 1.0 (pow t_1 3.0))
       (+ 1.0 (- (pow (fma t_0 0.5 0.5) 2.0) (* 1.0 t_1))))
      (+ 1.0 (sqrt (* (+ t_0 1.0) 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = fma(-0.5, t_0, -0.5);
	double tmp;
	if (x_m <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = ((1.0 + pow(t_1, 3.0)) / (1.0 + (pow(fma(t_0, 0.5, 0.5), 2.0) - (1.0 * t_1)))) / (1.0 + sqrt(((t_0 + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = fma(-0.5, t_0, -0.5)
	tmp = 0.0
	if (x_m <= 0.00013)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(Float64(1.0 + (t_1 ^ 3.0)) / Float64(1.0 + Float64((fma(t_0, 0.5, 0.5) ^ 2.0) - Float64(1.0 * t_1)))) / Float64(1.0 + sqrt(Float64(Float64(t_0 + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * t$95$0 + -0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00013], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(N[(1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 99.8%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]

   5. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := \mathsf{fma}\left(t\_0, 0.5, 0.5\right)\\ \mathbf{if}\;x\_m \leq 0.00013:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_1}^{3}}{\mathsf{fma}\left(1.5 + t\_0 \cdot 0.5, t\_1, 1\right) \cdot \left(\sqrt{t\_1} - -1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m))) (t_1 (fma t_0 0.5 0.5)))
   (if (<= x_m 0.00013)
     (* (* x_m x_m) 0.125)
     (/
      (- 1.0 (pow t_1 3.0))
      (* (fma (+ 1.5 (* t_0 0.5)) t_1 1.0) (- (sqrt t_1) -1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = fma(t_0, 0.5, 0.5);
	double tmp;
	if (x_m <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - pow(t_1, 3.0)) / (fma((1.5 + (t_0 * 0.5)), t_1, 1.0) * (sqrt(t_1) - -1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = fma(t_0, 0.5, 0.5)
	tmp = 0.0
	if (x_m <= 0.00013)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64(fma(Float64(1.5 + Float64(t_0 * 0.5)), t_1, 1.0) * Float64(sqrt(t_1) - -1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00013], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.5 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1 - {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 99.7

         \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{\color{blue}{{\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right)}^{2}}} \]

   7. Applied rewrites 99.7%

      \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{\color{blue}{{\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \mathsf{fma}\left(\sqrt{\cos \tan^{-1} x + 1}, \sqrt{0.5}, 1\right)\right)\right)}^{2}}} \]

   9. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m + 1\\ \mathbf{if}\;x\_m \leq 0.000155:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right)}^{1.5} \cdot \sqrt{0.125}}{1 + \mathsf{fma}\left(t\_0, 0.5, 1 \cdot \sqrt{t\_0 \cdot 0.5}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (cos (atan x_m)) 1.0)))
   (if (<= x_m 0.000155)
     (* (* x_m x_m) 0.125)
     (/
      (-
       1.0
       (* (pow (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 1.5) (sqrt 0.125)))
      (+ 1.0 (fma t_0 0.5 (* 1.0 (sqrt (* t_0 0.5)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m)) + 1.0;
	double tmp;
	if (x_m <= 0.000155) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - (pow((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0), 1.5) * sqrt(0.125))) / (1.0 + fma(t_0, 0.5, (1.0 * sqrt((t_0 * 0.5)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(cos(atan(x_m)) + 1.0)
	tmp = 0.0
	if (x_m <= 0.000155)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(1.0 - Float64((Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) ^ 1.5) * sqrt(0.125))) / Float64(1.0 + fma(t_0, 0.5, Float64(1.0 * sqrt(Float64(t_0 * 0.5))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.000155], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - N[(N[Power[N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision], 1.5], $MachinePrecision] * N[Sqrt[0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * 0.5 + N[(1.0 * N[Sqrt[N[(t$95$0 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.55e-4 < x

   1. Initial program 98.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1 - {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)}} \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lift-+.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. cos-atan-rev N/A

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

      8. +-commutative N/A

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

      9. cos-atan-rev N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow2 N/A

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

      14. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \frac{1 - \color{blue}{{\left(\cos \tan^{-1} x + 1\right)}^{1.5} \cdot \sqrt{0.125}}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]
   6. Step-by-step derivation

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.7

          \[\leadsto \frac{1 - {\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right)}^{1.5} \cdot \sqrt{0.125}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]

   7. Applied rewrites 99.7%

      \[\leadsto \frac{1 - {\left(\color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right)}^{1.5} \cdot \sqrt{0.125}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m + 1\\ \mathbf{if}\;x\_m \leq 0.00013:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(t\_0, 0.5, 1 \cdot \sqrt{t\_0 \cdot 0.5}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (cos (atan x_m)) 1.0)))
   (if (<= x_m 0.00013)
     (* (* x_m x_m) 0.125)
     (/
      (- 1.0 (pow (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5) 1.5))
      (+ 1.0 (fma t_0 0.5 (* 1.0 (sqrt (* t_0 0.5)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m)) + 1.0;
	double tmp;
	if (x_m <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - pow(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5), 1.5)) / (1.0 + fma(t_0, 0.5, (1.0 * sqrt((t_0 * 0.5)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(cos(atan(x_m)) + 1.0)
	tmp = 0.0
	if (x_m <= 0.00013)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(1.0 - (Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5) ^ 1.5)) / Float64(1.0 + fma(t_0, 0.5, Float64(1.0 * sqrt(Float64(t_0 * 0.5))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.00013], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 * 0.5 + N[(1.0 * N[Sqrt[N[(t$95$0 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1 - {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)}} \]
   4. Step-by-step derivation

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.7

          \[\leadsto \frac{1 - {\left(\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]

   5. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00013:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}{1 + \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00013)
   (* (* x_m x_m) 0.125)
   (/
    (- 1.0 (* (+ (cos (atan x_m)) 1.0) 0.5))
    (+ 1.0 (sqrt (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - ((cos(atan(x_m)) + 1.0) * 0.5)) / (1.0 + sqrt((((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00013)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00013], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 99.8%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00013:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00013)
   (* (* x_m x_m) 0.125)
   (- 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 <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.00013:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x_m)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00013)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.00013)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00013], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $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 < 1.29999999999999989e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.3%

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

Alternative 7: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00013:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00013)
   (* (* x_m x_m) 0.125)
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00013) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00013)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00013], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.6

         \[\leadsto 0 \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.1%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.3%

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

      1. lift-hypot.f64 N/A

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 98.3

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

   3. Applied rewrites 98.3%

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

Alternative 8: 98.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.5) (* (* x_m x_m) 0.125) (/ (- 1.0 0.5) (+ 1.0 (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - 0.5) / (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 <= 1.5d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = (1.0d0 - 0.5d0) / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - 0.5) / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.5:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = (1.0 - 0.5) / (1.0 + math.sqrt(0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(1.0 - 0.5) / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = (1.0 - 0.5) / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.5], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(1.0 - 0.5), $MachinePrecision] / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.0

         \[\leadsto 0 \]

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 52.9%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if 1.5 < x

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in x around inf

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

      1. metadata-eval 96.5

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

      2. *-commutative 96.5

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

      3. +-commutative 96.5

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

      4. cos-atan-rev 96.5

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

   7. Applied rewrites 96.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 97.9%

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

Alternative 9: 98.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.25) (* (* x_m x_m) 0.125) (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 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 <= 1.25d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(((0.5d0 / x_m) + 0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.25:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt(((0.5 / x_m) + 0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.25], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.1

         \[\leadsto 0 \]

   4. Applied rewrites 52.1%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 99.0

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

   9. Applied rewrites 99.0%

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

   if 1.25 < x

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

Alternative 10: 97.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.5) (* (* x_m x_m) 0.125) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = (x_m * x_m) * 0.125;
	} 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 <= 1.5d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.5:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.5], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 52.0

         \[\leadsto 0 \]

   4. Applied rewrites 52.0%

      \[\leadsto \color{blue}{0} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 52.9%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

      5. associate--l+ N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

      7. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if 1.5 < x

   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%

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

Alternative 11: 51.4% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

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 = (x_m * x_m) * 0.125d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

2. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{1} \]

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 27.5

      \[\leadsto 0 \]

4. Applied rewrites 27.5%

   \[\leadsto \color{blue}{0} \]

5. Taylor expanded in x around 0

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

7. Applied rewrites 28.5%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 1\right) - 1 \]

   5. associate--l+ N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{4} + 0 \]

   7. +-rgt-identity N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

   8. associate-*l* N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{4}\right)} \]

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 51.4

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

9. Applied rewrites 51.4%

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

Alternative 12: 27.5% accurate, 134.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

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 = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 76.1%

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

2. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{1} \]

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 27.5

      \[\leadsto 0 \]

4. Applied rewrites 27.5%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))