Given's Rotation SVD example, simplified

Percentage Accurate: 75.8% → 99.9%

Time: 5.3s

Alternatives: 15

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -1\right) \cdot 0.5\\ t_1 := \sqrt{t\_0} - -1\\ \mathbf{if}\;x\_m \leq 0.0029:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1} - \frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (/ 1.0 (sqrt (fma x_m x_m 1.0))) -1.0) 0.5))
        (t_1 (- (sqrt t_0) -1.0)))
   (if (<= x_m 0.0029)
     (* (fma -0.0859375 (* x_m x_m) 0.125) (* x_m x_m))
     (- (/ 1.0 t_1) (/ t_0 t_1)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((1.0 / sqrt(fma(x_m, x_m, 1.0))) - -1.0) * 0.5;
	double t_1 = sqrt(t_0) - -1.0;
	double tmp;
	if (x_m <= 0.0029) {
		tmp = fma(-0.0859375, (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 / t_1) - (t_0 / t_1);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) - -1.0) * 0.5)
	t_1 = Float64(sqrt(t_0) - -1.0)
	tmp = 0.0
	if (x_m <= 0.0029)
		tmp = Float64(fma(-0.0859375, Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 / t_1) - Float64(t_0 / t_1));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0029], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0029

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

   if 0.0029 < x

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.5%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0315:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\_0\right) \cdot \frac{1}{\sqrt{t\_0} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (/ 1.0 (sqrt (fma x_m x_m 1.0))) -1.0) 0.5)))
   (if (<= x_m 0.0315)
     (*
      (fma
       (fma
        (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
        (* x_m x_m)
        -0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (* (- 1.0 t_0) (/ 1.0 (- (sqrt t_0) -1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((1.0 / sqrt(fma(x_m, x_m, 1.0))) - -1.0) * 0.5;
	double tmp;
	if (x_m <= 0.0315) {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - t_0) * (1.0 / (sqrt(t_0) - -1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) - -1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.0315)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_0) * Float64(1.0 / Float64(sqrt(t_0) - -1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = 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]}, If[LessEqual[x$95$m, 0.0315], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[t$95$0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0315

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.3%

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

   if 0.0315 < x

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   7. Applied rewrites 76.6%

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

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.9995:\\ \;\;\;\;\frac{1 - \left(t\_0 + 1\right) \cdot 0.5}{\mathsf{fma}\left(\sqrt{t\_0 - -1}, \sqrt{0.5}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.9995)
     (/ (- 1.0 (* (+ t_0 1.0) 0.5)) (fma (sqrt (- t_0 -1.0)) (sqrt 0.5) 1.0))
     (*
      (fma
       (fma
        (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
        (* x_m x_m)
        -0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / sqrt(fma(x_m, x_m, 1.0));
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.9995) {
		tmp = (1.0 - ((t_0 + 1.0) * 0.5)) / fma(sqrt((t_0 - -1.0)), sqrt(0.5), 1.0);
	} else {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.9995)
		tmp = Float64(Float64(1.0 - Float64(Float64(t_0 + 1.0) * 0.5)) / fma(sqrt(Float64(t_0 - -1.0)), sqrt(0.5), 1.0));
	else
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.9995], N[(N[(1.0 - N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 - -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.99950000000000006

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. sqrt-prod N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.6%

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

   if 0.99950000000000006 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -1\right) \cdot 0.5\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.9995:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} - -1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (- (/ 1.0 (sqrt (fma x_m x_m 1.0))) -1.0) 0.5)))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.9995)
     (/ (- 1.0 t_0) (- (sqrt t_0) -1.0))
     (*
      (fma
       (fma
        (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
        (* x_m x_m)
        -0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((1.0 / sqrt(fma(x_m, x_m, 1.0))) - -1.0) * 0.5;
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.9995) {
		tmp = (1.0 - t_0) / (sqrt(t_0) - -1.0);
	} else {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) - -1.0) * 0.5)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.9995)
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) - -1.0));
	else
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = 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]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.9995], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.99950000000000006

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

   7. Applied rewrites 76.6%

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

   if 0.99950000000000006 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} - -0.5\\ t_1 := \sqrt{t\_0} + 1\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;\frac{1}{t\_1} - \frac{t\_0}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ 0.5 x_m) -0.5)) (t_1 (+ (sqrt t_0) 1.0)))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
     (- (/ 1.0 t_1) (/ t_0 t_1))
     (*
      (fma
       (fma
        (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
        (* x_m x_m)
        -0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) - -0.5;
	double t_1 = sqrt(t_0) + 1.0;
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = (1.0 / t_1) - (t_0 / t_1);
	} else {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) - -0.5)
	t_1 = Float64(sqrt(t_0) + 1.0)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(Float64(1.0 / t_1) - Float64(t_0 / t_1));
	else
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(N[(1.0 / t$95$1), $MachinePrecision] - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 50.7

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

   9. Applied rewrites 50.7%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. div-sub N/A

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

       4. lower--.f64 N/A

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

   11. Applied rewrites 50.7%

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

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

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right), x\_m \cdot x\_m, -0.0859375\right), x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x\_m}}{1 + \sqrt{\left(\frac{1}{x\_m} + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (*
    (fma
     (fma
      (fma -0.056243896484375 (* x_m x_m) 0.0673828125)
      (* x_m x_m)
      -0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))
   (/ (- 0.5 (/ 0.5 x_m)) (+ 1.0 (sqrt (* (+ (/ 1.0 x_m) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma(fma(fma(-0.056243896484375, (x_m * x_m), 0.0673828125), (x_m * x_m), -0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (0.5 - (0.5 / x_m)) / (1.0 + sqrt((((1.0 / x_m) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125), Float64(x_m * x_m), -0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x_m)) / Float64(1.0 + sqrt(Float64(Float64(Float64(1.0 / x_m) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(1.0 / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 50.3%

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

   if 1.1000000000000001 < x

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. lower-/.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 50.7

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

   9. Applied rewrites 50.7%

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

Alternative 7: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 50.2

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

   6. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1.1000000000000001 < x

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. lower-/.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 50.7

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

   9. Applied rewrites 50.7%

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

Alternative 8: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} - -0.5\\ \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ 0.5 x_m) -0.5)))
   (if (<= x_m 1.1)
     (* (* (fma (* x_m x_m) -0.0859375 0.125) x_m) x_m)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) - -0.5;
	double tmp;
	if (x_m <= 1.1) {
		tmp = (fma((x_m * x_m), -0.0859375, 0.125) * x_m) * x_m;
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) - -0.5)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 50.2

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

   6. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1.1000000000000001 < x

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 50.7

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

   9. Applied rewrites 50.7%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. metadata-eval N/A

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

       4. lower--.f64 50.7

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

       5. lower--.f64 N/A

          \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - \color{blue}{\frac{-1}{2}}\right)}{\mathsf{Rewrite=>}\left(lift-+.f64, \left(1 + \sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}}\right)\right)} \]

       6. lower--.f64 N/A

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

       7. lower--.f64 N/A

          \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - \color{blue}{\frac{-1}{2}}\right)}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(\sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}} + 1\right)\right)} \]

       8. lower--.f64 N/A

          \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - \color{blue}{\frac{-1}{2}}\right)}{\sqrt{\mathsf{Rewrite=>}\left(lift-+.f64, \left(\frac{\frac{1}{2}}{x} + \frac{1}{2}\right)\right)} + 1} \]

       9. lower--.f64 N/A

          \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - \color{blue}{\frac{-1}{2}}\right)}{\sqrt{\mathsf{Rewrite=>}\left(add-flip, \left(\frac{\frac{1}{2}}{x} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)} + 1} \]

       10. lower--.f64 N/A

           \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - \color{blue}{\frac{-1}{2}}\right)}{\sqrt{\frac{\frac{1}{2}}{x} - \mathsf{Rewrite=>}\left(metadata-eval, \frac{-1}{2}\right)} + 1} \]

       11. lower--.f64 N/A

           \[\leadsto \frac{1 - \left(\frac{\frac{1}{2}}{x} - \color{blue}{\frac{-1}{2}}\right)}{\sqrt{\mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{\frac{1}{2}}{x} - \frac{-1}{2}\right)\right)} + 1} \]

   11. Applied rewrites 50.7%

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

Alternative 9: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0024:\\ \;\;\;\;\mathsf{fma}\left(-0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} - -1\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0024)
   (* (fma -0.0859375 (* x_m x_m) 0.125) (* x_m x_m))
   (- 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 <= 0.0024) {
		tmp = fma(-0.0859375, (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 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 <= 0.0024)
		tmp = Float64(fma(-0.0859375, Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = 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, 0.0024], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00239999999999999979

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

   if 0.00239999999999999979 < x

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   5. Applied rewrites 75.8%

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

Alternative 10: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(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 < 1.1000000000000001

   1. Initial program 75.8%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 76.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 50.2

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

   6. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1.1000000000000001 < x

   1. Initial program 75.8%

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

   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. mult-flip-rev N/A

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

      4. lower-/.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. metadata-eval 50.0

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

      2. metadata-eval 50.0

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lift-/.f64 50.0

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

   6. Applied rewrites 50.0%

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

Alternative 11: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $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 < 1.1000000000000001

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 50.2

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

   4. Applied rewrites 50.2%

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

   if 1.1000000000000001 < x

   1. Initial program 75.8%

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

   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. mult-flip-rev N/A

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

      4. lower-/.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. metadata-eval 50.0

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

      2. metadata-eval 50.0

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lift-/.f64 50.0

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

   6. Applied rewrites 50.0%

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

Alternative 12: 98.4% accurate, 1.8× 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 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 51.7

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

   4. Applied rewrites 51.7%

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

   if 1.25 < x

   1. Initial program 75.8%

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

   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. mult-flip-rev N/A

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

      4. lower-/.f64 50.0

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

   4. Applied rewrites 50.0%

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

      1. metadata-eval 50.0

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

      2. metadata-eval 50.0

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lift-/.f64 50.0

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

   6. Applied rewrites 50.0%

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

Alternative 13: 97.7% accurate, 2.6× 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 75.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 51.7

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

   4. Applied rewrites 51.7%

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

   if 1.5 < x

   1. Initial program 75.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.3%

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

Alternative 14: 74.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;1 - 1\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2e-77) (- 1.0 1.0) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2e-77) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.2d-77) then
        tmp = 1.0d0 - 1.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2e-77) {
		tmp = 1.0 - 1.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2e-77:
		tmp = 1.0 - 1.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2e-77)
		tmp = Float64(1.0 - 1.0);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2e-77)
		tmp = 1.0 - 1.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2e-77], N[(1.0 - 1.0), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   1. Initial program 75.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 27.5%

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

   if 2.20000000000000007e-77 < x

   1. Initial program 75.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 50.3%

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

Alternative 15: 27.5% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2025140 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))