Given's Rotation SVD example, simplified

Percentage Accurate: 76.1% → 76.9%

Time: 6.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\\ t_1 := \mathsf{fma}\left(-0.5, t\_0, -0.5\right)\\ \frac{\frac{1 + {t\_1}^{3}}{1 + \left({\left(\mathsf{fma}\left(t\_0, 0.5, 0.5\right)\right)}^{2} - 1 \cdot t\_1\right)}}{1 + \sqrt{\left(t\_0 + 1\right) \cdot 0.5}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cos (atan x))) (t_1 (fma -0.5 t_0 -0.5)))
   (/
    (/
     (+ 1.0 (pow t_1 3.0))
     (+ 1.0 (- (pow (fma t_0 0.5 0.5) 2.0) (* 1.0 t_1))))
    (+ 1.0 (sqrt (* (+ t_0 1.0) 0.5))))))

C

double code(double x) {
	double t_0 = cos(atan(x));
	double t_1 = fma(-0.5, t_0, -0.5);
	return ((1.0 + pow(t_1, 3.0)) / (1.0 + (pow(fma(t_0, 0.5, 0.5), 2.0) - (1.0 * t_1)))) / (1.0 + sqrt(((t_0 + 1.0) * 0.5)));
}

Julia

function code(x)
	t_0 = cos(atan(x))
	t_1 = fma(-0.5, t_0, -0.5)
	return Float64(Float64(Float64(1.0 + (t_1 ^ 3.0)) / Float64(1.0 + Float64((fma(t_0, 0.5, 0.5) ^ 2.0) - Float64(1.0 * t_1)))) / Float64(1.0 + sqrt(Float64(Float64(t_0 + 1.0) * 0.5))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * t$95$0 + -0.5), $MachinePrecision]}, N[(N[(N[(1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision], 2.0], $MachinePrecision] - N[(1.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 76.1%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 76.9%

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

5. Applied rewrites 76.8%

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

Alternative 2: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\\ t_1 := \mathsf{fma}\left(t\_0, 0.5, 0.5\right)\\ \frac{1 - {t\_1}^{3}}{\mathsf{fma}\left(1.5 + t\_0 \cdot 0.5, t\_1, 1\right) \cdot \left(\sqrt{t\_1} - -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cos (atan x))) (t_1 (fma t_0 0.5 0.5)))
   (/
    (- 1.0 (pow t_1 3.0))
    (* (fma (+ 1.5 (* t_0 0.5)) t_1 1.0) (- (sqrt t_1) -1.0)))))

C

double code(double x) {
	double t_0 = cos(atan(x));
	double t_1 = fma(t_0, 0.5, 0.5);
	return (1.0 - pow(t_1, 3.0)) / (fma((1.5 + (t_0 * 0.5)), t_1, 1.0) * (sqrt(t_1) - -1.0));
}

Julia

function code(x)
	t_0 = cos(atan(x))
	t_1 = fma(t_0, 0.5, 0.5)
	return Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64(fma(Float64(1.5 + Float64(t_0 * 0.5)), t_1, 1.0) * Float64(sqrt(t_1) - -1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.5 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 76.1%

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

   1. lift--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-hypot.f64 N/A

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

   7. metadata-eval N/A

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

   8. flip3-- N/A

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

   9. lower-/.f64 N/A

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

3. Applied rewrites 76.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 76.8%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-log1p.f64 76.8

      \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{{\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)\right)}}\right)}^{2}} \]

7. Applied rewrites 76.8%

   \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{\color{blue}{{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)\right)}\right)}^{2}}} \]

9. Applied rewrites 76.8%

   \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)\right)}^{3}}{\mathsf{fma}\left(1.5 + \cos \tan^{-1} x \cdot 0.5, \mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right), 1\right) \cdot \left(\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1\right)}} \]
10. Add Preprocessing

Alternative 3: 76.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\\ \frac{\frac{1 - {\left(\mathsf{fma}\left(t\_0, 0.5, 0.5\right)\right)}^{2}}{1.5 + t\_0 \cdot 0.5}}{1 + \sqrt{\left(t\_0 + 1\right) \cdot 0.5}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cos (atan x))))
   (/
    (/ (- 1.0 (pow (fma t_0 0.5 0.5) 2.0)) (+ 1.5 (* t_0 0.5)))
    (+ 1.0 (sqrt (* (+ t_0 1.0) 0.5))))))

C

double code(double x) {
	double t_0 = cos(atan(x));
	return ((1.0 - pow(fma(t_0, 0.5, 0.5), 2.0)) / (1.5 + (t_0 * 0.5))) / (1.0 + sqrt(((t_0 + 1.0) * 0.5)));
}

Julia

function code(x)
	t_0 = cos(atan(x))
	return Float64(Float64(Float64(1.0 - (fma(t_0, 0.5, 0.5) ^ 2.0)) / Float64(1.5 + Float64(t_0 * 0.5))) / Float64(1.0 + sqrt(Float64(Float64(t_0 + 1.0) * 0.5))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.0 - N[Power[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.5 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.1%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 76.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

5. Applied rewrites 76.9%

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

Alternative 4: 76.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \tan^{-1} x + 1\\ t_1 := t\_0 \cdot 0.5\\ \frac{1 - {t\_1}^{1.5}}{\mathsf{fma}\left(t\_0, 0.5, \sqrt{t\_1}\right) + 1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cos (atan x)) 1.0)) (t_1 (* t_0 0.5)))
   (/ (- 1.0 (pow t_1 1.5)) (+ (fma t_0 0.5 (sqrt t_1)) 1.0))))

C

double code(double x) {
	double t_0 = cos(atan(x)) + 1.0;
	double t_1 = t_0 * 0.5;
	return (1.0 - pow(t_1, 1.5)) / (fma(t_0, 0.5, sqrt(t_1)) + 1.0);
}

Julia

function code(x)
	t_0 = Float64(cos(atan(x)) + 1.0)
	t_1 = Float64(t_0 * 0.5)
	return Float64(Float64(1.0 - (t_1 ^ 1.5)) / Float64(fma(t_0, 0.5, sqrt(t_1)) + 1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5), $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * 0.5 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 76.1%

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

   1. lift--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-hypot.f64 N/A

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

   7. metadata-eval N/A

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

   8. flip3-- N/A

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

   9. lower-/.f64 N/A

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

3. Applied rewrites 76.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 76.8

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity 76.8

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

5. Applied rewrites 76.8%

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

Alternative 5: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x + 1\right) \cdot 0.5\\ \frac{1 - t\_0}{1 + \sqrt{t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (+ (cos (atan x)) 1.0) 0.5)))
   (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0)))))

C

double code(double x) {
	double t_0 = (cos(atan(x)) + 1.0) * 0.5;
	return (1.0 - t_0) / (1.0 + sqrt(t_0));
}

Fortran

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (cos(atan(x)) + 1.0d0) * 0.5d0
    code = (1.0d0 - t_0) / (1.0d0 + sqrt(t_0))
end function

Java

public static double code(double x) {
	double t_0 = (Math.cos(Math.atan(x)) + 1.0) * 0.5;
	return (1.0 - t_0) / (1.0 + Math.sqrt(t_0));
}

Python

def code(x):
	t_0 = (math.cos(math.atan(x)) + 1.0) * 0.5
	return (1.0 - t_0) / (1.0 + math.sqrt(t_0))

Julia

function code(x)
	t_0 = Float64(Float64(cos(atan(x)) + 1.0) * 0.5)
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = (cos(atan(x)) + 1.0) * 0.5;
	tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.1%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 76.9%

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

Alternative 6: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\\ \frac{0.5 - t\_0 \cdot 0.5}{\sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right)} - -1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cos (atan x))))
   (/ (- 0.5 (* t_0 0.5)) (- (sqrt (fma t_0 0.5 0.5)) -1.0))))

C

double code(double x) {
	double t_0 = cos(atan(x));
	return (0.5 - (t_0 * 0.5)) / (sqrt(fma(t_0, 0.5, 0.5)) - -1.0);
}

Julia

function code(x)
	t_0 = cos(atan(x))
	return Float64(Float64(0.5 - Float64(t_0 * 0.5)) / Float64(sqrt(fma(t_0, 0.5, 0.5)) - -1.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision]}, N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.1%

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

   1. lift--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-hypot.f64 N/A

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

   7. metadata-eval N/A

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

   8. flip3-- N/A

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

   9. lower-/.f64 N/A

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

3. Applied rewrites 76.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 76.8%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-log1p.f64 76.8

      \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{{\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)\right)}}\right)}^{2}} \]

7. Applied rewrites 76.8%

   \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{\color{blue}{{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)\right)}\right)}^{2}}} \]

9. Applied rewrites 76.1%

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

Alternative 7: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

Alternative 8: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (sqrt (fma (cos (atan x)) 0.5 0.5))))

C

double code(double x) {
	return 1.0 - sqrt(fma(cos(atan(x)), 0.5, 0.5));
}

Julia

function code(x)
	return Float64(1.0 - sqrt(fma(cos(atan(x)), 0.5, 0.5)))
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

   1. lift--.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-hypot.f64 N/A

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

   7. metadata-eval N/A

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

   8. flip3-- N/A

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

   9. lower-/.f64 N/A

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

3. Applied rewrites 76.8%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 76.8%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-log1p.f64 76.8

      \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{{\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)\right)}}\right)}^{2}} \]

7. Applied rewrites 76.8%

   \[\leadsto \frac{1 \cdot \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) - \left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right) + 1\right) \cdot {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{\color{blue}{{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)\right)}\right)}^{2}}} \]

9. Applied rewrites 76.1%

   \[\leadsto \color{blue}{1 - \sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)}} \]
10. Add Preprocessing

Alternative 9: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \leq 0.8:\\ \;\;\;\;\frac{1 - 0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(-0.25, x \cdot x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x))))) 0.8)
   (/ (- 1.0 0.5) (+ 1.0 (sqrt 0.5)))
   (- 1.0 (sqrt (fma -0.25 (* x x) 1.0)))))

C

double code(double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))) <= 0.8) {
		tmp = (1.0 - 0.5) / (1.0 + sqrt(0.5));
	} else {
		tmp = 1.0 - sqrt(fma(-0.25, (x * x), 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))) <= 0.8)
		tmp = Float64(Float64(1.0 - 0.5) / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(1.0 - sqrt(fma(-0.25, Float64(x * x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(N[(1.0 - 0.5), $MachinePrecision] / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $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 98.5%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 96.5%

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

      1. metadata-eval 96.5

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 98.0%

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

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

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

Alternative 10: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2e-77) 0.0 (/ (- 1.0 0.5) (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = (1.0 - 0.5) / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d-77) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 - 0.5d0) / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = (1.0 - 0.5) / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2e-77:
		tmp = 0.0
	else:
		tmp = (1.0 - 0.5) / (1.0 + math.sqrt(0.5))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.2e-77], 0.0, N[(N[(1.0 - 0.5), $MachinePrecision] / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   1. Initial program 74.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 38.1

         \[\leadsto 0 \]

   4. Applied rewrites 38.1%

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

   if 2.20000000000000007e-77 < x

   1. Initial program 80.4%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 78.6

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 78.3%

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

      1. metadata-eval 78.3

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 79.5%

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

Alternative 11: 50.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.2e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.2e-77], 0.0, 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 74.2%

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

   2. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 38.1

         \[\leadsto 0 \]

   4. Applied rewrites 38.1%

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

   if 2.20000000000000007e-77 < x

   1. Initial program 80.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 78.3%

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

Alternative 12: 27.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 76.1%

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

2. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 27.5

      \[\leadsto 0 \]

4. Applied rewrites 27.5%

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

Reproduce

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