Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.9%

Time: 18.6s

Alternatives: 12

Speedup: 0.7×

• could not determine a ground truth

  1. x = -4.649053160556577e-141

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: 98.4% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \frac{0.5 + t\_0}{1 + \sqrt{0.5 - t\_0}} \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	return (0.5 + t_0) / (1.0 + sqrt((0.5 - t_0)));
}

Java

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

Python

def code(x):
	t_0 = -0.5 / math.hypot(1.0, x)
	return (0.5 + t_0) / (1.0 + math.sqrt((0.5 - t_0)))

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	return Float64(Float64(0.5 + t_0) / Float64(1.0 + sqrt(Float64(0.5 - t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = -0.5 / hypot(1.0, x);
	tmp = (0.5 + t_0) / (1.0 + sqrt((0.5 - t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.5 + t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right), \color{blue}{\left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}\right)}\right) \]

6. Applied egg-rr 99.9%

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

Alternative 2: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{\left(0.5 + \frac{-0.25}{x \cdot x}\right) + \frac{0.1875}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\\ \frac{1}{\frac{1 + \sqrt{t\_0}}{1 - t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          0.5
          (/ (+ (+ 0.5 (/ -0.25 (* x x))) (/ 0.1875 (* x (* x (* x x))))) x))))
   (/ 1.0 (/ (+ 1.0 (sqrt t_0)) (- 1.0 t_0)))))

C

double code(double x) {
	double t_0 = 0.5 + (((0.5 + (-0.25 / (x * x))) + (0.1875 / (x * (x * (x * x))))) / x);
	return 1.0 / ((1.0 + sqrt(t_0)) / (1.0 - t_0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 0.5d0 + (((0.5d0 + ((-0.25d0) / (x * x))) + (0.1875d0 / (x * (x * (x * x))))) / x)
    code = 1.0d0 / ((1.0d0 + sqrt(t_0)) / (1.0d0 - t_0))
end function

Java

public static double code(double x) {
	double t_0 = 0.5 + (((0.5 + (-0.25 / (x * x))) + (0.1875 / (x * (x * (x * x))))) / x);
	return 1.0 / ((1.0 + Math.sqrt(t_0)) / (1.0 - t_0));
}

Python

def code(x):
	t_0 = 0.5 + (((0.5 + (-0.25 / (x * x))) + (0.1875 / (x * (x * (x * x))))) / x)
	return 1.0 / ((1.0 + math.sqrt(t_0)) / (1.0 - t_0))

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(Float64(Float64(0.5 + Float64(-0.25 / Float64(x * x))) + Float64(0.1875 / Float64(x * Float64(x * Float64(x * x))))) / x))
	return Float64(1.0 / Float64(Float64(1.0 + sqrt(t_0)) / Float64(1.0 - t_0)))
end

MATLAB

function tmp = code(x)
	t_0 = 0.5 + (((0.5 + (-0.25 / (x * x))) + (0.1875 / (x * (x * (x * x))))) / x);
	tmp = 1.0 / ((1.0 + sqrt(t_0)) / (1.0 - t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(N[(N[(0.5 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\left(\frac{1}{2} + \frac{\frac{3}{16}}{{x}^{4}}\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right)\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\left(\frac{1}{2} + \frac{\frac{3}{16}}{{x}^{4}}\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

7. Simplified 96.5%

   \[\leadsto 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 + \left(\frac{0.1875}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{-0.25}{x \cdot x}\right)}{x}}} \]
8. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \frac{1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}}}{\color{blue}{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}}}} \]

   2. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}}}{1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}}}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}}}{1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2} + \left(\frac{\frac{3}{16}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \frac{\frac{-1}{4}}{x \cdot x}\right)}{x}}}\right)}\right) \]

9. Applied egg-rr 97.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{1 + \sqrt{0.5 + \frac{\left(0.5 + \frac{-0.25}{x \cdot x}\right) + \frac{0.1875}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}}}{1 - \left(0.5 + \frac{\left(0.5 + \frac{-0.25}{x \cdot x}\right) + \frac{0.1875}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}{x}\right)}}} \]
10. Add Preprocessing

Alternative 3: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}\\ \frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (+ 0.5 (/ -0.25 (* x x))) x)))
   (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0))))))

C

double code(double x) {
	double t_0 = (0.5 + (-0.25 / (x * x))) / x;
	return (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (0.5d0 + ((-0.25d0) / (x * x))) / x
    code = (0.5d0 - t_0) / (1.0d0 + sqrt((0.5d0 + t_0)))
end function

Java

public static double code(double x) {
	double t_0 = (0.5 + (-0.25 / (x * x))) / x;
	return (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
}

Python

def code(x):
	t_0 = (0.5 + (-0.25 / (x * x))) / x
	return (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))

Julia

function code(x)
	t_0 = Float64(Float64(0.5 + Float64(-0.25 / Float64(x * x))) / x)
	return Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = (0.5 + (-0.25 / (x * x))) / x;
	tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{4}}{{x}^{3}}\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{3}}\right)\right)\right)\right) \]

   2. unpow3 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{\left(x \cdot x\right) \cdot x}\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{2} \cdot x}\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left({x}^{2}\right)\right)\right), x\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left(x \cdot x\right)\right)\right), x\right)\right)\right)\right) \]

   17. *-lowering-*.f64 96.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right) \]

7. Simplified 96.4%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5 - \frac{0.25}{x \cdot x}}{x}}} \]
8. Step-by-step derivation

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}}\right), \color{blue}{\left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}}\right)}\right) \]

9. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{1 - \left(0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}\right)}{1 + \sqrt{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}}} \]
10. Step-by-step derivation

    1. associate--r+ N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - \frac{1}{2}\right) - \frac{\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}}{x}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}}{x}\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    3. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}}{x}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{4}}{x \cdot x}\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \left(x \cdot x\right)\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 97.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

11. Applied egg-rr 97.9%

    \[\leadsto \frac{\color{blue}{0.5 - \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}}{1 + \sqrt{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}} \]
12. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 - \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 0.5 (/ (+ 0.5 (/ -0.25 (* x x))) x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x))))))

C

double code(double x) {
	return (0.5 - ((0.5 + (-0.25 / (x * x))) / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 - ((0.5d0 + ((-0.25d0) / (x * x))) / x)) / (1.0d0 + sqrt((0.5d0 + (0.5d0 / x))))
end function

Java

public static double code(double x) {
	return (0.5 - ((0.5 + (-0.25 / (x * x))) / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
}

Python

def code(x):
	return (0.5 - ((0.5 + (-0.25 / (x * x))) / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))

Julia

function code(x)
	return Float64(Float64(0.5 - Float64(Float64(0.5 + Float64(-0.25 / Float64(x * x))) / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 - ((0.5 + (-0.25 / (x * x))) / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
end

Wolfram

code[x_] := N[(N[(0.5 - N[(N[(0.5 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{4}}{{x}^{3}}\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{3}}\right)\right)\right)\right) \]

   2. unpow3 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{\left(x \cdot x\right) \cdot x}\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{2} \cdot x}\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left({x}^{2}\right)\right)\right), x\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left(x \cdot x\right)\right)\right), x\right)\right)\right)\right) \]

   17. *-lowering-*.f64 96.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right) \]

7. Simplified 96.4%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5 - \frac{0.25}{x \cdot x}}{x}}} \]
8. Step-by-step derivation

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}}\right), \color{blue}{\left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}}\right)}\right) \]

9. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{1 - \left(0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}\right)}{1 + \sqrt{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}}} \]
10. Step-by-step derivation

    1. associate--r+ N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\left(1 - \frac{1}{2}\right) - \frac{\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}}{x}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}}{x}\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    3. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}}{x}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} + \frac{\frac{-1}{4}}{x \cdot x}\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    5. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{4}}{x \cdot x}\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \left(x \cdot x\right)\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 97.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

11. Applied egg-rr 97.9%

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

12. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{\frac{1}{2}}{x}\right)}\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. /-lowering-/.f64 97.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right)\right) \]

14. Simplified 97.6%

    \[\leadsto \frac{0.5 - \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}{1 + \sqrt{0.5 + \color{blue}{\frac{0.5}{x}}}} \]
15. Add Preprocessing

Alternative 5: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ (+ 0.5 (/ -0.25 (* x x))) x))))))

C

double code(double x) {
	return (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + ((-0.5d0) / x)) / (1.0d0 + sqrt((0.5d0 + ((0.5d0 + ((-0.25d0) / (x * x))) / x))))
end function

Java

public static double code(double x) {
	return (0.5 + (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x))));
}

Python

def code(x):
	return (0.5 + (-0.5 / x)) / (1.0 + math.sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x))))

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(-0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(Float64(0.5 + Float64(-0.25 / Float64(x * x))) / x)))))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + ((0.5 + (-0.25 / (x * x))) / x))));
end

Wolfram

code[x_] := N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(N[(0.5 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{4}}{{x}^{3}}\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{3}}\right)\right)\right)\right) \]

   2. unpow3 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{\left(x \cdot x\right) \cdot x}\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{2} \cdot x}\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left({x}^{2}\right)\right)\right), x\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left(x \cdot x\right)\right)\right), x\right)\right)\right)\right) \]

   17. *-lowering-*.f64 96.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right) \]

7. Simplified 96.4%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5 - \frac{0.25}{x \cdot x}}{x}}} \]
8. Step-by-step derivation

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}}\right), \color{blue}{\left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2} - \frac{\frac{1}{4}}{x \cdot x}}{x}}\right)}\right) \]

9. Applied egg-rr 97.8%

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

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}\right)}, \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    5. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{2}}{x}\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

    7. /-lowering-/.f64 97.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right)\right) \]

12. Simplified 97.6%

    \[\leadsto \frac{\color{blue}{0.5 + \frac{-0.5}{x}}}{1 + \sqrt{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}} \]
13. Add Preprocessing

Alternative 6: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ \frac{1 - t\_0}{1 + \sqrt{t\_0}} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 96.1%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right) \]

7. Simplified 96.1%

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

   1. flip-- N/A

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot 1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right), \color{blue}{\left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}} \cdot \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right), \left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)\right) \]

   4. rem-square-sqrt N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - \left(\frac{1}{2} + \frac{\frac{1}{2}}{x}\right)\right), \left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{2} + \frac{\frac{1}{2}}{x}\right)\right), \left(\color{blue}{1} + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right), \left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{x}}\right)}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2}}{x}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right)\right) \]

   11. /-lowering-/.f64 97.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right)\right)\right) \]

9. Applied egg-rr 97.5%

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

Alternative 7: 96.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5 + \frac{0.5 - \frac{0.25}{x \cdot x}}{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (+ 0.5 (/ (- 0.5 (/ 0.25 (* x x))) x)))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 + ((0.5 - (0.25 / (x * x))) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - sqrt((0.5d0 + ((0.5d0 - (0.25d0 / (x * x))) / x)))
end function

Java

public static double code(double x) {
	return 1.0 - Math.sqrt((0.5 + ((0.5 - (0.25 / (x * x))) / x)));
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 + ((0.5 - (0.25 / (x * x))) / x)))

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 + Float64(Float64(0.5 - Float64(0.25 / Float64(x * x))) / x))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 + ((0.5 - (0.25 / (x * x))) / x)));
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 + N[(N[(0.5 - N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{\frac{1}{4}}{{x}^{3}}\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{3}}\right)\right)\right)\right) \]

   2. unpow3 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{\left(x \cdot x\right) \cdot x}\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4}}{{x}^{2} \cdot x}\right)\right)\right)\right) \]

   4. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4}}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{\frac{1}{4} \cdot 1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{1}{2} \cdot \frac{1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2} \cdot 1}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right), x\right)\right)\right)\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4} \cdot 1}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{4}}{{x}^{2}}\right)\right), x\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left({x}^{2}\right)\right)\right), x\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \left(x \cdot x\right)\right)\right), x\right)\right)\right)\right) \]

   17. *-lowering-*.f64 96.4%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right)\right)\right) \]

7. Simplified 96.4%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5 - \frac{0.25}{x \cdot x}}{x}}} \]
8. Add Preprocessing

Alternative 8: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{1 + \sqrt{0.5}} \end{array} \]

FPCore

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

C

double code(double x) {
	return 0.5 / (1.0 + sqrt(0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / (1.0d0 + sqrt(0.5d0))
end function

Java

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

Python

def code(x):
	return 0.5 / (1.0 + math.sqrt(0.5))

Julia

function code(x)
	return Float64(0.5 / Float64(1.0 + sqrt(0.5)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (1.0 + sqrt(0.5));
end

Wolfram

code[x_] := N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right), \color{blue}{\left(1 + \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}}\right)}\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + \sqrt{\frac{1}{2}}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right)\right) \]

   3. sqrt-lowering-sqrt.f64 96.8%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right) \]

9. Simplified 96.8%

   \[\leadsto \color{blue}{\frac{0.5}{1 + \sqrt{0.5}}} \]
10. Add Preprocessing

Alternative 9: 95.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt{0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (sqrt 0.5)))

C

double code(double x) {
	return 1.0 - sqrt(0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - sqrt(0.5d0)
end function

Java

public static double code(double x) {
	return 1.0 - Math.sqrt(0.5);
}

Python

def code(x):
	return 1.0 - math.sqrt(0.5)

Julia

function code(x)
	return Float64(1.0 - sqrt(0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt(0.5);
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around inf

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2}}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 95.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right) \]

7. Simplified 95.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
8. Add Preprocessing

Alternative 10: 4.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(-1 - \left(x \cdot x\right) \cdot -0.125\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (- -1.0 (* (* x x) -0.125))))

C

double code(double x) {
	return 1.0 + (-1.0 - ((x * x) * -0.125));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((-1.0d0) - ((x * x) * (-0.125d0)))
end function

Java

public static double code(double x) {
	return 1.0 + (-1.0 - ((x * x) * -0.125));
}

Python

def code(x):
	return 1.0 + (-1.0 - ((x * x) * -0.125))

Julia

function code(x)
	return Float64(1.0 + Float64(-1.0 - Float64(Float64(x * x) * -0.125)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (-1.0 - ((x * x) * -0.125));
end

Wolfram

code[x_] := N[(1.0 + N[(-1.0 - N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(1 + \frac{-1}{8} \cdot {x}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{8} \cdot {x}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{8}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{8}\right)\right)\right) \]

   5. *-lowering-*.f64 4.5%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{8}\right)\right)\right) \]

7. Simplified 4.5%

   \[\leadsto 1 - \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.125\right)} \]

8. Final simplification 4.5%

   \[\leadsto 1 + \left(-1 - \left(x \cdot x\right) \cdot -0.125\right) \]
9. Add Preprocessing

Alternative 11: 4.5% accurate, 42.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* x x) 0.125))

C

double code(double x) {
	return (x * x) * 0.125;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * 0.125d0
end function

Java

public static double code(double x) {
	return (x * x) * 0.125;
}

Python

def code(x):
	return (x * x) * 0.125

Julia

function code(x)
	return Float64(Float64(x * x) * 0.125)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 4.5%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

7. Simplified 4.5%

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

8. Final simplification 4.5%

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

Alternative 12: 3.1% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    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 98.3%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right)\right) \]

   6. associate-*l/ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

   9. hypot-undefine N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

   10. hypot-lowering-hypot.f64 98.3%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

3. Simplified 98.3%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 3.1%

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

   1. metadata-eval 3.1%

      \[\leadsto 0 \]

9. Applied egg-rr 3.1%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))