Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.9%

Time: 17.8s

Alternatives: 13

Speedup: 0.7×

• could not determine a ground truth

  1. x = -4.328526064979254e-152

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot x\\ \frac{1}{1 + \sqrt{\frac{0.25 + \left(\frac{0.25}{t\_0} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}{0.125 + \frac{-0.125}{{t\_0}^{1.5}}}} \cdot \sqrt{0.25 + \frac{-0.25}{t\_0}}} \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x x))))
   (*
    (/
     1.0
     (+
      1.0
      (*
       (sqrt
        (/
         (+ 0.25 (+ (/ 0.25 t_0) (/ 0.25 (hypot 1.0 x))))
         (+ 0.125 (/ -0.125 (pow t_0 1.5)))))
       (sqrt (+ 0.25 (/ -0.25 t_0))))))
    (- 0.5 (/ 0.5 (hypot 1.0 x))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * x);
	return (1.0 / (1.0 + (sqrt(((0.25 + ((0.25 / t_0) + (0.25 / hypot(1.0, x)))) / (0.125 + (-0.125 / pow(t_0, 1.5))))) * sqrt((0.25 + (-0.25 / t_0)))))) * (0.5 - (0.5 / hypot(1.0, x)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * x);
	return (1.0 / (1.0 + (Math.sqrt(((0.25 + ((0.25 / t_0) + (0.25 / Math.hypot(1.0, x)))) / (0.125 + (-0.125 / Math.pow(t_0, 1.5))))) * Math.sqrt((0.25 + (-0.25 / t_0)))))) * (0.5 - (0.5 / Math.hypot(1.0, x)));
}

Python

def code(x):
	t_0 = 1.0 + (x * x)
	return (1.0 / (1.0 + (math.sqrt(((0.25 + ((0.25 / t_0) + (0.25 / math.hypot(1.0, x)))) / (0.125 + (-0.125 / math.pow(t_0, 1.5))))) * math.sqrt((0.25 + (-0.25 / t_0)))))) * (0.5 - (0.5 / math.hypot(1.0, x)))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * x))
	return Float64(Float64(1.0 / Float64(1.0 + Float64(sqrt(Float64(Float64(0.25 + Float64(Float64(0.25 / t_0) + Float64(0.25 / hypot(1.0, x)))) / Float64(0.125 + Float64(-0.125 / (t_0 ^ 1.5))))) * sqrt(Float64(0.25 + Float64(-0.25 / t_0)))))) * Float64(0.5 - Float64(0.5 / hypot(1.0, x))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (x * x);
	tmp = (1.0 / (1.0 + (sqrt(((0.25 + ((0.25 / t_0) + (0.25 / hypot(1.0, x)))) / (0.125 + (-0.125 / (t_0 ^ 1.5))))) * sqrt((0.25 + (-0.25 / t_0)))))) * (0.5 - (0.5 / hypot(1.0, x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / N[(1.0 + N[(N[Sqrt[N[(N[(0.25 + N[(N[(0.25 / t$95$0), $MachinePrecision] + N[(0.25 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.125 + N[(-0.125 / N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.25 + N[(-0.25 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $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. div-inv N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

6. Applied egg-rr 99.9%

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

8. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{0.125 + \frac{-0.125}{{\left(1 + x \cdot x\right)}^{1.5}}}} \cdot \sqrt{0.25 + \left(\frac{0.25}{1 + x \cdot x} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}}} \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \]
9. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\sqrt{\frac{\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}} \cdot \left(\frac{1}{4} + \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)}\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   2. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\left(\frac{\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}} \cdot \left(\frac{1}{4} + \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   3. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\left(\left(\left(\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}\right) \cdot \frac{1}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}}\right) \cdot \left(\frac{1}{4} + \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\left(\left(\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}\right) \cdot \left(\frac{1}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}} \cdot \left(\frac{1}{4} + \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   5. unpow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\left(\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}} \cdot \left(\frac{1}{4} + \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}}\right)\right)\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

10. Applied egg-rr 99.9%

    \[\leadsto \frac{1}{1 + \color{blue}{{\left(0.25 - \frac{0.25}{1 + x \cdot x}\right)}^{0.5} \cdot {\left(\frac{1}{0.125 + \frac{-0.125}{{\left(1 + x \cdot x\right)}^{1.5}}} \cdot \left(\frac{0.25}{1 + x \cdot x} + \left(\frac{0.25}{\mathsf{hypot}\left(1, x\right)} + 0.25\right)\right)\right)}^{0.5}}} \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\left(\frac{1}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}} \cdot \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \left(\frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}} + \frac{1}{4}\right)\right)\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}\right)}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\left(\frac{1}{\frac{1}{8} + \frac{\frac{-1}{8}}{{\left(1 + x \cdot x\right)}^{\frac{3}{2}}}} \cdot \left(\frac{\frac{1}{4}}{1 + x \cdot x} + \left(\frac{\frac{1}{4}}{\sqrt{1 \cdot 1 + x \cdot x}} + \frac{1}{4}\right)\right)\right)}^{\frac{1}{2}}\right), \left({\left(\frac{1}{4} - \frac{\frac{1}{4}}{1 + x \cdot x}\right)}^{\frac{1}{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

12. Applied egg-rr 99.9%

    \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{\frac{0.25 + \left(\frac{0.25}{1 + x \cdot x} + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}{0.125 + \frac{-0.125}{{\left(1 + x \cdot x\right)}^{1.5}}}} \cdot \sqrt{0.25 + \frac{-0.25}{1 + x \cdot x}}}} \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \]
13. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = (0.5 - t_0) * (1.0 / (1.0 + ((0.5 + t_0) ^ 0.5)));
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[(1.0 + N[Power[N[(0.5 + t$95$0), $MachinePrecision], 0.5], $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. div-inv N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

6. Applied egg-rr 99.9%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

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

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

   4. sub-neg N/A

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

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

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

   6. distribute-neg-frac N/A

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

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

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

   8. metadata-eval N/A

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

   9. hypot-undefine N/A

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

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

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

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

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

   12. unpow1/2 N/A

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

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

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

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

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

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

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

8. Applied egg-rr 99.8%

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

Alternative 4: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + {\left(0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}\right)}^{0.5}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(0.5 - Float64(0.5 / hypot(1.0, x))) * Float64(1.0 / Float64(1.0 + (Float64(0.5 + Float64(Float64(0.5 + Float64(-0.25 / Float64(x * x))) / x)) ^ 0.5))))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Power[N[(0.5 + N[(N[(0.5 + N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.5], $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. div-inv N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf

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

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

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

   2. sub-neg N/A

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

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

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 97.9%

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

9. Simplified 97.9%

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

10. Final simplification 97.9%

    \[\leadsto \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + {\left(0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}\right)}^{0.5}} \]
11. Add Preprocessing

Alternative 5: 97.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ (+ 0.5 (/ -0.25 (* x x))) 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))) / 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))) / 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))) / 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))) / 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(0.5 + Float64(-0.25 / 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))) / 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[(0.5 + N[(-0.25 / N[(x * x), $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

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

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

7. Simplified 96.2%

   \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5 + \frac{-0.25}{x \cdot x}}{x}}} \]
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. clear-num N/A

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

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

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

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

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

9. Applied egg-rr 97.6%

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

Alternative 6: 97.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(1.0 + N[Power[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 / x), $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. div-inv N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 97.5%

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

9. Simplified 97.5%

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

10. Taylor expanded in x around inf

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

    1. /-lowering-/.f64 97.4%

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

12. Simplified 97.4%

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

Alternative 7: 96.3% 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

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

7. Simplified 96.2%

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

Alternative 8: 96.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / 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

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}\right)}\right)\right) \]
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 96.0%

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

7. Simplified 96.0%

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

Alternative 9: 97.1% 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. 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 95.3%

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

7. Simplified 95.3%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. metadata-eval N/A

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

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

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

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

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

   7. 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. Applied egg-rr 96.8%

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

Alternative 10: 95.6% 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 95.3%

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

7. Simplified 95.3%

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

Alternative 11: 22.6% accurate, 30.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (-0.25 + (0.25 * x)) / x;
}

Python

def code(x):
	return (-0.25 + (0.25 * x)) / x

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(-0.25 + N[(0.25 * x), $MachinePrecision]), $MachinePrecision] / x), $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. div-inv N/A

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 97.5%

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

9. Simplified 97.5%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot x - \frac{1}{4}}{x}} \]
11. Step-by-step derivation

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. +-commutative N/A

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

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

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 22.6%

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

12. Simplified 22.6%

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

13. Final simplification 22.6%

    \[\leadsto \frac{-0.25 + 0.25 \cdot x}{x} \]
14. Add Preprocessing

Alternative 12: 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 4.7%

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

7. Simplified 4.7%

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

8. Final simplification 4.7%

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

Alternative 13: 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 2024158 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))