Given's Rotation SVD example, simplified

Percentage Accurate: 98.4% → 99.9%

Time: 17.1s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. x = -3.089644090851901e-186

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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

   1. flip-+ N/A

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

   2. div-inv N/A

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

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

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

8. Applied egg-rr 99.9%

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

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

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

   9. hypot-undefine N/A

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

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

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

10. Applied egg-rr 99.9%

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

    1. associate-*l/ N/A

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

    2. metadata-eval N/A

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

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

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

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

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

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

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

    6. hypot-undefine N/A

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

    7. hypot-lowering-hypot.f64 99.9%

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

12. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

   1. flip-+ N/A

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

   2. div-inv N/A

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

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

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(0.25 / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] - 0.5), $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 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

   1. +-commutative N/A

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

   2. flip-+ N/A

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

   3. frac-times N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. frac-times N/A

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

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

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

8. Applied egg-rr 99.9%

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

Alternative 4: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

   1. clear-num N/A

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

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

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. cancel-sign-sub-inv N/A

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

   6. div-inv N/A

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

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

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

8. Applied egg-rr 99.9%

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

Alternative 5: 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 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

Alternative 7: 97.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = 0.5 + ((0.5 + (-0.25 / (x * x))) / x);
	tmp = (1.0 - t_0) / (1.0 + sqrt(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[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. unpow3 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ 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 96.0%

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

   1. flip-- N/A

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

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

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

   3. metadata-eval N/A

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

   4. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 97.4%

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

Alternative 8: 97.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 95.6%

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

7. Simplified 95.6%

   \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
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 97.1%

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

9. Applied egg-rr 97.1%

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

Alternative 9: 95.8% 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 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 95.6%

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

7. Simplified 95.6%

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

Alternative 10: 22.8% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 + \frac{-0.5}{1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)}}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   0.5
   (/
    -0.5
    (+ 1.0 (* x (* x (+ 0.5 (* x (* x (+ -0.125 (* (* x x) 0.0625))))))))))
  2.0))

C

double code(double x) {
	return (0.5 + (-0.5 / (1.0 + (x * (x * (0.5 + (x * (x * (-0.125 + ((x * x) * 0.0625)))))))))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + ((-0.5d0) / (1.0d0 + (x * (x * (0.5d0 + (x * (x * ((-0.125d0) + ((x * x) * 0.0625d0)))))))))) / 2.0d0
end function

Java

public static double code(double x) {
	return (0.5 + (-0.5 / (1.0 + (x * (x * (0.5 + (x * (x * (-0.125 + ((x * x) * 0.0625)))))))))) / 2.0;
}

Python

def code(x):
	return (0.5 + (-0.5 / (1.0 + (x * (x * (0.5 + (x * (x * (-0.125 + ((x * x) * 0.0625)))))))))) / 2.0

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(-0.5 / Float64(1.0 + Float64(x * Float64(x * Float64(0.5 + Float64(x * Float64(x * Float64(-0.125 + Float64(Float64(x * x) * 0.0625)))))))))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (-0.5 / (1.0 + (x * (x * (0.5 + (x * (x * (-0.125 + ((x * x) * 0.0625)))))))))) / 2.0;
end

Wolfram

code[x_] := N[(N[(0.5 + N[(-0.5 / N[(1.0 + N[(x * N[(x * N[(0.5 + N[(x * N[(x * N[(-0.125 + N[(N[(x * x), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 22.8%

   \[\leadsto \frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)}\right)\right), 2\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), 2\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right), 2\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left({x}^{2} \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot {x}^{2} - \frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    11. sub-neg N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot {x}^{2} + \frac{-1}{8}\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    13. +-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

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

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

    18. *-lowering-*.f64 22.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{16}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

12. Simplified 22.8%

    \[\leadsto \frac{0.5 + \frac{-0.5}{\color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.125 + \left(x \cdot x\right) \cdot 0.0625\right)\right)\right)\right)}}}{2} \]
13. Add Preprocessing

Alternative 11: 22.8% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 + \frac{-0.5}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)}}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ 0.5 (/ -0.5 (+ 1.0 (* (* x x) (+ 0.5 (* (* x x) -0.125)))))) 2.0))

C

double code(double x) {
	return (0.5 + (-0.5 / (1.0 + ((x * x) * (0.5 + ((x * x) * -0.125)))))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + ((-0.5d0) / (1.0d0 + ((x * x) * (0.5d0 + ((x * x) * (-0.125d0))))))) / 2.0d0
end function

Java

public static double code(double x) {
	return (0.5 + (-0.5 / (1.0 + ((x * x) * (0.5 + ((x * x) * -0.125)))))) / 2.0;
}

Python

def code(x):
	return (0.5 + (-0.5 / (1.0 + ((x * x) * (0.5 + ((x * x) * -0.125)))))) / 2.0

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(-0.5 / Float64(1.0 + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * -0.125)))))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (-0.5 / (1.0 + ((x * x) * (0.5 + ((x * x) * -0.125)))))) / 2.0;
end

Wolfram

code[x_] := N[(N[(0.5 + N[(-0.5 / N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 22.8%

   \[\leadsto \frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

10. Taylor expanded in x around 0

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

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

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

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

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

    3. unpow2 N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. unpow2 N/A

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

    9. *-lowering-*.f64 22.8%

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

12. Simplified 22.8%

    \[\leadsto \frac{0.5 + \frac{-0.5}{\color{blue}{1 + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)}}}{2} \]
13. Add Preprocessing

Alternative 12: 22.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 + \frac{-0.5}{1 + 0.5 \cdot \left(x \cdot x\right)}}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ 0.5 (/ -0.5 (+ 1.0 (* 0.5 (* x x))))) 2.0))

C

double code(double x) {
	return (0.5 + (-0.5 / (1.0 + (0.5 * (x * x))))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 + ((-0.5d0) / (1.0d0 + (0.5d0 * (x * x))))) / 2.0d0
end function

Java

public static double code(double x) {
	return (0.5 + (-0.5 / (1.0 + (0.5 * (x * x))))) / 2.0;
}

Python

def code(x):
	return (0.5 + (-0.5 / (1.0 + (0.5 * (x * x))))) / 2.0

Julia

function code(x)
	return Float64(Float64(0.5 + Float64(-0.5 / Float64(1.0 + Float64(0.5 * Float64(x * x))))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 + (-0.5 / (1.0 + (0.5 * (x * x))))) / 2.0;
end

Wolfram

code[x_] := N[(N[(0.5 + N[(-0.5 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 22.8%

   \[\leadsto \frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

10. Taylor expanded in x around 0

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

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

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

    2. *-commutative N/A

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

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

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

    4. unpow2 N/A

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

    5. *-lowering-*.f64 22.8%

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

12. Simplified 22.8%

    \[\leadsto \frac{0.5 + \frac{-0.5}{\color{blue}{1 + \left(x \cdot x\right) \cdot 0.5}}}{2} \]

13. Final simplification 22.8%

    \[\leadsto \frac{0.5 + \frac{-0.5}{1 + 0.5 \cdot \left(x \cdot x\right)}}{2} \]
14. Add Preprocessing

Alternative 13: 22.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ 0.25 + \frac{-0.25 + \frac{0.125}{x \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 0.25 (/ (+ -0.25 (/ 0.125 (* x x))) x)))

C

double code(double x) {
	return 0.25 + ((-0.25 + (0.125 / (x * x))) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0 + (((-0.25d0) + (0.125d0 / (x * x))) / x)
end function

Java

public static double code(double x) {
	return 0.25 + ((-0.25 + (0.125 / (x * x))) / x);
}

Python

def code(x):
	return 0.25 + ((-0.25 + (0.125 / (x * x))) / x)

Julia

function code(x)
	return Float64(0.25 + Float64(Float64(-0.25 + Float64(0.125 / Float64(x * x))) / x))
end

MATLAB

function tmp = code(x)
	tmp = 0.25 + ((-0.25 + (0.125 / (x * x))) / x);
end

Wolfram

code[x_] := N[(0.25 + N[(N[(-0.25 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 22.8%

   \[\leadsto \frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\left(\frac{1}{4} + \frac{\frac{1}{8}}{{x}^{3}}\right) - \frac{1}{4} \cdot \frac{1}{x}} \]
11. Step-by-step derivation

    1. associate--l+ N/A

       \[\leadsto \frac{1}{4} + \color{blue}{\left(\frac{\frac{1}{8}}{{x}^{3}} - \frac{1}{4} \cdot \frac{1}{x}\right)} \]

    2. unpow3 N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{1}{4} \cdot \frac{1}{x}\right) \]

    3. unpow2 N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{1}{8}}{{x}^{2} \cdot x} - \frac{1}{4} \cdot \frac{1}{x}\right) \]

    4. associate-/r* N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{\frac{1}{8}}{{x}^{2}}}{x} - \color{blue}{\frac{1}{4}} \cdot \frac{1}{x}\right) \]

    5. metadata-eval N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}{x} - \frac{1}{4} \cdot \frac{1}{x}\right) \]

    6. associate-*r/ N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x} - \frac{1}{4} \cdot \frac{1}{x}\right) \]

    7. associate-*r/ N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x} - \frac{\frac{1}{4} \cdot 1}{\color{blue}{x}}\right) \]

    8. metadata-eval N/A

       \[\leadsto \frac{1}{4} + \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x} - \frac{\frac{1}{4}}{x}\right) \]

    9. div-sub N/A

       \[\leadsto \frac{1}{4} + \frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{4}}{\color{blue}{x}} \]

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

        \[\leadsto \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{4}}{x}\right)}\right) \]

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

        \[\leadsto \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{4}\right), \color{blue}{x}\right)\right) \]

12. Simplified 22.6%

    \[\leadsto \color{blue}{0.25 + \frac{-0.25 + \frac{0.125}{x \cdot x}}{x}} \]
13. Add Preprocessing

Alternative 14: 22.7% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. rem-square-sqrt N/A

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

   4. associate--r+ N/A

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

   5. metadata-eval N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 22.8%

   \[\leadsto \frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

10. Taylor expanded in x around inf

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

12. Simplified 22.6%

    \[\leadsto \color{blue}{0.25} \]
13. Add Preprocessing

Alternative 15: 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 98.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 98.4%

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

3. Simplified 98.4%

   \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
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 2024138 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))