Numeric.Log:$clog1p from log-domain-0.10.2.1, B

Percentage Accurate: 99.7% → 99.7%

Time: 10.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.0390625, 0.0625\right), -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 5e-5)
     (* x (fma x (fma x (fma x -0.0390625 0.0625) -0.125) 0.5))
     (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((x / (1.0 + t_0)) <= 5e-5) {
		tmp = x * fma(x, fma(x, fma(x, -0.0390625, 0.0625), -0.125), 0.5);
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(x / Float64(1.0 + t_0)) <= 5e-5)
		tmp = Float64(x * fma(x, fma(x, fma(x, -0.0390625, 0.0625), -0.125), 0.5));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 5e-5], N[(x * N[(x * N[(x * N[(x * -0.0390625 + 0.0625), $MachinePrecision] + -0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.00000000000000024e-5

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) + \color{blue}{\frac{-1}{8}}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{16} + \frac{-5}{128} \cdot x, \frac{-1}{8}\right)}, \frac{1}{2}\right) \]

      7. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-5}{128} \cdot x + \frac{1}{16}}, \frac{-1}{8}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-5}{128}} + \frac{1}{16}, \frac{-1}{8}\right), \frac{1}{2}\right) \]

      9. lower-fma.f64 100.0

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.0390625, 0.0625\right)}, -0.125\right), 0.5\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.0390625, 0.0625\right), -0.125\right), 0.5\right)} \]

   if 5.00000000000000024e-5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{\mathsf{neg}\left(\left(1 + \sqrt{x + 1}\right)\right)} \]

      6. metadata-eval N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-+.f64 N/A

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

      16. flip-- N/A

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

      17. lower-neg.f64 N/A

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

      18. lower--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]

      6. sub-neg N/A

         \[\leadsto 0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right) + 1\right)} \]

      8. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right) - 1} \]

      9. neg-sub0 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 100.0

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.0390625, 0.0625\right), -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0625, -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 5e-5)
     (* x (fma x (fma x 0.0625 -0.125) 0.5))
     (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((x / (1.0 + t_0)) <= 5e-5) {
		tmp = x * fma(x, fma(x, 0.0625, -0.125), 0.5);
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(x / Float64(1.0 + t_0)) <= 5e-5)
		tmp = Float64(x * fma(x, fma(x, 0.0625, -0.125), 0.5));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 5e-5], N[(x * N[(x * N[(x * 0.0625 + -0.125), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.00000000000000024e-5

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right) + \frac{1}{2}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{16} \cdot x - \frac{1}{8}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{16}} + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right), \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{16} + \color{blue}{\frac{-1}{8}}, \frac{1}{2}\right) \]

      7. lower-fma.f64 99.9

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.0625, -0.125\right)}, 0.5\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0625, -0.125\right), 0.5\right)} \]

   if 5.00000000000000024e-5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{\mathsf{neg}\left(\left(1 + \sqrt{x + 1}\right)\right)} \]

      6. metadata-eval N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-+.f64 N/A

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

      16. flip-- N/A

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

      17. lower-neg.f64 N/A

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

      18. lower--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]

      6. sub-neg N/A

         \[\leadsto 0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right) + 1\right)} \]

      8. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right) - 1} \]

      9. neg-sub0 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 100.0

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt{x + 1} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0625, -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 4e-7)
     (fma (* x x) -0.125 (* x 0.5))
     (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((x / (1.0 + t_0)) <= 4e-7) {
		tmp = fma((x * x), -0.125, (x * 0.5));
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(x / Float64(1.0 + t_0)) <= 4e-7)
		tmp = fma(Float64(x * x), -0.125, Float64(x * 0.5));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[(x * x), $MachinePrecision] * -0.125 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.6

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 99.6

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.125, 0.5 \cdot x\right) \]

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.6

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.125, \color{blue}{x \cdot 0.5}\right) \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.125, x \cdot 0.5\right)} \]

   if 3.9999999999999998e-7 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{\mathsf{neg}\left(\left(1 + \sqrt{x + 1}\right)\right)} \]

      6. metadata-eval N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-+.f64 N/A

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

      16. flip-- N/A

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

      17. lower-neg.f64 N/A

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

      18. lower--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]

      6. sub-neg N/A

         \[\leadsto 0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right) + 1\right)} \]

      8. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right) - 1} \]

      9. neg-sub0 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 99.8

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 4e-7) (* x (fma x -0.125 0.5)) (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((x / (1.0 + t_0)) <= 4e-7) {
		tmp = x * fma(x, -0.125, 0.5);
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(x / Float64(1.0 + t_0)) <= 4e-7)
		tmp = Float64(x * fma(x, -0.125, 0.5));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 4e-7], N[(x * N[(x * -0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 3.9999999999999998e-7

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.6

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   if 3.9999999999999998e-7 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{\mathsf{neg}\left(\left(1 + \sqrt{x + 1}\right)\right)} \]

      6. metadata-eval N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-+.f64 N/A

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

      16. flip-- N/A

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

      17. lower-neg.f64 N/A

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

      18. lower--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]

      6. sub-neg N/A

         \[\leadsto 0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right) + 1\right)} \]

      8. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right) - 1} \]

      9. neg-sub0 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 99.8

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\sqrt{x + 1} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 5e-5)
   (* x (fma x -0.125 0.5))
   (+ (sqrt x) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-5) {
		tmp = x * fma(x, -0.125, 0.5);
	} else {
		tmp = sqrt(x) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 5e-5)
		tmp = Float64(x * fma(x, -0.125, 0.5));
	else
		tmp = Float64(sqrt(x) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], N[(x * N[(x * -0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.00000000000000024e-5

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.4

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   if 5.00000000000000024e-5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 98.3

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

   5. Simplified 98.3%

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

Alternative 6: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 5e-5)
   (* x (fma x -0.125 0.5))
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-5) {
		tmp = x * fma(x, -0.125, 0.5);
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 5e-5)
		tmp = Float64(x * fma(x, -0.125, 0.5));
	else
		tmp = sqrt(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], N[(x * N[(x * -0.125 + 0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.00000000000000024e-5

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 99.4

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)} \]

   if 5.00000000000000024e-5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{x}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f64 96.7

         \[\leadsto \color{blue}{\sqrt{x}} \]

   5. Simplified 96.7%

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

Alternative 7: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 5e-5) (* x 0.5) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-5) {
		tmp = x * 0.5;
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 + sqrt((x + 1.0d0)))) <= 5d-5) then
        tmp = x * 0.5d0
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 + Math.sqrt((x + 1.0)))) <= 5e-5) {
		tmp = x * 0.5;
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 + math.sqrt((x + 1.0)))) <= 5e-5:
		tmp = x * 0.5
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 5e-5)
		tmp = Float64(x * 0.5);
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-5)
		tmp = x * 0.5;
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-5], N[(x * 0.5), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.00000000000000024e-5

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 98.3

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if 5.00000000000000024e-5 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.2%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{x}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f64 96.7

         \[\leadsto \color{blue}{\sqrt{x}} \]

   5. Simplified 96.7%

      \[\leadsto \color{blue}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 6.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 1.5 \cdot 10^{-154}:\\ \;\;\;\;1 + -1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 1.5e-154) (+ 1.0 -1.0) 2.0))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 1.5e-154) {
		tmp = 1.0 + -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 + sqrt((x + 1.0d0)))) <= 1.5d-154) then
        tmp = 1.0d0 + (-1.0d0)
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 + Math.sqrt((x + 1.0)))) <= 1.5e-154) {
		tmp = 1.0 + -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 + math.sqrt((x + 1.0)))) <= 1.5e-154:
		tmp = 1.0 + -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 1.5e-154)
		tmp = Float64(1.0 + -1.0);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 1.5e-154)
		tmp = 1.0 + -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5e-154], N[(1.0 + -1.0), $MachinePrecision], 2.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 1.5000000000000001e-154

   1. Initial program 100.0%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - x}}{\mathsf{neg}\left(\left(1 + \sqrt{x + 1}\right)\right)} \]

      6. metadata-eval N/A

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

      7. associate--r+ N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lift-+.f64 N/A

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

      16. flip-- N/A

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

      17. lower-neg.f64 N/A

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

      18. lower--.f64 7.8

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

   4. Applied egg-rr 7.8%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]

      5. lift--.f64 N/A

         \[\leadsto 0 - \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]

      6. sub-neg N/A

         \[\leadsto 0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right) + 1\right)} \]

      8. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - \left(\mathsf{neg}\left(\sqrt{x + 1}\right)\right)\right) - 1} \]

      9. neg-sub0 N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 7.8

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

   6. Applied egg-rr 7.8%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 6.1%

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

   if 1.5000000000000001e-154 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.4%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 31.7

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

   5. Simplified 31.7%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2} \]
   7. Step-by-step derivation

   8. Simplified 6.7%

      \[\leadsto \color{blue}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 1.5 \cdot 10^{-154}:\\ \;\;\;\;1 + -1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 10: 67.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-*.f64 64.7

      \[\leadsto \color{blue}{0.5 \cdot x} \]

5. Simplified 64.7%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

6. Final simplification 64.7%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

Alternative 11: 4.9% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 65.4

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

5. Simplified 65.4%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.5, x, 2\right)}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2} \]
7. Step-by-step derivation

8. Simplified 4.8%

   \[\leadsto \color{blue}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))