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

Percentage Accurate: 99.7% → 99.7%

Time: 7.3s

Alternatives: 10

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, 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 99.8%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 4e-9)
     (/ x (fma x (fma x -0.125 0.5) 2.0))
     (+ 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-9) {
		tmp = x / fma(x, fma(x, -0.125, 0.5), 2.0);
	} 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-9)
		tmp = Float64(x / fma(x, fma(x, -0.125, 0.5), 2.0));
	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-9], N[(x / N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + 2.0), $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))))) < 4.00000000000000025e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.6

         \[\leadsto \frac{x}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)}, 2\right)} \]

   5. Simplified 99.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 2\right)}} \]

   if 4.00000000000000025e-9 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   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

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0625, -0.125\right), x \cdot x, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 4e-9)
     (fma (fma x 0.0625 -0.125) (* x x) (* 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-9) {
		tmp = fma(fma(x, 0.0625, -0.125), (x * x), (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-9)
		tmp = fma(fma(x, 0.0625, -0.125), Float64(x * x), 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-9], N[(N[(x * 0.0625 + -0.125), $MachinePrecision] * N[(x * x), $MachinePrecision] + 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))))) < 4.00000000000000025e-9

   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 99.6

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0625, -0.125\right), 0.5\right)} \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   7. Applied egg-rr 99.6%

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

   if 4.00000000000000025e-9 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   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

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.0625, -0.125\right), x \cdot x, x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0625, -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 4e-9)
     (* 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)) <= 4e-9) {
		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)) <= 4e-9)
		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], 4e-9], 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))))) < 4.00000000000000025e-9

   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 99.6

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

   5. Simplified 99.6%

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

   if 4.00000000000000025e-9 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   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

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0625, -0.125\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{x}{1 + t\_0} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (/ x (+ 1.0 t_0)) 4e-9) (* 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-9) {
		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-9)
		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-9], 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))))) < 4.00000000000000025e-9

   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 4.00000000000000025e-9 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   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

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.001:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.001)
   (* 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)))) <= 0.001) {
		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)))) <= 0.001)
		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], 0.001], 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))))) < 1e-3

   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 99.3

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

   5. Simplified 99.3%

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

   if 1e-3 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

      \[\frac{x}{1 + \sqrt{x + 1}} \]
   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 99.6

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

   5. Simplified 99.6%

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

Alternative 7: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.001:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.001)
   (* x (fma x -0.125 0.5))
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.001) {
		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)))) <= 0.001)
		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], 0.001], 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))))) < 1e-3

   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 99.3

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

   5. Simplified 99.3%

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

   if 1e-3 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 8: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.001:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.001) (* x 0.5) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.001) {
		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)))) <= 0.001d0) 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)))) <= 0.001) {
		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)))) <= 0.001:
		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)))) <= 0.001)
		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)))) <= 0.001)
		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], 0.001], 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))))) < 1e-3

   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 98.5

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

   5. Simplified 98.5%

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

   if 1e-3 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-sqrt.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.001:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% 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 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
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 74.4

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

5. Simplified 74.4%

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

6. Final simplification 74.4%

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

Alternative 10: 4.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 + -1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{x}{1 + \sqrt{x + 1}} \]
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 31.8

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

4. Applied egg-rr 31.8%

   \[\leadsto \color{blue}{-\left(1 - \sqrt{x + 1}\right)} \]
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 31.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 31.8

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

6. Applied egg-rr 31.8%

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

7. Taylor expanded in x around 0

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

9. Simplified 4.5%

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

10. Final simplification 4.5%

    \[\leadsto 1 + -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))