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

Percentage Accurate: 99.7% → 99.8%

Time: 5.6s

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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{x}{t\_0 + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0390625, x, 0.0625\right), x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (/ x (+ t_0 1.0)) 0.0002)
     (* (fma (fma (fma -0.0390625 x 0.0625) x -0.125) x 0.5) x)
     (- t_0 1.0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((x / (t_0 + 1.0)) <= 0.0002) {
		tmp = fma(fma(fma(-0.0390625, x, 0.0625), x, -0.125), x, 0.5) * x;
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(x / Float64(t_0 + 1.0)) <= 0.0002)
		tmp = Float64(fma(fma(fma(-0.0390625, x, 0.0625), x, -0.125), x, 0.5) * x);
	else
		tmp = Float64(t_0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(N[(N[(-0.0390625 * x + 0.0625), $MachinePrecision] * x + -0.125), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $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))))) < 2.0000000000000001e-4

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 2.0000000000000001e-4 < (/.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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate--l+ N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. rem-exp-log N/A

          \[\leadsto \frac{x}{x + 0} \cdot \left(\color{blue}{e^{\log \left(\sqrt{x + 1}\right)}} - 1\right) \]

      17. lower-expm1.f64 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \color{blue}{\mathsf{expm1}\left(\log \left(\sqrt{x + 1}\right)\right)} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{x + 1}\right)}\right) \]

      19. pow1/2 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left({\left(x + 1\right)}^{\frac{1}{2}}\right)}\right) \]

      20. pow-to-exp N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{2}}\right)}\right) \]

   4. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{x}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      5. *-inverses N/A

         \[\leadsto \color{blue}{1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      6. lift-expm1.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} - 1\right)} \]

      7. sub-neg N/A

         \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 1 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. lift-log1p.f64 N/A

         \[\leadsto 1 \cdot \left(e^{\color{blue}{\log \left(1 + x\right)} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. exp-to-pow N/A

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

      11. +-commutative N/A

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

      12. pow1/2 N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0390625, x, 0.0625\right), x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{x}{t\_0 + 1} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (/ x (+ t_0 1.0)) 2e-14)
     (* (fma (fma 0.0625 x -0.125) x 0.5) x)
     (- t_0 1.0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((x / (t_0 + 1.0)) <= 2e-14) {
		tmp = fma(fma(0.0625, x, -0.125), x, 0.5) * x;
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(x / Float64(t_0 + 1.0)) <= 2e-14)
		tmp = Float64(fma(fma(0.0625, x, -0.125), x, 0.5) * x);
	else
		tmp = Float64(t_0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], 2e-14], N[(N[(N[(0.0625 * x + -0.125), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $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))))) < 2e-14

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 2e-14 < (/.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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate--l+ N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. rem-exp-log N/A

          \[\leadsto \frac{x}{x + 0} \cdot \left(\color{blue}{e^{\log \left(\sqrt{x + 1}\right)}} - 1\right) \]

      17. lower-expm1.f64 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \color{blue}{\mathsf{expm1}\left(\log \left(\sqrt{x + 1}\right)\right)} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{x + 1}\right)}\right) \]

      19. pow1/2 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left({\left(x + 1\right)}^{\frac{1}{2}}\right)}\right) \]

      20. pow-to-exp N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{2}}\right)}\right) \]

   4. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{x}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      5. *-inverses N/A

         \[\leadsto \color{blue}{1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      6. lift-expm1.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} - 1\right)} \]

      7. sub-neg N/A

         \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 1 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. lift-log1p.f64 N/A

         \[\leadsto 1 \cdot \left(e^{\color{blue}{\log \left(1 + x\right)} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. exp-to-pow N/A

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

      11. +-commutative N/A

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

      12. pow1/2 N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{x}{t\_0 + 1} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot x, x, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (/ x (+ t_0 1.0)) 2e-14)
     (fma (* -0.125 x) x (* 0.5 x))
     (- t_0 1.0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((x / (t_0 + 1.0)) <= 2e-14) {
		tmp = fma((-0.125 * x), x, (0.5 * x));
	} else {
		tmp = t_0 - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(x / Float64(t_0 + 1.0)) <= 2e-14)
		tmp = fma(Float64(-0.125 * x), x, Float64(0.5 * x));
	else
		tmp = Float64(t_0 - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], 2e-14], N[(N[(-0.125 * x), $MachinePrecision] * x + N[(0.5 * x), $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))))) < 2e-14

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

   if 2e-14 < (/.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

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-+.f64 N/A

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

      12. metadata-eval N/A

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

      13. associate--l+ N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 N/A

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

      16. rem-exp-log N/A

          \[\leadsto \frac{x}{x + 0} \cdot \left(\color{blue}{e^{\log \left(\sqrt{x + 1}\right)}} - 1\right) \]

      17. lower-expm1.f64 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \color{blue}{\mathsf{expm1}\left(\log \left(\sqrt{x + 1}\right)\right)} \]

      18. lift-sqrt.f64 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{x + 1}\right)}\right) \]

      19. pow1/2 N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left({\left(x + 1\right)}^{\frac{1}{2}}\right)}\right) \]

      20. pow-to-exp N/A

          \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{2}}\right)}\right) \]

   4. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{x}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      5. *-inverses N/A

         \[\leadsto \color{blue}{1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

      6. lift-expm1.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} - 1\right)} \]

      7. sub-neg N/A

         \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto 1 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      9. lift-log1p.f64 N/A

         \[\leadsto 1 \cdot \left(e^{\color{blue}{\log \left(1 + x\right)} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

      10. exp-to-pow N/A

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

      11. +-commutative N/A

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

      12. pow1/2 N/A

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

      13. distribute-rgt-in N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. +-commutative N/A

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

      22. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot x, x, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot x, x, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 0.0002)
   (fma (* -0.125 x) x (* 0.5 x))
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 0.0002) {
		tmp = fma((-0.125 * x), x, (0.5 * x));
	} else {
		tmp = sqrt(x) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(sqrt(Float64(1.0 + x)) + 1.0)) <= 0.0002)
		tmp = fma(Float64(-0.125 * x), x, Float64(0.5 * x));
	else
		tmp = Float64(sqrt(x) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(-0.125 * x), $MachinePrecision] * x + N[(0.5 * x), $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))))) < 2.0000000000000001e-4

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.2%

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

   if 2.0000000000000001e-4 < (/.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. lower--.f64 N/A

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

      2. lower-sqrt.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot x, x, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 0.0002)
   (* (fma -0.125 x 0.5) x)
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 0.0002) {
		tmp = fma(-0.125, x, 0.5) * x;
	} else {
		tmp = sqrt(x) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(sqrt(Float64(1.0 + x)) + 1.0)) <= 0.0002)
		tmp = Float64(fma(-0.125, x, 0.5) * x);
	else
		tmp = Float64(sqrt(x) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x), $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))))) < 2.0000000000000001e-4

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.0000000000000001e-4 < (/.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. lower--.f64 N/A

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

      2. lower-sqrt.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 0.0002)
   (* (fma -0.125 x 0.5) x)
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 0.0002) {
		tmp = fma(-0.125, x, 0.5) * x;
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(sqrt(Float64(1.0 + x)) + 1.0)) <= 0.0002)
		tmp = Float64(fma(-0.125, x, 0.5) * x);
	else
		tmp = sqrt(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x), $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))))) < 2.0000000000000001e-4

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 2.0000000000000001e-4 < (/.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 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 0.0002) (* 0.5 x) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 0.0002) {
		tmp = 0.5 * x;
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (sqrt((1.0d0 + x)) + 1.0d0)) <= 0.0002d0) then
        tmp = 0.5d0 * x
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (Math.sqrt((1.0 + x)) + 1.0)) <= 0.0002) {
		tmp = 0.5 * x;
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (math.sqrt((1.0 + x)) + 1.0)) <= 0.0002:
		tmp = 0.5 * x
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(sqrt(Float64(1.0 + x)) + 1.0)) <= 0.0002)
		tmp = Float64(0.5 * x);
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 0.0002)
		tmp = 0.5 * x;
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 0.0002], N[(0.5 * x), $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))))) < 2.0000000000000001e-4

   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 2.0000000000000001e-4 < (/.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 95.8

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

   5. Applied rewrites 95.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 0.0002:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 9: 68.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.5 * x

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * x), $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 69.5

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

5. Applied rewrites 69.5%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. 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 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-+.f64 N/A

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

   12. metadata-eval N/A

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

   13. associate--l+ N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f64 N/A

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

   16. rem-exp-log N/A

       \[\leadsto \frac{x}{x + 0} \cdot \left(\color{blue}{e^{\log \left(\sqrt{x + 1}\right)}} - 1\right) \]

   17. lower-expm1.f64 N/A

       \[\leadsto \frac{x}{x + 0} \cdot \color{blue}{\mathsf{expm1}\left(\log \left(\sqrt{x + 1}\right)\right)} \]

   18. lift-sqrt.f64 N/A

       \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\sqrt{x + 1}\right)}\right) \]

   19. pow1/2 N/A

       \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left({\left(x + 1\right)}^{\frac{1}{2}}\right)}\right) \]

   20. pow-to-exp N/A

       \[\leadsto \frac{x}{x + 0} \cdot \mathsf{expm1}\left(\log \color{blue}{\left(e^{\log \left(x + 1\right) \cdot \frac{1}{2}}\right)}\right) \]

4. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot 0.5\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + 0} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{x + 0}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

   4. +-rgt-identity N/A

      \[\leadsto \frac{x}{\color{blue}{x}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

   5. *-inverses N/A

      \[\leadsto \color{blue}{1} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}\right) \]

   6. lift-expm1.f64 N/A

      \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} - 1\right)} \]

   7. sub-neg N/A

      \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto 1 \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{2}}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   9. lift-log1p.f64 N/A

      \[\leadsto 1 \cdot \left(e^{\color{blue}{\log \left(1 + x\right)} \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right) \]

   10. exp-to-pow N/A

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

   11. +-commutative N/A

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

   12. pow1/2 N/A

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

   13. distribute-rgt-in N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. sub-neg N/A

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

   19. lower--.f64 N/A

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

   20. lower-sqrt.f64 N/A

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

   21. +-commutative N/A

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

   22. lower-+.f64 37.0

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

6. Applied rewrites 37.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 4.5%

   \[\leadsto \color{blue}{1} - 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))