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

Percentage Accurate: 99.7% → 99.7%

Time: 4.8s

Alternatives: 9

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

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

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 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{x}{t\_0 + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (/ x (+ t_0 1.0)) 5e-6)
     (* (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)) <= 5e-6) {
		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)) <= 5e-6)
		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], 5e-6], 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))))) < 5.00000000000000041e-6

   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 99.6

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

   5. Applied rewrites 99.6%

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

   if 5.00000000000000041e-6 < (/.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. frac-2neg N/A

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

      3. neg-sub0 N/A

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

      4. metadata-eval N/A

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

      5. associate--r+ N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lift-+.f64 N/A

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

      14. flip-- N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 100.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. lift--.f64 N/A

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

      4. associate--r- N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0625, x, -0.125\right), x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{x}{t\_0 + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (/ x (+ t_0 1.0)) 5e-6)
     (fma (* x x) -0.125 (* 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)) <= 5e-6) {
		tmp = fma((x * x), -0.125, (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)) <= 5e-6)
		tmp = fma(Float64(x * x), -0.125, 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], 5e-6], N[(N[(x * x), $MachinePrecision] * -0.125 + 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))))) < 5.00000000000000041e-6

   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 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (/.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. frac-2neg N/A

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

      3. neg-sub0 N/A

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

      4. metadata-eval N/A

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

      5. associate--r+ N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. lift-+.f64 N/A

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

      14. flip-- N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 100.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. lift--.f64 N/A

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

      4. associate--r- N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 5e-6)
   (fma (* x x) -0.125 (* 0.5 x))
   (- (sqrt x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 5e-6) {
		tmp = fma((x * x), -0.125, (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)) <= 5e-6)
		tmp = fma(Float64(x * x), -0.125, 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], 5e-6], N[(N[(x * x), $MachinePrecision] * -0.125 + 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))))) < 5.00000000000000041e-6

   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 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.4%

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

   if 5.00000000000000041e-6 < (/.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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.125, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 5e-6)
   (* (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)) <= 5e-6) {
		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)) <= 5e-6)
		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], 5e-6], 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))))) < 5.00000000000000041e-6

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 5.00000000000000041e-6 < (/.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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} - 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 5e-6)
   (* (fma -0.125 x 0.5) x)
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 5e-6) {
		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)) <= 5e-6)
		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], 5e-6], 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))))) < 5.00000000000000041e-6

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 5.00000000000000041e-6 < (/.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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x, 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ x (+ (sqrt (+ 1.0 x)) 1.0)) 5e-6) (* 0.5 x) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (sqrt((1.0 + x)) + 1.0)) <= 5e-6) {
		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)) <= 5d-6) 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)) <= 5e-6) {
		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)) <= 5e-6:
		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)) <= 5e-6)
		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)) <= 5e-6)
		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], 5e-6], 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))))) < 5.00000000000000041e-6

   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. *-commutative N/A

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

      2. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 5.00000000000000041e-6 < (/.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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{1 + x} + 1} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% 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. *-commutative N/A

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

   2. lower-*.f64 68.8

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

5. Applied rewrites 68.8%

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

6. Final simplification 68.8%

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

Alternative 9: 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. frac-2neg N/A

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

   3. neg-sub0 N/A

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

   4. metadata-eval N/A

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

   5. associate--r+ N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. distribute-neg-frac2 N/A

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

   13. lift-+.f64 N/A

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

   14. flip-- N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 37.8

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 37.8

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

4. Applied rewrites 37.8%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. lift--.f64 N/A

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

   4. associate--r- N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 37.8

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

6. Applied rewrites 37.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 4.5%

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

10. Final simplification 4.5%

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

Reproduce

herbie shell --seed 2024257 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))