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

Percentage Accurate: 99.7% → 99.8%

Time: 8.4s

Alternatives: 8

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00115:\\ \;\;\;\;\frac{x}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{0.5} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00115)
   (/ x (+ (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625))))) 2.0))
   (+ (pow (+ x 1.0) 0.5) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00115) {
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	} else {
		tmp = pow((x + 1.0), 0.5) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00115d0) then
        tmp = x / ((x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0))))) + 2.0d0)
    else
        tmp = ((x + 1.0d0) ** 0.5d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00115) {
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	} else {
		tmp = Math.pow((x + 1.0), 0.5) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00115:
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0)
	else:
		tmp = math.pow((x + 1.0), 0.5) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00115)
		tmp = Float64(x / Float64(Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625))))) + 2.0));
	else
		tmp = Float64((Float64(x + 1.0) ^ 0.5) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00115)
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	else
		tmp = ((x + 1.0) ^ 0.5) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00115], N[(x / N[(N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.5], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00115

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.8%

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

   5. Simplified 99.8%

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

   if 0.00115 < x

   1. Initial program 99.2%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

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

      3. metadata-eval N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. rem-square-sqrt N/A

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

      8. distribute-neg-frac2 N/A

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

      9. flip-- N/A

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

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

          \[\leadsto \mathsf{neg.f64}\left(\left(1 - \sqrt{x + 1}\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{\_.f64}\left(1, \left(\sqrt{x + 1}\right)\right)\right) \]

      12. pow1/2 N/A

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

      13. pow-lowering-pow.f64 N/A

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

      14. +-lowering-+.f64 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{-\left(1 - {\left(x + 1\right)}^{0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00115:\\ \;\;\;\;\frac{x}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{0.5} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x}{1 + \left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.0)
   (/ x (+ 1.0 (+ 1.0 (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625))))))))
   (+ (sqrt x) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))));
	} else {
		tmp = sqrt(x) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.0d0) then
        tmp = x / (1.0d0 + (1.0d0 + (x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0)))))))
    else
        tmp = sqrt(x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))));
	} else {
		tmp = Math.sqrt(x) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.0:
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))))
	else:
		tmp = math.sqrt(x) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(x / Float64(1.0 + Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625))))))));
	else
		tmp = Float64(sqrt(x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.0)
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))));
	else
		tmp = sqrt(x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.0], N[(x / N[(1.0 + N[(1.0 + N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.3%

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

   5. Simplified 99.3%

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

   if 3 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. sqrt-lowering-sqrt.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(x\right), -1\right) \]

   5. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x}{1 + \left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6:\\ \;\;\;\;\frac{x}{1 + \left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.6)
   (/ x (+ 1.0 (+ 1.0 (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625))))))))
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.6) {
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))));
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.6d0) then
        tmp = x / (1.0d0 + (1.0d0 + (x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0)))))))
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.6) {
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))));
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.6:
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))))
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.6)
		tmp = Float64(x / Float64(1.0 + Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625))))))));
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.6)
		tmp = x / (1.0 + (1.0 + (x * (0.5 + (x * (-0.125 + (x * 0.0625)))))));
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.6], N[(x / N[(1.0 + N[(1.0 + N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.3%

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

   5. Simplified 99.3%

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

   if 3.60000000000000009 < x

   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. sqrt-lowering-sqrt.f64 97.8%

         \[\leadsto \mathsf{sqrt.f64}\left(x\right) \]

   5. Simplified 97.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6:\\ \;\;\;\;\frac{x}{1 + \left(1 + x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. *-lowering-*.f64 67.7%

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

5. Simplified 67.7%

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

6. Final simplification 67.7%

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

Alternative 6: 68.6% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{2 + x \cdot 0.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. *-lowering-*.f64 67.7%

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

5. Simplified 67.7%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]

6. Final simplification 67.7%

   \[\leadsto \frac{x}{2 + x \cdot 0.5} \]
7. Add Preprocessing

Alternative 7: 67.9% accurate, 35.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ x 2.0))

C

double code(double x) {
	return x / 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return x / 2.0

Julia

function code(x)
	return Float64(x / 2.0)
end

MATLAB

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

Wolfram

code[x_] := N[(x / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 66.8%

   \[\leadsto \frac{x}{\color{blue}{2}} \]
6. Add Preprocessing

Alternative 8: 4.8% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

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

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 66.6%

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

5. Simplified 66.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 67.7%

   \[\leadsto \frac{x}{x \cdot \color{blue}{0.5} + 2} \]

9. Taylor expanded in x around inf

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

11. Simplified 4.8%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))