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

Average Accuracy: 99.7% → 99.7%

Time: 8.7s

Precision: binary64

Cost: 6852

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot -0.125\right) + x \cdot \left(x \cdot \left(x \cdot 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 0.000205)
   (+ (* x (+ 0.5 (* x -0.125))) (* x (* x (* x 0.0625))))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 0.000205) {
		tmp = (x * (0.5 + (x * -0.125))) + (x * (x * (x * 0.0625)));
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000205d0) then
        tmp = (x * (0.5d0 + (x * (-0.125d0)))) + (x * (x * (x * 0.0625d0)))
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= 0.000205) {
		tmp = (x * (0.5 + (x * -0.125))) + (x * (x * (x * 0.0625)));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= 0.000205:
		tmp = (x * (0.5 + (x * -0.125))) + (x * (x * (x * 0.0625)))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= 0.000205)
		tmp = Float64(Float64(x * Float64(0.5 + Float64(x * -0.125))) + Float64(x * Float64(x * Float64(x * 0.0625))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000205)
		tmp = (x * (0.5 + (x * -0.125))) + (x * (x * (x * 0.0625)));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, 0.000205], N[(N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.05e-4

   1. Initial program 100.0%

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

   2. Applied egg-rr 8.0%

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

      [Start] 100.0	\[ \frac{x}{1 + \sqrt{x + 1}} \]
      flip-+ [=>] 4.0	\[ \frac{x}{\color{blue}{\frac{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}{1 - \sqrt{x + 1}}}} \]
      associate-/r/ [=>] 4.0	\[ \color{blue}{\frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(1 - \sqrt{x + 1}\right)} \]
      sub-neg [=>] 4.0	\[ \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \color{blue}{\left(1 + \left(-\sqrt{x + 1}\right)\right)} \]
      distribute-lft-in [=>] 2.8	\[ \color{blue}{\frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right)} \]
      metadata-eval [=>] 2.8	\[ \frac{x}{\color{blue}{1} - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      add-sqr-sqrt [<=] 2.1	\[ \frac{x}{1 - \color{blue}{\left(x + 1\right)}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      +-commutative [=>] 2.1	\[ \frac{x}{1 - \color{blue}{\left(1 + x\right)}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      associate--r+ [=>] 1.4	\[ \frac{x}{\color{blue}{\left(1 - 1\right) - x}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      metadata-eval [=>] 1.4	\[ \frac{x}{\color{blue}{0} - x} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      neg-sub0 [<=] 1.4	\[ \frac{x}{\color{blue}{-x}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]

   3. Simplified 8.0%

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

      [Start] 8.0	\[ \frac{x}{-x} \cdot 1 + \frac{x}{-x} \cdot \left(-\sqrt{x + 1}\right) \]
      distribute-lft-out [=>] 8.0	\[ \color{blue}{\frac{x}{-x} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right)} \]
      remove-double-neg [<=] 8.0	\[ \frac{\color{blue}{-\left(-x\right)}}{-x} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      neg-mul-1 [=>] 8.0	\[ \frac{\color{blue}{-1 \cdot \left(-x\right)}}{-x} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      neg-mul-1 [=>] 8.0	\[ \frac{-1 \cdot \left(-x\right)}{\color{blue}{-1 \cdot x}} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      times-frac [=>] 8.0	\[ \color{blue}{\left(\frac{-1}{-1} \cdot \frac{-x}{x}\right)} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      metadata-eval [=>] 8.0	\[ \left(\color{blue}{1} \cdot \frac{-x}{x}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      remove-double-neg [<=] 8.0	\[ \left(1 \cdot \frac{-x}{\color{blue}{-\left(-x\right)}}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      distribute-frac-neg [=>] 8.0	\[ \left(1 \cdot \color{blue}{\left(-\frac{x}{-\left(-x\right)}\right)}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      remove-double-neg [=>] 8.0	\[ \left(1 \cdot \left(-\frac{x}{\color{blue}{x}}\right)\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      *-inverses [=>] 8.0	\[ \left(1 \cdot \left(-\color{blue}{1}\right)\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      metadata-eval [=>] 8.0	\[ \left(1 \cdot \color{blue}{-1}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      metadata-eval [=>] 8.0	\[ \color{blue}{-1} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      sub-neg [<=] 8.0	\[ -1 \cdot \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]
      neg-mul-1 [<=] 8.0	\[ \color{blue}{-\left(1 - \sqrt{x + 1}\right)} \]
      neg-sub0 [=>] 8.0	\[ \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]
      associate--r- [=>] 8.0	\[ \color{blue}{\left(0 - 1\right) + \sqrt{x + 1}} \]
      metadata-eval [=>] 8.0	\[ \color{blue}{-1} + \sqrt{x + 1} \]
      +-commutative [<=] 8.0	\[ \color{blue}{\sqrt{x + 1} + -1} \]

   4. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-0.125 \cdot {x}^{2} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right)} \]

   5. Simplified 99.6%

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

      [Start] 99.6	\[ -0.125 \cdot {x}^{2} + \left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right) \]
      +-commutative [=>] 99.6	\[ \color{blue}{\left(0.5 \cdot x + 0.0625 \cdot {x}^{3}\right) + -0.125 \cdot {x}^{2}} \]
      associate-+l+ [=>] 99.6	\[ \color{blue}{0.5 \cdot x + \left(0.0625 \cdot {x}^{3} + -0.125 \cdot {x}^{2}\right)} \]
      *-commutative [=>] 99.6	\[ \color{blue}{x \cdot 0.5} + \left(0.0625 \cdot {x}^{3} + -0.125 \cdot {x}^{2}\right) \]
      unpow3 [=>] 99.6	\[ x \cdot 0.5 + \left(0.0625 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + -0.125 \cdot {x}^{2}\right) \]
      unpow2 [<=] 99.6	\[ x \cdot 0.5 + \left(0.0625 \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + -0.125 \cdot {x}^{2}\right) \]
      associate-*r* [=>] 99.6	\[ x \cdot 0.5 + \left(\color{blue}{\left(0.0625 \cdot {x}^{2}\right) \cdot x} + -0.125 \cdot {x}^{2}\right) \]
      unpow2 [=>] 99.6	\[ x \cdot 0.5 + \left(\left(0.0625 \cdot {x}^{2}\right) \cdot x + -0.125 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
      associate-*r* [=>] 99.6	\[ x \cdot 0.5 + \left(\left(0.0625 \cdot {x}^{2}\right) \cdot x + \color{blue}{\left(-0.125 \cdot x\right) \cdot x}\right) \]
      distribute-rgt-out [=>] 99.6	\[ x \cdot 0.5 + \color{blue}{x \cdot \left(0.0625 \cdot {x}^{2} + -0.125 \cdot x\right)} \]
      distribute-lft-out [=>] 99.6	\[ \color{blue}{x \cdot \left(0.5 + \left(0.0625 \cdot {x}^{2} + -0.125 \cdot x\right)\right)} \]
      unpow2 [=>] 99.6	\[ x \cdot \left(0.5 + \left(0.0625 \cdot \color{blue}{\left(x \cdot x\right)} + -0.125 \cdot x\right)\right) \]
      associate-*r* [=>] 99.6	\[ x \cdot \left(0.5 + \left(\color{blue}{\left(0.0625 \cdot x\right) \cdot x} + -0.125 \cdot x\right)\right) \]
      *-commutative [<=] 99.6	\[ x \cdot \left(0.5 + \left(\color{blue}{\left(x \cdot 0.0625\right)} \cdot x + -0.125 \cdot x\right)\right) \]
      distribute-rgt-out [=>] 99.6	\[ x \cdot \left(0.5 + \color{blue}{x \cdot \left(x \cdot 0.0625 + -0.125\right)}\right) \]
      +-commutative [=>] 99.6	\[ x \cdot \left(0.5 + x \cdot \color{blue}{\left(-0.125 + x \cdot 0.0625\right)}\right) \]

   6. Applied egg-rr 99.6%

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

      [Start] 99.6	\[ x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) \]
      distribute-rgt-in [=>] 99.6	\[ x \cdot \left(0.5 + \color{blue}{\left(-0.125 \cdot x + \left(x \cdot 0.0625\right) \cdot x\right)}\right) \]
      associate-+r+ [=>] 99.6	\[ x \cdot \color{blue}{\left(\left(0.5 + -0.125 \cdot x\right) + \left(x \cdot 0.0625\right) \cdot x\right)} \]
      distribute-lft-in [=>] 99.6	\[ \color{blue}{x \cdot \left(0.5 + -0.125 \cdot x\right) + x \cdot \left(\left(x \cdot 0.0625\right) \cdot x\right)} \]
      *-commutative [=>] 99.6	\[ x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right) + x \cdot \left(\left(x \cdot 0.0625\right) \cdot x\right) \]
      *-commutative [=>] 99.6	\[ x \cdot \left(0.5 + x \cdot -0.125\right) + x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.0625\right)\right)} \]

   if 2.05e-4 < x

   1. Initial program 99.2%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.2	\[ \frac{x}{1 + \sqrt{x + 1}} \]
      flip-+ [=>] 99.2	\[ \frac{x}{\color{blue}{\frac{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}{1 - \sqrt{x + 1}}}} \]
      associate-/r/ [=>] 99.1	\[ \color{blue}{\frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(1 - \sqrt{x + 1}\right)} \]
      sub-neg [=>] 99.1	\[ \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \color{blue}{\left(1 + \left(-\sqrt{x + 1}\right)\right)} \]
      distribute-lft-in [=>] 99.0	\[ \color{blue}{\frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right)} \]
      metadata-eval [=>] 99.0	\[ \frac{x}{\color{blue}{1} - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      add-sqr-sqrt [<=] 99.0	\[ \frac{x}{1 - \color{blue}{\left(x + 1\right)}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      +-commutative [=>] 99.0	\[ \frac{x}{1 - \color{blue}{\left(1 + x\right)}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      associate--r+ [=>] 98.9	\[ \frac{x}{\color{blue}{\left(1 - 1\right) - x}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      metadata-eval [=>] 98.9	\[ \frac{x}{\color{blue}{0} - x} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]
      neg-sub0 [<=] 98.9	\[ \frac{x}{\color{blue}{-x}} \cdot 1 + \frac{x}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}} \cdot \left(-\sqrt{x + 1}\right) \]

   3. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{x}{-x} \cdot 1 + \frac{x}{-x} \cdot \left(-\sqrt{x + 1}\right) \]
      distribute-lft-out [=>] 99.9	\[ \color{blue}{\frac{x}{-x} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right)} \]
      remove-double-neg [<=] 99.9	\[ \frac{\color{blue}{-\left(-x\right)}}{-x} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      neg-mul-1 [=>] 99.9	\[ \frac{\color{blue}{-1 \cdot \left(-x\right)}}{-x} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      neg-mul-1 [=>] 99.9	\[ \frac{-1 \cdot \left(-x\right)}{\color{blue}{-1 \cdot x}} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      times-frac [=>] 99.9	\[ \color{blue}{\left(\frac{-1}{-1} \cdot \frac{-x}{x}\right)} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      metadata-eval [=>] 99.9	\[ \left(\color{blue}{1} \cdot \frac{-x}{x}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      remove-double-neg [<=] 99.9	\[ \left(1 \cdot \frac{-x}{\color{blue}{-\left(-x\right)}}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      distribute-frac-neg [=>] 99.9	\[ \left(1 \cdot \color{blue}{\left(-\frac{x}{-\left(-x\right)}\right)}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      remove-double-neg [=>] 99.9	\[ \left(1 \cdot \left(-\frac{x}{\color{blue}{x}}\right)\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      *-inverses [=>] 99.9	\[ \left(1 \cdot \left(-\color{blue}{1}\right)\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      metadata-eval [=>] 99.9	\[ \left(1 \cdot \color{blue}{-1}\right) \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      metadata-eval [=>] 99.9	\[ \color{blue}{-1} \cdot \left(1 + \left(-\sqrt{x + 1}\right)\right) \]
      sub-neg [<=] 99.9	\[ -1 \cdot \color{blue}{\left(1 - \sqrt{x + 1}\right)} \]
      neg-mul-1 [<=] 99.9	\[ \color{blue}{-\left(1 - \sqrt{x + 1}\right)} \]
      neg-sub0 [=>] 99.9	\[ \color{blue}{0 - \left(1 - \sqrt{x + 1}\right)} \]
      associate--r- [=>] 99.9	\[ \color{blue}{\left(0 - 1\right) + \sqrt{x + 1}} \]
      metadata-eval [=>] 99.9	\[ \color{blue}{-1} + \sqrt{x + 1} \]
      +-commutative [<=] 99.9	\[ \color{blue}{\sqrt{x + 1} + -1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot -0.125\right) + x \cdot \left(x \cdot \left(x \cdot 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	6848

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

Alternative 2
Accuracy	68.3%
Cost	576

\[x \cdot \frac{1}{2 + x \cdot 0.5} \]

Alternative 3
Accuracy	68.3%
Cost	448

\[\frac{x}{2 + x \cdot 0.5} \]

Alternative 4
Accuracy	67.6%
Cost	192

\[x \cdot 0.5 \]

Alternative 5
Accuracy	4.9%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))