Average Error: 0.2 → 0.2
Time: 4.7s
Precision: binary64
Cost: 6852
\[\frac{x}{1 + \sqrt{x + 1}} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]
(FPCore (x) :precision binary64 (/ x (+ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x 9.346374982514411e-7)
   (* x (+ 0.5 (+ (* x (* x 0.0625)) (* x -0.125))))
   (+ (sqrt (+ x 1.0)) -1.0)))
double code(double x) {
	return x / (1.0 + sqrt((x + 1.0)));
}
↓
double code(double x) {
	double tmp;
	if (x <= 9.346374982514411e-7) {
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

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 <= 9.346374982514411d-7) then
        tmp = x * (0.5d0 + ((x * (x * 0.0625d0)) + (x * (-0.125d0))))
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

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 <= 9.346374982514411e-7) {
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

def code(x):
	return x / (1.0 + math.sqrt((x + 1.0)))
↓
def code(x):
	tmp = 0
	if x <= 9.346374982514411e-7:
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

function code(x)
	return Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	tmp = 0.0
	if (x <= 9.346374982514411e-7)
		tmp = Float64(x * Float64(0.5 + Float64(Float64(x * Float64(x * 0.0625)) + Float64(x * -0.125))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

function tmp = code(x)
	tmp = x / (1.0 + sqrt((x + 1.0)));
end
↓
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.346374982514411e-7)
		tmp = x * (0.5 + ((x * (x * 0.0625)) + (x * -0.125)));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

code[x_] := N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := If[LessEqual[x, 9.346374982514411e-7], N[(x * N[(0.5 + N[(N[(x * N[(x * 0.0625), $MachinePrecision]), $MachinePrecision] + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

\frac{x}{1 + \sqrt{x + 1}}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\
\;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{x + 1} + -1\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 9.3463749825144108e-7

    1. Initial program 0.0

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

    2. Taylor expanded in x around 0 0.2

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

    3. Simplified0.2

      \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \mathsf{fma}\left(x, 0.0625, -0.125\right)\right)} \]
      Proof
      (*.f64 x (+.f64 1/2 (*.f64 x (fma.f64 x 1/16 -1/8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (*.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x 1/16) -1/8))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (*.f64 x (Rewrite=> +-commutative_binary64 (+.f64 -1/8 (*.f64 x 1/16)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x -1/8) (*.f64 x (*.f64 x 1/16)))))): 2 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (+.f64 (*.f64 x -1/8) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) 1/16))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (+.f64 (*.f64 x -1/8) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 1/16)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x 1/2) (*.f64 x (+.f64 (*.f64 x -1/8) (*.f64 (pow.f64 x 2) 1/16))))): 1 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 x)) (*.f64 x (+.f64 (*.f64 x -1/8) (*.f64 (pow.f64 x 2) 1/16)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x (*.f64 x -1/8)) (*.f64 x (*.f64 (pow.f64 x 2) 1/16))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) -1/8)) (*.f64 x (*.f64 (pow.f64 x 2) 1/16)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) -1/8) (*.f64 x (*.f64 (pow.f64 x 2) 1/16)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/8 (pow.f64 x 2))) (*.f64 x (*.f64 (pow.f64 x 2) 1/16)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 -1/8 (pow.f64 x 2)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (pow.f64 x 2)) 1/16)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 -1/8 (pow.f64 x 2)) (*.f64 (*.f64 x (Rewrite=> unpow2_binary64 (*.f64 x x))) 1/16))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 -1/8 (pow.f64 x 2)) (*.f64 (Rewrite<= cube-mult_binary64 (pow.f64 x 3)) 1/16))): 2 points increase in error, 6 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 -1/8 (pow.f64 x 2)) (Rewrite<= *-commutative_binary64 (*.f64 1/16 (pow.f64 x 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/16 (pow.f64 x 3)) (*.f64 -1/8 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 1/2 x) (*.f64 1/16 (pow.f64 x 3))) (*.f64 -1/8 (pow.f64 x 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/8 (pow.f64 x 2)) (+.f64 (*.f64 1/2 x) (*.f64 1/16 (pow.f64 x 3))))): 0 points increase in error, 0 points decrease in error

    4. Applied egg-rr0.2

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

    if 9.3463749825144108e-7 < x

    1. Initial program 0.5

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

    2. Applied egg-rr0.2

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

    3. Applied egg-rr0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.346374982514411 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot \left(x \cdot 0.0625\right) + x \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternatives

Alternative 1
Error0.2
Cost6848
\[\frac{x}{1 + \sqrt{x + 1}} \]
Alternative 2
Error20.1
Cost448
\[\frac{x}{x \cdot 0.5 + 2} \]
Alternative 3
Error20.5
Cost192
\[\frac{x}{2} \]
Alternative 4
Error60.9
Cost64
\[2 \]

Error

Reproduce

herbie shell --seed 2022294 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))