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

Average Error: 0.2 → 0.2

Time: 3.7s

Precision: binary64

Cost: 6852

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 0.00018:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot -0.125\right) + x \cdot \left(x \cdot \left(x \cdot 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 0.00018)
   (+ (* 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 <= 0.00018) {
		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 <= 0.00018d0) 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 <= 0.00018) {
		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 <= 0.00018:
		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 <= 0.00018)
		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 <= 0.00018)
		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, 0.00018], 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 < 1.80000000000000011e-4

   1. Initial program 0.0

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

   2. Applied egg-rr 58.9

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

   3. Simplified 58.9

      \[\leadsto \color{blue}{\sqrt{x + 1} + -1} \]
      Proof
      (+.f64 (sqrt.f64 (+.f64 x 1)) -1): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 -1 (sqrt.f64 (+.f64 x 1)))): 14 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (sqrt.f64 (+.f64 x 1))): 0 points increase in error, 14 points decrease in error
      (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (sqrt.f64 (+.f64 x 1))))): 14 points increase in error, 0 points decrease in error
      (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 (sqrt.f64 (+.f64 x 1))))): 14 points increase in error, 0 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 (sqrt.f64 (+.f64 x 1))))): 14 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (neg.f64 (Rewrite<= *-inverses_binary64 (/.f64 (neg.f64 x) (neg.f64 x)))) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 14 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (neg.f64 x)) (neg.f64 x))) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 x)) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 x (neg.f64 x)) 1)) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 x (neg.f64 x))) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 14 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x (neg.f64 x)) (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 (sqrt.f64 (+.f64 x 1)))))): 14 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 x (neg.f64 x)) 1) (*.f64 (/.f64 x (neg.f64 x)) (neg.f64 (sqrt.f64 (+.f64 x 1)))))): 0 points increase in error, 14 points decrease in error

   4. Taylor expanded in x around 0 0.3

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right)} \]
      Proof
      (*.f64 x (+.f64 1/2 (*.f64 x (+.f64 -1/8 (*.f64 x 1/16))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x 1/16) -1/8))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 x 1/16) x) (*.f64 -1/8 x))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (+.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 1/16 x)) x) (*.f64 -1/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/16 (*.f64 x x))) (*.f64 -1/8 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (+.f64 1/2 (+.f64 (*.f64 1/16 (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (*.f64 -1/8 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x 1/2) (*.f64 x (+.f64 (*.f64 1/16 (pow.f64 x 2)) (*.f64 -1/8 x))))): 16 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 x)) (*.f64 x (+.f64 (*.f64 1/16 (pow.f64 x 2)) (*.f64 -1/8 x)))): 0 points increase in error, 16 points decrease in error
      (+.f64 (*.f64 1/2 x) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/16 (pow.f64 x 2)) x) (*.f64 (*.f64 -1/8 x) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/16 (*.f64 (pow.f64 x 2) x))) (*.f64 (*.f64 -1/8 x) x))): 6 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 1/16 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 x x)) x)) (*.f64 (*.f64 -1/8 x) x))): 16 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 1/16 (Rewrite<= unpow3_binary64 (pow.f64 x 3))) (*.f64 (*.f64 -1/8 x) x))): 0 points increase in error, 16 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 1/16 (pow.f64 x 3)) (Rewrite<= associate-*r*_binary64 (*.f64 -1/8 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/2 x) (+.f64 (*.f64 1/16 (pow.f64 x 3)) (*.f64 -1/8 (Rewrite<= unpow2_binary64 (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

   6. Applied egg-rr 0.3

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

   if 1.80000000000000011e-4 < x

   1. Initial program 0.5

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

   2. Applied egg-rr 0.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\sqrt{x + 1} + -1} \]
      Proof
      (+.f64 (sqrt.f64 (+.f64 x 1)) -1): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 -1 (sqrt.f64 (+.f64 x 1)))): 14 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (sqrt.f64 (+.f64 x 1))): 0 points increase in error, 14 points decrease in error
      (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 (sqrt.f64 (+.f64 x 1))))): 14 points increase in error, 0 points decrease in error
      (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 (sqrt.f64 (+.f64 x 1))))): 14 points increase in error, 0 points decrease in error
      (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 (sqrt.f64 (+.f64 x 1))))): 14 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= metadata-eval (neg.f64 1)) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (neg.f64 (Rewrite<= *-inverses_binary64 (/.f64 (neg.f64 x) (neg.f64 x)))) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 14 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (neg.f64 x)) (neg.f64 x))) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 (Rewrite=> remove-double-neg_binary64 x) (neg.f64 x)) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 x (neg.f64 x)) 1)) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 0 points increase in error, 14 points decrease in error
      (*.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 x (neg.f64 x))) (-.f64 1 (sqrt.f64 (+.f64 x 1)))): 14 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x (neg.f64 x)) (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 (sqrt.f64 (+.f64 x 1)))))): 14 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 x (neg.f64 x)) 1) (*.f64 (/.f64 x (neg.f64 x)) (neg.f64 (sqrt.f64 (+.f64 x 1)))))): 0 points increase in error, 14 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00018:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot -0.125\right) + x \cdot \left(x \cdot \left(x \cdot 0.0625\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	6848

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

Alternative 2
Error	20.3
Cost	448

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

Alternative 3
Error	20.7
Cost	192

\[\frac{x}{2} \]

Alternative 4
Error	60.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))