ln(1 + x)

Average Error: 38.6 → 0.0

Time: 2.5s

Precision: binary64

Math

\[\log \left(1 + x\right) \]

↓

\[\mathsf{log1p}\left(x\right) \]

FPCore

(FPCore (x) :precision binary64 (log (+ 1.0 x)))

↓

(FPCore (x) :precision binary64 (log1p x))

C

double code(double x) {
	return log((1.0 + x));
}

↓

double code(double x) {
	return log1p(x);
}

Java

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

↓

public static double code(double x) {
	return Math.log1p(x);
}

Python

def code(x):
	return math.log((1.0 + x))

↓

def code(x):
	return math.log1p(x)

Julia

function code(x)
	return log(Float64(1.0 + x))
end

↓

function code(x)
	return log1p(x)
end

Wolfram

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

↓

code[x_] := N[Log[1 + x], $MachinePrecision]

Error

Bits error versus x

Target

Original	38.6
Target	0.2
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array} \]

Derivation

1. Initial program 38.6

   \[\log \left(1 + x\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]

3. Final simplification 0.0

   \[\leadsto \mathsf{log1p}\left(x\right) \]

Reproduce

herbie shell --seed 2022156 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))