qlog (example 3.10)

Average Error: 61.6 → 0.0

Time: 11.3s

Precision: binary64

Cost: 19520

\[-1 < x \land x < 1\]

Math

\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]

↓

\[{\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (pow (/ (log1p x) (log1p (- x))) -1.0))

C

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

↓

double code(double x) {
	return pow((log1p(x) / log1p(-x)), -1.0);
}

Java

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

↓

public static double code(double x) {
	return Math.pow((Math.log1p(x) / Math.log1p(-x)), -1.0);
}

Python

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

↓

def code(x):
	return math.pow((math.log1p(x) / math.log1p(-x)), -1.0)

Julia

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

↓

function code(x)
	return Float64(log1p(x) / log1p(Float64(-x))) ^ -1.0
end

Wolfram

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

↓

code[x_] := N[Power[N[(N[Log[1 + x], $MachinePrecision] / N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Target

Original	61.6
Target	0.3
Herbie	0.0

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right) \]

Derivation

1. Initial program 61.6

   \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{{\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1}} \]

3. Final simplification 0.0

   \[\leadsto {\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1} \]

Alternatives

Alternative 1
Error	0.0
Cost	13056

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

Alternative 2
Error	0.3
Cost	7296

\[{x}^{3} \cdot -0.4166666666666667 + \left(\left(-1 + x \cdot \left(x \cdot -0.5\right)\right) - x\right) \]

Alternative 3
Error	0.4
Cost	6848

\[\mathsf{fma}\left(x, x \cdot -0.5, -1\right) - x \]

Alternative 4
Error	0.6
Cost	192

\[-1 - x \]

Alternative 5
Error	1.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (- (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))