qlog (example 3.10)

Average Error: 61.3 → 0.3

Time: 8.0s

Precision: binary64

Cost: 832

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

Math

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

↓

\[-1 + x \cdot \left(-1 - x \cdot \left(x \cdot 0.4166666666666667 + 0.5\right)\right)\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+ -1.0 (* x (- -1.0 (* x (+ (* x 0.4166666666666667) 0.5))))))

C

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

↓

double code(double x) {
	return -1.0 + (x * (-1.0 - (x * ((x * 0.4166666666666667) + 0.5))));
}

Error

Bits error versus x

Target

Original	61.3
Target	0.3
Herbie	0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	576

\[-1 - \left(x + 0.5 \cdot \left(x \cdot x\right)\right)\]

Alternative 2
Error	0.7
Cost	192

\[-1 - x\]

Alternative 3
Error	1.4
Cost	64

\[-1\]

Alternative 4
Error	62.0
Cost	64

\[0\]

Alternative 5
Error	63.0
Cost	64

\[1\]

Derivation

1. Initial program 61.3

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

2. Taylor expanded around 0 0.3

   \[\leadsto \color{blue}{-\left(x + \left(0.5 \cdot {x}^{2} + \left(0.4166666666666667 \cdot {x}^{3} + 1\right)\right)\right)}\]

3. Simplified 0.3

   \[\leadsto \color{blue}{-1 + x \cdot \left(-1 - x \cdot \left(0.5 + x \cdot 0.4166666666666667\right)\right)}\]

4. Simplified 0.3

   \[\leadsto \color{blue}{-1 + x \cdot \left(-1 - x \cdot \left(x \cdot 0.4166666666666667 + 0.5\right)\right)}\]

5. Final simplification 0.3

   \[\leadsto -1 + x \cdot \left(-1 - x \cdot \left(x \cdot 0.4166666666666667 + 0.5\right)\right)\]

Reproduce

herbie shell --seed 2021044 
(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))))