Average Error: 60.7 → 0.5

Time: 1.2m

Precision: 64

Internal Precision: 1408

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

↓

\[\left(\left(-x\right) \cdot \left(\frac{2}{3} + x \cdot \frac{2}{9}\right) + \left(-1\right)\right) \cdot \left(\frac{1}{18} \cdot {x}^{2} + \left(1 + \frac{1}{3} \cdot x\right)\right)\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	60.7
Target	0.4
Herbie	0.5

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

Initial program 60.7
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
Taylor expanded around 0 0.5
\[\leadsto \color{blue}{-\left(\frac{1}{2} \cdot {x}^{2} + \left(1 + x\right)\right)}\]
Using strategy rm
Applied add-cube-cbrt0.5
\[\leadsto -\color{blue}{\left(\sqrt[3]{\frac{1}{2} \cdot {x}^{2} + \left(1 + x\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot {x}^{2} + \left(1 + x\right)}\right) \cdot \sqrt[3]{\frac{1}{2} \cdot {x}^{2} + \left(1 + x\right)}}\]
Taylor expanded around 0 0.5
\[\leadsto -\color{blue}{\left(\frac{2}{9} \cdot {x}^{2} + \left(1 + \frac{2}{3} \cdot x\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot {x}^{2} + \left(1 + x\right)}\]
Applied simplify0.5
\[\leadsto \color{blue}{\left(\left(-x\right) \cdot \left(\frac{2}{3} + x \cdot \frac{2}{9}\right) + \left(-1\right)\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \left(x \cdot x\right) + \left(x + 1\right)}}\]
Taylor expanded around 0 0.5
\[\leadsto \left(\left(-x\right) \cdot \left(\frac{2}{3} + x \cdot \frac{2}{9}\right) + \left(-1\right)\right) \cdot \color{blue}{\left(\frac{1}{18} \cdot {x}^{2} + \left(1 + \frac{1}{3} \cdot x\right)\right)}\]
Removed slow pow expressions.

Runtime

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))