qlog (example 3.10)

Average Error: 61.3 → 0.4

Time: 12.0s

Precision: binary64

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

Math

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

↓

\[\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \left(\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}\right)\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (*
  (cbrt
   (/
    (- (log 1.0) (+ (* 1.0 x) (* 0.5 (/ (* x x) (* 1.0 1.0)))))
    (- (+ (log 1.0) (* 1.0 x)) (* 0.5 (/ (* x x) (* 1.0 1.0))))))
  (*
   (cbrt
    (/
     (- (log 1.0) (+ (* 1.0 x) (* 0.5 (/ (* x x) (* 1.0 1.0)))))
     (- (+ (log 1.0) (* 1.0 x)) (* 0.5 (/ (* x x) (* 1.0 1.0))))))
   (cbrt
    (/
     (- (log 1.0) (+ (* 1.0 x) (* 0.5 (/ (* x x) (* 1.0 1.0)))))
     (- (+ (log 1.0) (* 1.0 x)) (* 0.5 (/ (* x x) (* 1.0 1.0)))))))))

C

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

↓

double code(double x) {
	return ((double) (((double) cbrt((((double) (((double) log(1.0)) - ((double) (((double) (1.0 * x)) + ((double) (0.5 * (((double) (x * x)) / ((double) (1.0 * 1.0))))))))) / ((double) (((double) (((double) log(1.0)) + ((double) (1.0 * x)))) - ((double) (0.5 * (((double) (x * x)) / ((double) (1.0 * 1.0)))))))))) * ((double) (((double) cbrt((((double) (((double) log(1.0)) - ((double) (((double) (1.0 * x)) + ((double) (0.5 * (((double) (x * x)) / ((double) (1.0 * 1.0))))))))) / ((double) (((double) (((double) log(1.0)) + ((double) (1.0 * x)))) - ((double) (0.5 * (((double) (x * x)) / ((double) (1.0 * 1.0)))))))))) * ((double) cbrt((((double) (((double) log(1.0)) - ((double) (((double) (1.0 * x)) + ((double) (0.5 * (((double) (x * x)) / ((double) (1.0 * 1.0))))))))) / ((double) (((double) (((double) log(1.0)) + ((double) (1.0 * x)))) - ((double) (0.5 * (((double) (x * x)) / ((double) (1.0 * 1.0))))))))))))));
}

Error

Bits error versus x

Target

Original	61.3
Target	0.3
Herbie	0.4

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

Derivation

1. Initial program 61.3

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

2. Taylor expanded around 0 60.4

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

3. Simplified 60.4

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

4. Taylor expanded around 0 0.4

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

5. Simplified 0.4

   \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}}{\left(1 \cdot x + \log 1\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}\]
6. Using strategy rm

7. Applied add-cube-cbrt_binary64 0.4

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(1 \cdot x + \log 1\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(1 \cdot x + \log 1\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}\right) \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(1 \cdot x + \log 1\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}}\]

8. Simplified 0.4

   \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}\right)} \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(1 \cdot x + \log 1\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}\]

9. Simplified 0.4

   \[\leadsto \left(\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}}\]

10. Final simplification 0.4

    \[\leadsto \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \left(\sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}} \cdot \sqrt[3]{\frac{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}\right)}{\left(\log 1 + 1 \cdot x\right) - 0.5 \cdot \frac{x \cdot x}{1 \cdot 1}}}\right)\]

Reproduce

herbie shell --seed 2020210 
(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))))