Average Error: 58.5 → 0.2

Time: 10.4s

Precision: binary64

Cost: 14144

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

↓

\[\left(\left(\varepsilon \cdot -2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)

↓

\left(\left(\varepsilon \cdot -2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}

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

↓

(FPCore (eps)
 :precision binary64
 (-
  (-
   (- (* eps -2.0) (* (* eps eps) (* eps 0.6666666666666666)))
   (* 0.4 (pow eps 5.0)))
  (* 0.2857142857142857 (pow eps 7.0))))

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

↓

double code(double eps) {
	return (((eps * -2.0) - ((eps * eps) * (eps * 0.6666666666666666))) - (0.4 * pow(eps, 5.0))) - (0.2857142857142857 * pow(eps, 7.0));
}

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	58.5
Target	0.2
Herbie	0.2

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

Alternatives

Alternative 1
Error	61.1
Cost	45888

\[\left(\log \left(1 + \sqrt{\varepsilon}\right) + -2 \cdot \log \left(\sqrt[3]{\varepsilon + 1}\right)\right) + \log \left(\frac{1 - \sqrt{\varepsilon}}{\sqrt[3]{\varepsilon + 1}}\right)\]

Alternative 2
Error	58.6
Cost	45888

\[\left(-2 \cdot \log \left(\sqrt[3]{\varepsilon + 1}\right) + \log \left(\sqrt{1 - \varepsilon}\right)\right) + \log \left(\frac{\sqrt{1 - \varepsilon}}{\sqrt[3]{\varepsilon + 1}}\right)\]

Alternative 3
Error	58.5
Cost	39872

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

Alternative 4
Error	61.2
Cost	39616

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

Alternative 5
Error	61.1
Cost	39360

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

Alternative 6
Error	61.2
Cost	32960

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

Alternative 7
Error	58.6
Cost	26688

\[\log \left(\sqrt[3]{\frac{1 - \varepsilon}{\varepsilon + 1}}\right) + \log \left(\sqrt[3]{\frac{1 - \varepsilon}{\varepsilon + 1}}\right) \cdot 2\]

Alternative 8
Error	58.5
Cost	26560

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

Alternative 9
Error	61.1
Cost	26304

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

Alternative 10
Error	58.5
Cost	26304

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

Alternative 11
Error	58.5
Cost	26304

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

Alternative 12
Error	0.2
Cost	20352

\[\left(\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

Alternative 13
Error	61.2
Cost	19904

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

Alternative 14
Error	58.5
Cost	19712

\[\log \left(\sqrt[3]{{\left(\frac{1 - \varepsilon}{\varepsilon + 1}\right)}^{3}}\right)\]

Alternative 15
Error	58.5
Cost	19648

\[\log \log \left(e^{\frac{1 - \varepsilon}{\varepsilon + 1}}\right)\]

Alternative 16
Error	0.2
Cost	13632

\[\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\]

Alternative 17
Error	58.5
Cost	13248

\[\log \left(1 - \varepsilon\right) - \log \left(\varepsilon + 1\right)\]

Alternative 18
Error	0.3
Cost	6912

\[\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\]

Alternative 19
Error	58.5
Cost	6848

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

Alternative 20
Error	0.7
Cost	192

\[\varepsilon \cdot -2\]

Alternative 21
Error	61.6
Cost	64

\[1\]

Alternative 22
Error	60.6
Cost	64

\[0\]

Alternative 23
Error	61.5
Cost	64

\[-1\]

Derivation

Initial program 58.5
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
Taylor expanded around 0 0.2
\[\leadsto \color{blue}{-\left(0.4 \cdot {\varepsilon}^{5} + \left(0.2857142857142857 \cdot {\varepsilon}^{7} + \left(2 \cdot \varepsilon + 0.6666666666666666 \cdot {\varepsilon}^{3}\right)\right)\right)}\]
Simplified0.2
\[\leadsto \color{blue}{\left(\left(\varepsilon \cdot -2 - {\varepsilon}^{3} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}}\]
Using strategy rm
Applied unpow3_binary64_21350.2
\[\leadsto \left(\left(\varepsilon \cdot -2 - \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \cdot 0.6666666666666666\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]
Applied associate-*l*_binary64_20100.2
\[\leadsto \left(\left(\varepsilon \cdot -2 - \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.6666666666666666\right)}\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]
Simplified0.2
\[\leadsto \color{blue}{\left(\left(\varepsilon \cdot -2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}}\]
Final simplification0.2
\[\leadsto \left(\left(\varepsilon \cdot -2 - \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.6666666666666666\right)\right) - 0.4 \cdot {\varepsilon}^{5}\right) - 0.2857142857142857 \cdot {\varepsilon}^{7}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))