Average Error: 58.4 → 0.7
Time: 16.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\frac{\varepsilon \cdot \varepsilon}{1 \cdot 1} + \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\frac{\varepsilon \cdot \varepsilon}{1 \cdot 1} + \varepsilon\right)\right)
double f(double eps) {
        double r6087243 = 1.0;
        double r6087244 = eps;
        double r6087245 = r6087243 - r6087244;
        double r6087246 = r6087243 + r6087244;
        double r6087247 = r6087245 / r6087246;
        double r6087248 = log(r6087247);
        return r6087248;
}
↓
double f(double eps) {
        double r6087249 = 1.0;
        double r6087250 = log(r6087249);
        double r6087251 = 2.0;
        double r6087252 = eps;
        double r6087253 = r6087252 * r6087252;
        double r6087254 = r6087249 * r6087249;
        double r6087255 = r6087253 / r6087254;
        double r6087256 = r6087255 + r6087252;
        double r6087257 = r6087253 - r6087256;
        double r6087258 = r6087251 * r6087257;
        double r6087259 = r6087250 + r6087258;
        return r6087259;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019174 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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