Average Error: 29.9 → 0.3
Time: 20.4s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00024460776635084816:\\ \;\;\;\;\frac{\left(\sqrt{e^{a \cdot x}} + e^{a \cdot x}\right) \cdot e^{a \cdot x} - \left(1 + \sqrt{e^{a \cdot x}}\right)}{\left(1 + \sqrt{e^{a \cdot x}}\right) + \sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{48}\right)\right) \cdot x + \left(\left(\left(a \cdot x\right) \cdot \frac{1}{8}\right) \cdot \left(a \cdot x\right) + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.00024460776635084816:\\
\;\;\;\;\frac{\left(\sqrt{e^{a \cdot x}} + e^{a \cdot x}\right) \cdot e^{a \cdot x} - \left(1 + \sqrt{e^{a \cdot x}}\right)}{\left(1 + \sqrt{e^{a \cdot x}}\right) + \sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}}\\

\mathbf{else}:\\
\;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{48}\right)\right) \cdot x + \left(\left(\left(a \cdot x\right) \cdot \frac{1}{8}\right) \cdot \left(a \cdot x\right) + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\

\end{array}
double f(double a, double x) {
        double r5982231 = a;
        double r5982232 = x;
        double r5982233 = r5982231 * r5982232;
        double r5982234 = exp(r5982233);
        double r5982235 = 1.0;
        double r5982236 = r5982234 - r5982235;
        return r5982236;
}
↓
double f(double a, double x) {
        double r5982237 = a;
        double r5982238 = x;
        double r5982239 = r5982237 * r5982238;
        double r5982240 = -0.00024460776635084816;
        bool r5982241 = r5982239 <= r5982240;
        double r5982242 = exp(r5982239);
        double r5982243 = sqrt(r5982242);
        double r5982244 = r5982243 + r5982242;
        double r5982245 = r5982244 * r5982242;
        double r5982246 = 1.0;
        double r5982247 = r5982246 + r5982243;
        double r5982248 = r5982245 - r5982247;
        double r5982249 = r5982243 * r5982243;
        double r5982250 = r5982247 + r5982249;
        double r5982251 = r5982248 / r5982250;
        double r5982252 = r5982239 * r5982239;
        double r5982253 = 0.020833333333333332;
        double r5982254 = r5982237 * r5982253;
        double r5982255 = r5982252 * r5982254;
        double r5982256 = r5982255 * r5982238;
        double r5982257 = 0.125;
        double r5982258 = r5982239 * r5982257;
        double r5982259 = r5982258 * r5982239;
        double r5982260 = 0.5;
        double r5982261 = r5982239 * r5982260;
        double r5982262 = r5982259 + r5982261;
        double r5982263 = r5982256 + r5982262;
        double r5982264 = r5982247 * r5982263;
        double r5982265 = r5982241 ? r5982251 : r5982264;
        return r5982265;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.00024460776635084816

    1. Initial program 0.0

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.0

      \[\leadsto e^{a \cdot x} - \color{blue}{1 \cdot 1}\]

    4. Applied add-sqr-sqrt0.0

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1 \cdot 1\]

    5. Applied difference-of-squares0.0

      \[\leadsto \color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\]
    6. Using strategy rm

    7. Applied flip3--0.0

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

    8. Applied associate-*r/0.0

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

    9. Simplified0.0

      \[\leadsto \frac{\color{blue}{e^{x \cdot a} \cdot \left(e^{x \cdot a} + \sqrt{e^{x \cdot a}}\right) - \left(\sqrt{e^{x \cdot a}} + 1\right)}}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}} + \left(1 \cdot 1 + \sqrt{e^{a \cdot x}} \cdot 1\right)}\]

    if -0.00024460776635084816 < (* a x)

    1. Initial program 44.7

      \[e^{a \cdot x} - 1\]
    2. Using strategy rm

    3. Applied *-un-lft-identity44.7

      \[\leadsto e^{a \cdot x} - \color{blue}{1 \cdot 1}\]

    4. Applied add-sqr-sqrt44.7

      \[\leadsto \color{blue}{\sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}} - 1 \cdot 1\]

    5. Applied difference-of-squares44.7

      \[\leadsto \color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\]

    6. Taylor expanded around 0 13.8

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{8} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right) + \frac{1}{48} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)\right)}\]

    7. Simplified0.5

      \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \left(x \cdot a\right) + \left(\frac{1}{8} \cdot \left(x \cdot a\right)\right) \cdot \left(x \cdot a\right)\right) + \left(\left(\frac{1}{48} \cdot a\right) \cdot \left(\left(x \cdot a\right) \cdot \left(x \cdot a\right)\right)\right) \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.00024460776635084816:\\ \;\;\;\;\frac{\left(\sqrt{e^{a \cdot x}} + e^{a \cdot x}\right) \cdot e^{a \cdot x} - \left(1 + \sqrt{e^{a \cdot x}}\right)}{\left(1 + \sqrt{e^{a \cdot x}}\right) + \sqrt{e^{a \cdot x}} \cdot \sqrt{e^{a \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot \frac{1}{48}\right)\right) \cdot x + \left(\left(\left(a \cdot x\right) \cdot \frac{1}{8}\right) \cdot \left(a \cdot x\right) + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))