Average Error: 29.4 → 0.4
Time: 11.9s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|a\right| \cdot \left|x\right|\right) \cdot \left(\left|a\right| \cdot \left|x\right|\right)\right) + a \cdot x\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \left(\left(\left|a\right| \cdot \left|x\right|\right) \cdot \left(\left|a\right| \cdot \left|x\right|\right)\right) + a \cdot x\\

\end{array}
double f(double a, double x) {
        double r143241 = a;
        double r143242 = x;
        double r143243 = r143241 * r143242;
        double r143244 = exp(r143243);
        double r143245 = 1.0;
        double r143246 = r143244 - r143245;
        return r143246;
}
↓
double f(double a, double x) {
        double r143247 = a;
        double r143248 = x;
        double r143249 = r143247 * r143248;
        double r143250 = -0.0009427648001274835;
        bool r143251 = r143249 <= r143250;
        double r143252 = exp(r143249);
        double r143253 = sqrt(r143252);
        double r143254 = 1.0;
        double r143255 = sqrt(r143254);
        double r143256 = r143253 + r143255;
        double r143257 = r143253 - r143255;
        double r143258 = r143256 * r143257;
        double r143259 = 0.5;
        double r143260 = fabs(r143247);
        double r143261 = fabs(r143248);
        double r143262 = r143260 * r143261;
        double r143263 = r143262 * r143262;
        double r143264 = r143259 * r143263;
        double r143265 = r143264 + r143249;
        double r143266 = r143251 ? r143258 : r143265;
        return r143266;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0009427648001274835

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    if -0.0009427648001274835 < (* a x)

    1. Initial program 44.3

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

    2. Taylor expanded around 0 14.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified7.5

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\left(\frac{1}{6} \cdot {a}^{3}\right) \cdot x + \frac{1}{2} \cdot {a}^{2}\right) + a\right)}\]

    4. Taylor expanded around 0 8.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt8.4

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

    7. Applied add-sqr-sqrt8.4

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

    8. Applied unswap-sqr8.4

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

    9. Simplified8.4

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

    10. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -9.4276480012748349 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{a \cdot x}} - \sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(\left|a\right| \cdot \left|x\right|\right) \cdot \left(\left|a\right| \cdot \left|x\right|\right)\right) + a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

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