Average Error: 29.1 → 0.3
Time: 38.4s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.009717306350483434:\\ \;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{\left(1 + e^{a \cdot x}\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.009717306350483434:\\
\;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{\left(1 + e^{a \cdot x}\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)\\

\end{array}
double f(double a, double x) {
        double r5772191 = a;
        double r5772192 = x;
        double r5772193 = r5772191 * r5772192;
        double r5772194 = exp(r5772193);
        double r5772195 = 1.0;
        double r5772196 = r5772194 - r5772195;
        return r5772196;
}
↓
double f(double a, double x) {
        double r5772197 = a;
        double r5772198 = x;
        double r5772199 = r5772197 * r5772198;
        double r5772200 = -0.009717306350483434;
        bool r5772201 = r5772199 <= r5772200;
        double r5772202 = -1.0;
        double r5772203 = 3.0;
        double r5772204 = r5772197 * r5772203;
        double r5772205 = r5772198 * r5772204;
        double r5772206 = exp(r5772205);
        double r5772207 = r5772202 + r5772206;
        double r5772208 = 1.0;
        double r5772209 = exp(r5772199);
        double r5772210 = r5772208 + r5772209;
        double r5772211 = r5772209 * r5772209;
        double r5772212 = r5772210 + r5772211;
        double r5772213 = r5772207 / r5772212;
        double r5772214 = 0.16666666666666666;
        double r5772215 = r5772199 * r5772199;
        double r5772216 = r5772199 * r5772215;
        double r5772217 = r5772214 * r5772216;
        double r5772218 = 0.5;
        double r5772219 = r5772218 * r5772199;
        double r5772220 = r5772199 * r5772219;
        double r5772221 = r5772199 + r5772220;
        double r5772222 = r5772217 + r5772221;
        double r5772223 = r5772201 ? r5772213 : r5772222;
        return r5772223;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.2
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) < -0.009717306350483434

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if -0.009717306350483434 < (* a x)

    1. Initial program 44.0

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) + a \cdot x\right) + \left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right) \cdot \frac{1}{6}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.009717306350483434:\\ \;\;\;\;\frac{-1 + e^{x \cdot \left(a \cdot 3\right)}}{\left(1 + e^{a \cdot x}\right) + e^{a \cdot x} \cdot e^{a \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot \left(\left(a \cdot x\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))