Average Error: 29.5 → 9.6
Time: 5.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.003644452020865184:\\ \;\;\;\;\log \left(e^{\frac{{\left({\left(e^{a \cdot x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + {\left(e^{a \cdot x}\right)}^{3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.003644452020865184:\\
\;\;\;\;\log \left(e^{\frac{{\left({\left(e^{a \cdot x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + {\left(e^{a \cdot x}\right)}^{3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\

\end{array}
double f(double a, double x) {
        double r128040 = a;
        double r128041 = x;
        double r128042 = r128040 * r128041;
        double r128043 = exp(r128042);
        double r128044 = 1.0;
        double r128045 = r128043 - r128044;
        return r128045;
}
↓
double f(double a, double x) {
        double r128046 = a;
        double r128047 = x;
        double r128048 = r128046 * r128047;
        double r128049 = -0.003644452020865184;
        bool r128050 = r128048 <= r128049;
        double r128051 = exp(r128048);
        double r128052 = 3.0;
        double r128053 = pow(r128051, r128052);
        double r128054 = pow(r128053, r128052);
        double r128055 = 1.0;
        double r128056 = pow(r128055, r128052);
        double r128057 = pow(r128056, r128052);
        double r128058 = r128054 - r128057;
        double r128059 = 6.0;
        double r128060 = pow(r128051, r128059);
        double r128061 = pow(r128055, r128059);
        double r128062 = r128060 + r128061;
        double r128063 = r128053 * r128056;
        double r128064 = r128062 + r128063;
        double r128065 = r128051 + r128055;
        double r128066 = r128051 * r128065;
        double r128067 = r128055 * r128055;
        double r128068 = r128066 + r128067;
        double r128069 = r128064 * r128068;
        double r128070 = r128058 / r128069;
        double r128071 = exp(r128070);
        double r128072 = log(r128071);
        double r128073 = 0.5;
        double r128074 = 2.0;
        double r128075 = pow(r128046, r128074);
        double r128076 = r128073 * r128075;
        double r128077 = r128076 * r128047;
        double r128078 = r128046 + r128077;
        double r128079 = r128047 * r128078;
        double r128080 = 0.16666666666666666;
        double r128081 = pow(r128046, r128052);
        double r128082 = pow(r128047, r128052);
        double r128083 = r128081 * r128082;
        double r128084 = r128080 * r128083;
        double r128085 = r128079 + r128084;
        double r128086 = r128050 ? r128072 : r128085;
        return r128086;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.2
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.003644452020865184

    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

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

    6. Applied flip3--0.0

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

    7. Applied associate-/l/0.0

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

    8. Simplified0.0

      \[\leadsto \frac{{\left({\left(e^{a \cdot x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\color{blue}{\left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + {\left(e^{a \cdot x}\right)}^{3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}}\]
    9. Using strategy rm

    10. Applied add-log-exp0.0

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

    if -0.003644452020865184 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 14.5

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

    3. Simplified14.5

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.003644452020865184:\\ \;\;\;\;\log \left(e^{\frac{{\left({\left(e^{a \cdot x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{\left(\left({\left(e^{a \cdot x}\right)}^{6} + {1}^{6}\right) + {\left(e^{a \cdot x}\right)}^{3} \cdot {1}^{3}\right) \cdot \left(e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))