Average Error: 29.6 → 3.4
Time: 16.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -2.126354662771189659841580832377871956851 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{elif}\;a \cdot x \le 7.692055924467544744602805206092219669968 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \log \left({\left(e^{x}\right)}^{\left(x \cdot \mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(\frac{1}{6} \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;a \cdot x \le 7.182914139653070480268595225980441913815 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left({a}^{3} \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -2.126354662771189659841580832377871956851 \cdot 10^{-4}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{elif}\;a \cdot x \le 7.692055924467544744602805206092219669968 \cdot 10^{-34}:\\
\;\;\;\;\mathsf{fma}\left(a, x, \log \left({\left(e^{x}\right)}^{\left(x \cdot \mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(\frac{1}{6} \cdot x\right)\right)\right)}\right)\right)\\

\mathbf{elif}\;a \cdot x \le 7.182914139653070480268595225980441913815 \cdot 10^{-18}:\\
\;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left({a}^{3} \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\

\end{array}
double f(double a, double x) {
        double r74055 = a;
        double r74056 = x;
        double r74057 = r74055 * r74056;
        double r74058 = exp(r74057);
        double r74059 = 1.0;
        double r74060 = r74058 - r74059;
        return r74060;
}
↓
double f(double a, double x) {
        double r74061 = a;
        double r74062 = x;
        double r74063 = r74061 * r74062;
        double r74064 = -0.00021263546627711897;
        bool r74065 = r74063 <= r74064;
        double r74066 = exp(r74063);
        double r74067 = 1.0;
        double r74068 = r74066 - r74067;
        double r74069 = exp(r74068);
        double r74070 = log(r74069);
        double r74071 = 7.692055924467545e-34;
        bool r74072 = r74063 <= r74071;
        double r74073 = exp(r74062);
        double r74074 = r74061 * r74061;
        double r74075 = 0.5;
        double r74076 = 3.0;
        double r74077 = pow(r74061, r74076);
        double r74078 = 0.16666666666666666;
        double r74079 = r74078 * r74062;
        double r74080 = r74077 * r74079;
        double r74081 = fma(r74074, r74075, r74080);
        double r74082 = r74062 * r74081;
        double r74083 = pow(r74073, r74082);
        double r74084 = log(r74083);
        double r74085 = fma(r74061, r74062, r74084);
        double r74086 = 7.18291413965307e-18;
        bool r74087 = r74063 <= r74086;
        double r74088 = r74077 * r74062;
        double r74089 = r74078 * r74088;
        double r74090 = fma(r74075, r74074, r74089);
        double r74091 = r74062 * r74062;
        double r74092 = r74090 * r74091;
        double r74093 = fma(r74061, r74062, r74092);
        double r74094 = sqrt(r74066);
        double r74095 = -r74067;
        double r74096 = fma(r74094, r74094, r74095);
        double r74097 = r74087 ? r74093 : r74096;
        double r74098 = r74072 ? r74085 : r74097;
        double r74099 = r74065 ? r74070 : r74098;
        return r74099;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.6
Target0.2
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (* a x) < -0.00021263546627711897

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied fma-neg0.1

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

    6. Applied add-log-exp0.1

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

    7. Simplified0.1

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

    if -0.00021263546627711897 < (* a x) < 7.692055924467545e-34

    1. Initial program 44.2

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

    2. Taylor expanded around 0 12.7

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

    3. Simplified9.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left({a}^{3} \cdot x\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp9.8

      \[\leadsto \mathsf{fma}\left(a, x, \color{blue}{\log \left(e^{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left({a}^{3} \cdot x\right)\right)}\right)}\right)\]

    6. Simplified3.7

      \[\leadsto \mathsf{fma}\left(a, x, \log \color{blue}{\left({\left(e^{x}\right)}^{\left(x \cdot \mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(x \cdot \frac{1}{6}\right)\right)\right)}\right)}\right)\]

    if 7.692055924467545e-34 < (* a x) < 7.18291413965307e-18

    1. Initial program 61.9

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

    2. Taylor expanded around 0 35.4

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

    3. Simplified29.6

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

    if 7.18291413965307e-18 < (* a x)

    1. Initial program 23.7

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

    3. Applied add-sqr-sqrt24.9

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

    4. Applied fma-neg24.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -2.126354662771189659841580832377871956851 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{elif}\;a \cdot x \le 7.692055924467544744602805206092219669968 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \log \left({\left(e^{x}\right)}^{\left(x \cdot \mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(\frac{1}{6} \cdot x\right)\right)\right)}\right)\right)\\ \mathbf{elif}\;a \cdot x \le 7.182914139653070480268595225980441913815 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(\frac{1}{2}, a \cdot a, \frac{1}{6} \cdot \left({a}^{3} \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a x)
  :name "expax (section 3.5)"
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))