Average Error: 30.0 → 0.5
Time: 3.5s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001626719355739147552727952295015256822808:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.001626719355739147552727952295015256822808:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

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

\end{array}
double f(double a, double x) {
        double r93859 = a;
        double r93860 = x;
        double r93861 = r93859 * r93860;
        double r93862 = exp(r93861);
        double r93863 = 1.0;
        double r93864 = r93862 - r93863;
        return r93864;
}
↓
double f(double a, double x) {
        double r93865 = a;
        double r93866 = x;
        double r93867 = r93865 * r93866;
        double r93868 = -0.0016267193557391476;
        bool r93869 = r93867 <= r93868;
        double r93870 = exp(r93867);
        double r93871 = 1.0;
        double r93872 = r93870 - r93871;
        double r93873 = exp(r93872);
        double r93874 = log(r93873);
        double r93875 = 0.5;
        double r93876 = r93866 * r93865;
        double r93877 = fabs(r93876);
        double r93878 = r93877 * r93877;
        double r93879 = fma(r93875, r93878, r93867);
        double r93880 = r93869 ? r93874 : r93879;
        return r93880;
}

Error

Bits error versus a

Bits error versus x

Target

Original30.0
Target0.2
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.0016267193557391476

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    if -0.0016267193557391476 < (* a x)

    1. Initial program 44.7

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

    2. Taylor expanded around 0 14.7

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

    3. Simplified14.7

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

    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. Simplified8.4

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

    7. Applied add-sqr-sqrt8.4

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

    8. Simplified8.4

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

    9. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.001626719355739147552727952295015256822808:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \left|x \cdot a\right| \cdot \left|x \cdot a\right|, a \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))