Average Error: 29.0 → 3.8
Time: 16.0s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -4.062951262301690436022485283190371518911 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \log \left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -4.062951262301690436022485283190371518911 \cdot 10^{-21}:\\
\;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(a, x, \log \left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right)\right)\\

\end{array}
double f(double a, double x) {
        double r75022 = a;
        double r75023 = x;
        double r75024 = r75022 * r75023;
        double r75025 = exp(r75024);
        double r75026 = 1.0;
        double r75027 = r75025 - r75026;
        return r75027;
}
↓
double f(double a, double x) {
        double r75028 = a;
        double r75029 = x;
        double r75030 = r75028 * r75029;
        double r75031 = -4.0629512623016904e-21;
        bool r75032 = r75030 <= r75031;
        double r75033 = exp(r75030);
        double r75034 = 1.0;
        double r75035 = r75033 - r75034;
        double r75036 = cbrt(r75035);
        double r75037 = r75036 * r75036;
        double r75038 = r75037 * r75036;
        double r75039 = exp(r75029);
        double r75040 = r75028 * r75028;
        double r75041 = 0.5;
        double r75042 = 3.0;
        double r75043 = pow(r75028, r75042);
        double r75044 = 0.16666666666666666;
        double r75045 = r75044 * r75029;
        double r75046 = r75043 * r75045;
        double r75047 = fma(r75040, r75041, r75046);
        double r75048 = r75047 * r75029;
        double r75049 = pow(r75039, r75048);
        double r75050 = log(r75049);
        double r75051 = fma(r75028, r75029, r75050);
        double r75052 = r75032 ? r75038 : r75051;
        return r75052;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.0
Target0.2
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -4.0629512623016904e-21

    1. Initial program 1.8

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

    3. Applied add-sqr-sqrt1.9

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

    4. Applied fma-neg1.9

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

    6. Applied add-cube-cbrt1.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}}\]

    7. Simplified1.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{e^{a \cdot x}}, \sqrt{e^{a \cdot x}}, -1\right)}\]

    8. Simplified1.8

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

    if -4.0629512623016904e-21 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 13.4

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

    3. Simplified10.3

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

    5. Applied add-log-exp10.4

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

    6. Simplified4.9

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

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -4.062951262301690436022485283190371518911 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt[3]{e^{a \cdot x} - 1} \cdot \sqrt[3]{e^{a \cdot x} - 1}\right) \cdot \sqrt[3]{e^{a \cdot x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \log \left({\left(e^{x}\right)}^{\left(\mathsf{fma}\left(a \cdot a, \frac{1}{2}, {a}^{3} \cdot \left(\frac{1}{6} \cdot x\right)\right) \cdot x\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))