Average Error: 29.5 → 5.1
Time: 19.4s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.3967409376004402132842585615435382351279:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{9}{2}, \left(a \cdot \left(a \cdot \left(\sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)}\right)\right)\right) \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.3967409376004402132842585615435382351279:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{9}{2}, \left(a \cdot \left(a \cdot \left(\sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)}\right)\right)\right) \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\

\end{array}
double f(double a, double x) {
        double r84251 = a;
        double r84252 = x;
        double r84253 = r84251 * r84252;
        double r84254 = exp(r84253);
        double r84255 = 1.0;
        double r84256 = r84254 - r84255;
        return r84256;
}
↓
double f(double a, double x) {
        double r84257 = a;
        double r84258 = x;
        double r84259 = r84257 * r84258;
        double r84260 = -0.3967409376004402;
        bool r84261 = r84259 <= r84260;
        double r84262 = exp(r84259);
        double r84263 = 3.0;
        double r84264 = pow(r84262, r84263);
        double r84265 = 1.0;
        double r84266 = pow(r84265, r84263);
        double r84267 = r84264 - r84266;
        double r84268 = r84262 + r84265;
        double r84269 = 2.0;
        double r84270 = r84269 * r84259;
        double r84271 = exp(r84270);
        double r84272 = fma(r84265, r84268, r84271);
        double r84273 = r84267 / r84272;
        double r84274 = 4.5;
        double r84275 = pow(r84258, r84263);
        double r84276 = r84275 * r84257;
        double r84277 = fma(r84258, r84258, r84276);
        double r84278 = cbrt(r84277);
        double r84279 = r84278 * r84278;
        double r84280 = r84257 * r84279;
        double r84281 = r84257 * r84280;
        double r84282 = pow(r84258, r84269);
        double r84283 = r84282 + r84276;
        double r84284 = cbrt(r84283);
        double r84285 = r84281 * r84284;
        double r84286 = r84263 * r84259;
        double r84287 = fma(r84274, r84285, r84286);
        double r84288 = r84287 / r84272;
        double r84289 = r84261 ? r84273 : r84288;
        return r84289;
}

Error

Bits error versus a

Bits error versus x

Target

Original29.5
Target0.2
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -0.3967409376004402

    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

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

    if -0.3967409376004402 < (* a x)

    1. Initial program 44.1

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

    3. Applied flip3--44.2

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

    4. Simplified44.2

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

    5. Taylor expanded around 0 14.5

      \[\leadsto \frac{\color{blue}{\frac{9}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{9}{2} \cdot \left({a}^{3} \cdot {x}^{3}\right) + 3 \cdot \left(a \cdot x\right)\right)}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

    6. Simplified11.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{9}{2}, {a}^{2} \cdot \left({x}^{2} + {x}^{3} \cdot a\right), 3 \cdot \left(a \cdot x\right)\right)}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt11.4

      \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, {a}^{2} \cdot \color{blue}{\left(\left(\sqrt[3]{{x}^{2} + {x}^{3} \cdot a} \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}\right) \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}\right)}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

    9. Applied associate-*r*11.4

      \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, \color{blue}{\left({a}^{2} \cdot \left(\sqrt[3]{{x}^{2} + {x}^{3} \cdot a} \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}\right)\right) \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]

    10. Simplified7.7

      \[\leadsto \frac{\mathsf{fma}\left(\frac{9}{2}, \color{blue}{\left(a \cdot \left(a \cdot \left(\sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)}\right)\right)\right)} \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.3967409376004402132842585615435382351279:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{9}{2}, \left(a \cdot \left(a \cdot \left(\sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, {x}^{3} \cdot a\right)}\right)\right)\right) \cdot \sqrt[3]{{x}^{2} + {x}^{3} \cdot a}, 3 \cdot \left(a \cdot x\right)\right)}{\mathsf{fma}\left(1, e^{a \cdot x} + 1, e^{2 \cdot \left(a \cdot x\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))