Average Error: 29.7 → 9.8
Time: 3.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} - \sqrt{{1}^{3}}\right)}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + {1}^{\frac{3}{2}}\right) \cdot e^{\log \left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} - {1}^{\frac{3}{2}}\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19}:\\
\;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} - \sqrt{{1}^{3}}\right)}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

\mathbf{elif}\;a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}:\\
\;\;\;\;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)\\

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

\end{array}
double f(double a, double x) {
        double r123367 = a;
        double r123368 = x;
        double r123369 = r123367 * r123368;
        double r123370 = exp(r123369);
        double r123371 = 1.0;
        double r123372 = r123370 - r123371;
        return r123372;
}
↓
double f(double a, double x) {
        double r123373 = a;
        double r123374 = x;
        double r123375 = r123373 * r123374;
        double r123376 = -6.883670130696454e-19;
        bool r123377 = r123375 <= r123376;
        double r123378 = exp(r123375);
        double r123379 = 3.0;
        double r123380 = pow(r123378, r123379);
        double r123381 = sqrt(r123380);
        double r123382 = 1.0;
        double r123383 = pow(r123382, r123379);
        double r123384 = sqrt(r123383);
        double r123385 = r123381 + r123384;
        double r123386 = r123381 - r123384;
        double r123387 = r123385 * r123386;
        double r123388 = r123378 + r123382;
        double r123389 = r123378 * r123388;
        double r123390 = r123382 * r123382;
        double r123391 = r123389 + r123390;
        double r123392 = r123387 / r123391;
        double r123393 = 5.1452723817109815e-33;
        bool r123394 = r123375 <= r123393;
        double r123395 = 0.5;
        double r123396 = 2.0;
        double r123397 = pow(r123373, r123396);
        double r123398 = r123395 * r123397;
        double r123399 = r123398 * r123374;
        double r123400 = r123373 + r123399;
        double r123401 = r123374 * r123400;
        double r123402 = 0.16666666666666666;
        double r123403 = pow(r123373, r123379);
        double r123404 = pow(r123374, r123379);
        double r123405 = r123403 * r123404;
        double r123406 = r123402 * r123405;
        double r123407 = r123401 + r123406;
        double r123408 = 1.5;
        double r123409 = pow(r123378, r123408);
        double r123410 = pow(r123382, r123408);
        double r123411 = r123409 + r123410;
        double r123412 = r123409 - r123410;
        double r123413 = log(r123412);
        double r123414 = exp(r123413);
        double r123415 = r123411 * r123414;
        double r123416 = r123415 / r123391;
        double r123417 = r123394 ? r123407 : r123416;
        double r123418 = r123377 ? r123392 : r123417;
        return r123418;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.1000000000000000055511151231257827021182:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -6.883670130696454e-19

    1. Initial program 1.4

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

    3. Applied flip3--1.4

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

    4. Simplified1.4

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

    6. Applied add-sqr-sqrt1.4

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

    7. Applied add-sqr-sqrt1.5

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

    8. Applied difference-of-squares1.4

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

    if -6.883670130696454e-19 < (* a x) < 5.1452723817109815e-33

    1. Initial program 45.3

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

    2. Taylor expanded around 0 12.6

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

    3. Simplified12.6

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

    if 5.1452723817109815e-33 < (* a x)

    1. Initial program 38.5

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

    3. Applied flip3--39.2

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

    4. Simplified39.2

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

    6. Applied sqr-pow39.2

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

    7. Applied sqr-pow39.2

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

    8. Applied difference-of-squares39.3

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

    9. Simplified39.3

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

    10. Simplified39.3

      \[\leadsto \frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + {1}^{\frac{3}{2}}\right) \cdot \color{blue}{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} - {1}^{\frac{3}{2}}\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\]
    11. Using strategy rm

    12. Applied add-exp-log39.3

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.883670130696454110254072018681462839622 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} + \sqrt{{1}^{3}}\right) \cdot \left(\sqrt{{\left(e^{a \cdot x}\right)}^{3}} - \sqrt{{1}^{3}}\right)}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 5.145272381710981521082629097103889956475 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} + {1}^{\frac{3}{2}}\right) \cdot e^{\log \left({\left(e^{a \cdot x}\right)}^{\frac{3}{2}} - {1}^{\frac{3}{2}}\right)}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))