Average Error: 29.2 → 9.2
Time: 3.7s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{{\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + \left(-{1}^{6}\right)\right)}^{3}}{{\left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{{\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + \left(-{1}^{6}\right)\right)}^{3}}{{\left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

\mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\
\;\;\;\;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{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\

\end{array}
double f(double a, double x) {
        double r76371 = a;
        double r76372 = x;
        double r76373 = r76371 * r76372;
        double r76374 = exp(r76373);
        double r76375 = 1.0;
        double r76376 = r76374 - r76375;
        return r76376;
}
↓
double f(double a, double x) {
        double r76377 = a;
        double r76378 = x;
        double r76379 = r76377 * r76378;
        double r76380 = -3.569795479894664e-07;
        bool r76381 = r76379 <= r76380;
        double r76382 = 3.0;
        double r76383 = r76379 * r76382;
        double r76384 = r76383 + r76383;
        double r76385 = exp(r76384);
        double r76386 = 1.0;
        double r76387 = 6.0;
        double r76388 = pow(r76386, r76387);
        double r76389 = -r76388;
        double r76390 = r76385 + r76389;
        double r76391 = pow(r76390, r76382);
        double r76392 = exp(r76383);
        double r76393 = pow(r76386, r76382);
        double r76394 = r76392 + r76393;
        double r76395 = pow(r76394, r76382);
        double r76396 = r76391 / r76395;
        double r76397 = cbrt(r76396);
        double r76398 = exp(r76379);
        double r76399 = r76398 + r76386;
        double r76400 = r76398 * r76399;
        double r76401 = r76386 * r76386;
        double r76402 = r76400 + r76401;
        double r76403 = r76397 / r76402;
        double r76404 = 2.9747332644534994e-19;
        bool r76405 = r76379 <= r76404;
        double r76406 = 0.5;
        double r76407 = 2.0;
        double r76408 = pow(r76377, r76407);
        double r76409 = r76406 * r76408;
        double r76410 = r76409 * r76378;
        double r76411 = r76377 + r76410;
        double r76412 = r76378 * r76411;
        double r76413 = 0.16666666666666666;
        double r76414 = pow(r76377, r76382);
        double r76415 = pow(r76378, r76382);
        double r76416 = r76414 * r76415;
        double r76417 = r76413 * r76416;
        double r76418 = r76412 + r76417;
        double r76419 = r76392 - r76393;
        double r76420 = pow(r76419, r76382);
        double r76421 = cbrt(r76420);
        double r76422 = r76421 / r76402;
        double r76423 = r76405 ? r76418 : r76422;
        double r76424 = r76381 ? r76403 : r76423;
        return r76424;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -3.569795479894664e-07

    1. Initial program 0.2

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

    3. Applied flip3--0.2

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

    4. Simplified0.2

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

    6. Applied pow-exp0.2

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

    8. Applied add-cbrt-cube0.2

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

    9. Simplified0.2

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

    11. Applied flip--0.2

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

    12. Applied cube-div0.2

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

    13. Simplified0.2

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

    if -3.569795479894664e-07 < (* a x) < 2.9747332644534994e-19

    1. Initial program 45.1

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

    2. Taylor expanded around 0 13.5

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

    3. Simplified13.5

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

    if 2.9747332644534994e-19 < (* a x)

    1. Initial program 25.4

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

    3. Applied flip3--26.4

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

    4. Simplified26.4

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

    6. Applied pow-exp25.0

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

    8. Applied add-cbrt-cube26.9

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

    9. Simplified26.9

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.56979547989466417 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{{\left(e^{\left(a \cdot x\right) \cdot 3 + \left(a \cdot x\right) \cdot 3} + \left(-{1}^{6}\right)\right)}^{3}}{{\left(e^{\left(a \cdot x\right) \cdot 3} + {1}^{3}\right)}^{3}}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \mathbf{elif}\;a \cdot x \le 2.97473326445349939 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}\right)}^{3}}}{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))