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 r100638 = a;
        double r100639 = x;
        double r100640 = r100638 * r100639;
        double r100641 = exp(r100640);
        double r100642 = 1.0;
        double r100643 = r100641 - r100642;
        return r100643;
}
↓
double f(double a, double x) {
        double r100644 = a;
        double r100645 = x;
        double r100646 = r100644 * r100645;
        double r100647 = -3.569795479894664e-07;
        bool r100648 = r100646 <= r100647;
        double r100649 = 3.0;
        double r100650 = r100646 * r100649;
        double r100651 = r100650 + r100650;
        double r100652 = exp(r100651);
        double r100653 = 1.0;
        double r100654 = 6.0;
        double r100655 = pow(r100653, r100654);
        double r100656 = -r100655;
        double r100657 = r100652 + r100656;
        double r100658 = pow(r100657, r100649);
        double r100659 = exp(r100650);
        double r100660 = pow(r100653, r100649);
        double r100661 = r100659 + r100660;
        double r100662 = pow(r100661, r100649);
        double r100663 = r100658 / r100662;
        double r100664 = cbrt(r100663);
        double r100665 = exp(r100646);
        double r100666 = r100665 + r100653;
        double r100667 = r100665 * r100666;
        double r100668 = r100653 * r100653;
        double r100669 = r100667 + r100668;
        double r100670 = r100664 / r100669;
        double r100671 = 2.9747332644534994e-19;
        bool r100672 = r100646 <= r100671;
        double r100673 = 0.5;
        double r100674 = 2.0;
        double r100675 = pow(r100644, r100674);
        double r100676 = r100673 * r100675;
        double r100677 = r100676 * r100645;
        double r100678 = r100644 + r100677;
        double r100679 = r100645 * r100678;
        double r100680 = 0.16666666666666666;
        double r100681 = pow(r100644, r100649);
        double r100682 = pow(r100645, r100649);
        double r100683 = r100681 * r100682;
        double r100684 = r100680 * r100683;
        double r100685 = r100679 + r100684;
        double r100686 = r100659 - r100660;
        double r100687 = pow(r100686, r100649);
        double r100688 = cbrt(r100687);
        double r100689 = r100688 / r100669;
        double r100690 = r100672 ? r100685 : r100689;
        double r100691 = r100648 ? r100670 : r100690;
        return r100691;
}

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))