Average Error: 29.4 → 10.2
Time: 4.6s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.2540867210978604 \cdot 10^{-60}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(e^{a \cdot \left(x + x\right)} - 1 \cdot 1\right)}^{3}}{{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right)}^{3}}} \cdot \left(1 \cdot \left(1 - e^{a \cdot x}\right) + e^{a \cdot x + a \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.2540867210978604 \cdot 10^{-60}:\\
\;\;\;\;\sqrt[3]{\frac{{\left(e^{a \cdot \left(x + x\right)} - 1 \cdot 1\right)}^{3}}{{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right)}^{3}}} \cdot \left(1 \cdot \left(1 - e^{a \cdot x}\right) + e^{a \cdot x + a \cdot x}\right)\\

\mathbf{else}:\\
\;\;\;\;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)\\

\end{array}
double f(double a, double x) {
        double r108781 = a;
        double r108782 = x;
        double r108783 = r108781 * r108782;
        double r108784 = exp(r108783);
        double r108785 = 1.0;
        double r108786 = r108784 - r108785;
        return r108786;
}
↓
double f(double a, double x) {
        double r108787 = a;
        double r108788 = x;
        double r108789 = r108787 * r108788;
        double r108790 = -6.25408672109786e-60;
        bool r108791 = r108789 <= r108790;
        double r108792 = r108788 + r108788;
        double r108793 = r108787 * r108792;
        double r108794 = exp(r108793);
        double r108795 = 1.0;
        double r108796 = r108795 * r108795;
        double r108797 = r108794 - r108796;
        double r108798 = 3.0;
        double r108799 = pow(r108797, r108798);
        double r108800 = exp(r108789);
        double r108801 = pow(r108800, r108798);
        double r108802 = pow(r108795, r108798);
        double r108803 = r108801 + r108802;
        double r108804 = pow(r108803, r108798);
        double r108805 = r108799 / r108804;
        double r108806 = cbrt(r108805);
        double r108807 = r108795 - r108800;
        double r108808 = r108795 * r108807;
        double r108809 = r108789 + r108789;
        double r108810 = exp(r108809);
        double r108811 = r108808 + r108810;
        double r108812 = r108806 * r108811;
        double r108813 = 0.5;
        double r108814 = 2.0;
        double r108815 = pow(r108787, r108814);
        double r108816 = r108813 * r108815;
        double r108817 = r108816 * r108788;
        double r108818 = r108787 + r108817;
        double r108819 = r108788 * r108818;
        double r108820 = 0.16666666666666666;
        double r108821 = pow(r108787, r108798);
        double r108822 = pow(r108788, r108798);
        double r108823 = r108821 * r108822;
        double r108824 = r108820 * r108823;
        double r108825 = r108819 + r108824;
        double r108826 = r108791 ? r108812 : r108825;
        return r108826;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.2
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -6.25408672109786e-60

    1. Initial program 7.9

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

    3. Applied add-cbrt-cube7.9

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

    4. Simplified7.9

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

    6. Applied flip--7.9

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

    7. Applied cube-div7.9

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

    9. Applied prod-exp7.8

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

    10. Simplified7.8

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

    12. Applied flip3-+7.8

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

    13. Applied cube-div7.8

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

    14. Applied associate-/r/7.8

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

    15. Applied cbrt-prod7.8

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

    16. Simplified7.8

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

    if -6.25408672109786e-60 < (* a x)

    1. Initial program 43.3

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

    2. Taylor expanded around 0 11.7

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

    3. Simplified11.7

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.2540867210978604 \cdot 10^{-60}:\\ \;\;\;\;\sqrt[3]{\frac{{\left(e^{a \cdot \left(x + x\right)} - 1 \cdot 1\right)}^{3}}{{\left({\left(e^{a \cdot x}\right)}^{3} + {1}^{3}\right)}^{3}}} \cdot \left(1 \cdot \left(1 - e^{a \cdot x}\right) + e^{a \cdot x + a \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(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))