Average Error: 28.7 → 8.7
Time: 3.8s
Precision: 64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.98282351915828164 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{elif}\;a \cdot x \le 2.1553377009136073 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.98282351915828164 \cdot 10^{-7}:\\
\;\;\;\;\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{e^{a \cdot x} + 1}}\\

\mathbf{elif}\;a \cdot x \le 2.1553377009136073 \cdot 10^{-16}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\

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

\end{array}
double f(double a, double x) {
        double r176743 = a;
        double r176744 = x;
        double r176745 = r176743 * r176744;
        double r176746 = exp(r176745);
        double r176747 = 1.0;
        double r176748 = r176746 - r176747;
        return r176748;
}
↓
double f(double a, double x) {
        double r176749 = a;
        double r176750 = x;
        double r176751 = r176749 * r176750;
        double r176752 = -3.9828235191582816e-07;
        bool r176753 = r176751 <= r176752;
        double r176754 = exp(r176751);
        double r176755 = 1.0;
        double r176756 = r176754 - r176755;
        double r176757 = cbrt(r176756);
        double r176758 = r176754 * r176754;
        double r176759 = r176755 * r176755;
        double r176760 = r176758 - r176759;
        double r176761 = r176760 * r176756;
        double r176762 = cbrt(r176761);
        double r176763 = r176754 + r176755;
        double r176764 = cbrt(r176763);
        double r176765 = r176762 / r176764;
        double r176766 = r176757 * r176765;
        double r176767 = 2.1553377009136073e-16;
        bool r176768 = r176751 <= r176767;
        double r176769 = 0.5;
        double r176770 = 2.0;
        double r176771 = pow(r176749, r176770);
        double r176772 = pow(r176750, r176770);
        double r176773 = r176771 * r176772;
        double r176774 = 0.16666666666666666;
        double r176775 = 3.0;
        double r176776 = pow(r176749, r176775);
        double r176777 = pow(r176750, r176775);
        double r176778 = r176776 * r176777;
        double r176779 = fma(r176774, r176778, r176751);
        double r176780 = fma(r176769, r176773, r176779);
        double r176781 = exp(r176756);
        double r176782 = log(r176781);
        double r176783 = r176768 ? r176780 : r176782;
        double r176784 = r176753 ? r176766 : r176783;
        return r176784;
}

Error

Bits error versus a

Bits error versus x

Target

Original28.7
Target0.2
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* a x) < -3.9828235191582816e-07

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    6. Applied cube-mult0.1

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

    7. Applied cbrt-prod0.1

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

    9. Applied flip--0.1

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

    10. Applied associate-*r/0.1

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

    11. Applied cbrt-div0.1

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

    12. Simplified0.1

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

    if -3.9828235191582816e-07 < (* a x) < 2.1553377009136073e-16

    1. Initial program 44.9

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

    2. Taylor expanded around 0 13.0

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

    3. Simplified13.0

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

    if 2.1553377009136073e-16 < (* a x)

    1. Initial program 17.4

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

    3. Applied add-log-exp17.4

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

    4. Applied add-log-exp24.5

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

    5. Applied diff-log24.9

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

    6. Simplified24.5

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.98282351915828164 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{e^{a \cdot x} - 1} \cdot \frac{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{e^{a \cdot x} + 1}}\\ \mathbf{elif}\;a \cdot x \le 2.1553377009136073 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {x}^{3}, a \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))