Average Error: 29.1 → 9.6
Time: 4.5s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -3.21301735111499177 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{e^{a \cdot x} \cdot \left(\left(\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} + 1}}\right)\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -3.21301735111499177 \cdot 10^{-23}:\\
\;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{e^{a \cdot x} \cdot \left(\left(\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} + 1}}\right)\right) + 1 \cdot 1}\\

\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 code(double a, double x) {
	return ((double) (((double) exp(((double) (a * x)))) - 1.0));
}
↓
double code(double a, double x) {
	double VAR;
	if ((((double) (a * x)) <= -3.213017351114992e-23)) {
		VAR = ((double) (((double) (((double) exp(((double) (((double) (a * x)) * 3.0)))) - ((double) pow(1.0, 3.0)))) / ((double) (((double) (((double) exp(((double) (a * x)))) * ((double) (((double) (((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0)))) * ((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0)))))) * ((double) (((double) cbrt(((double) (((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0)))) * ((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0)))))))) * ((double) cbrt(((double) cbrt(((double) (((double) exp(((double) (a * x)))) + 1.0)))))))))))) + ((double) (1.0 * 1.0))))));
	} else {
		VAR = ((double) (((double) (x * ((double) (a + ((double) (((double) (0.5 * ((double) pow(a, 2.0)))) * x)))))) + ((double) (0.16666666666666666 * ((double) (((double) pow(a, 3.0)) * ((double) pow(x, 3.0))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -3.21301735111499177e-23

    1. Initial program 2.0

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

    3. Applied flip3--2.0

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

    4. Simplified2.0

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

    6. Applied pow-exp1.9

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

    8. Applied add-cube-cbrt1.9

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

    10. Applied add-cube-cbrt1.9

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

    11. Applied cbrt-prod1.9

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

    if -3.21301735111499177e-23 < (* a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 13.9

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

    3. Simplified13.9

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -3.21301735111499177 \cdot 10^{-23}:\\ \;\;\;\;\frac{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}{e^{a \cdot x} \cdot \left(\left(\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}\right) \cdot \left(\sqrt[3]{\sqrt[3]{e^{a \cdot x} + 1} \cdot \sqrt[3]{e^{a \cdot x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{a \cdot x} + 1}}\right)\right) + 1 \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))