Average Error: 29.2 → 0.5
Time: 2.7s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.533591632706496 \cdot 10^{-05}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + 0.5 \cdot \sqrt[3]{{\left(a \cdot x\right)}^{6}}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -2.533591632706496 \cdot 10^{-05}:\\
\;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + 0.5 \cdot \sqrt[3]{{\left(a \cdot x\right)}^{6}}\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -2.533591632706496e-05)
   (log (exp (- (exp (* a x)) 1.0)))
   (+ (* a x) (* 0.5 (cbrt (pow (* a x) 6.0))))))
double code(double a, double x) {
	return ((double) (((double) exp(((double) (a * x)))) - 1.0));
}
↓
double code(double a, double x) {
	double tmp;
	if ((((double) (a * x)) <= -2.533591632706496e-05)) {
		tmp = ((double) log(((double) exp(((double) (((double) exp(((double) (a * x)))) - 1.0))))));
	} else {
		tmp = ((double) (((double) (a * x)) + ((double) (0.5 * ((double) cbrt(((double) pow(((double) (a * x)), 6.0))))))));
	}
	return tmp;
}

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -2.533591632706496e-5

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary640.1

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

    4. Applied add-log-exp_binary640.1

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

    5. Applied diff-log_binary640.1

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

    6. Simplified0.1

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

    if -2.533591632706496e-5 < (*.f64 a x)

    1. Initial program 44.1

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

    2. Taylor expanded around 0 15.0

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified8.0

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]

    4. Taylor expanded around 0 8.5

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x}\]

    5. Simplified4.8

      \[\leadsto \color{blue}{x \cdot a + 0.5 \cdot \left(x \cdot \left(x \cdot \left(a \cdot a\right)\right)\right)}\]
    6. Using strategy rm

    7. Applied add-cbrt-cube_binary648.1

      \[\leadsto x \cdot a + 0.5 \cdot \left(x \cdot \left(x \cdot \left(a \cdot \color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}}\right)\right)\right)\]

    8. Applied add-cbrt-cube_binary648.1

      \[\leadsto x \cdot a + 0.5 \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot a}\right)\right)\right)\]

    9. Applied cbrt-unprod_binary6412.0

      \[\leadsto x \cdot a + 0.5 \cdot \left(x \cdot \left(x \cdot \color{blue}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}\right)\right)\]

    10. Applied add-cbrt-cube_binary6418.9

      \[\leadsto x \cdot a + 0.5 \cdot \left(x \cdot \left(\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \cdot \sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}\right)\right)\]

    11. Applied cbrt-unprod_binary6418.9

      \[\leadsto x \cdot a + 0.5 \cdot \left(x \cdot \color{blue}{\sqrt[3]{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right)}}\right)\]

    12. Applied add-cbrt-cube_binary6418.9

      \[\leadsto x \cdot a + 0.5 \cdot \left(\color{blue}{\sqrt[3]{\left(x \cdot x\right) \cdot x}} \cdot \sqrt[3]{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right)}\right)\]

    13. Applied cbrt-unprod_binary6418.9

      \[\leadsto x \cdot a + 0.5 \cdot \color{blue}{\sqrt[3]{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right)\right)}}\]

    14. Simplified0.7

      \[\leadsto x \cdot a + 0.5 \cdot \sqrt[3]{\color{blue}{{\left(x \cdot a\right)}^{6}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.533591632706496 \cdot 10^{-05}:\\ \;\;\;\;\log \left(e^{e^{a \cdot x} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + 0.5 \cdot \sqrt[3]{{\left(a \cdot x\right)}^{6}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))