Average Error: 29.1 → 3.0
Time: 4.0s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -7.848923535993033 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{a \cdot x}\right)}^{2} + -1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -7.848923535993033 \cdot 10^{-11}:\\
\;\;\;\;\frac{\log \left(e^{{\left(e^{a \cdot x}\right)}^{2} + -1}\right)}{e^{a \cdot x} + 1}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -7.848923535993033e-11)
   (/ (log (exp (+ (pow (exp (* a x)) 2.0) -1.0))) (+ (exp (* a x)) 1.0))
   (* x (+ a (* x (* (* a a) (+ 0.5 (* a (* x 0.16666666666666666)))))))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -7.848923535993033e-11) {
		tmp = log(exp(pow(exp(a * x), 2.0) + -1.0)) / (exp(a * x) + 1.0);
	} else {
		tmp = x * (a + (x * ((a * a) * (0.5 + (a * (x * 0.16666666666666666))))));
	}
	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

Original29.1
Target0.2
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -7.8489235359930334e-11

    1. Initial program 0.4

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

    3. Applied flip--_binary64_14300.4

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

    4. Simplified0.4

      \[\leadsto \frac{\color{blue}{{\left(e^{a \cdot x}\right)}^{2} + -1}}{e^{a \cdot x} + 1}\]
    5. Using strategy rm

    6. Applied add-log-exp_binary64_14940.4

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

    7. Applied add-log-exp_binary64_14940.4

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

    8. Applied sum-log_binary64_15460.4

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

    9. Simplified0.4

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

    if -7.8489235359930334e-11 < (*.f64 a x)

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.2

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

    3. Simplified7.6

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

    4. Taylor expanded around 0 7.6

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

    5. Simplified4.4

      \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -7.848923535993033 \cdot 10^{-11}:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{a \cdot x}\right)}^{2} + -1}\right)}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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))