Average Error: 29.5 → 0.3
Time: 4.4s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.005563956516231315:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -0.005563956516231315:\\
\;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.005563956516231315)
   (+
    (log (sqrt (exp (- (exp (* a x)) 1.0))))
    (log (sqrt (exp (- (exp (* a x)) 1.0)))))
   (+ (* a x) (* (pow (* a x) 2.0) (+ 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) <= -0.005563956516231315) {
		tmp = log(sqrt(exp(exp(a * x) - 1.0))) + log(sqrt(exp(exp(a * x) - 1.0)));
	} else {
		tmp = (a * x) + (pow((a * x), 2.0) * (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

Original29.5
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -0.0055639565162313148

    1. Initial program 0.0

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

    3. Applied add-log-exp_binary64_11400.0

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

    4. Applied add-log-exp_binary64_11400.0

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

    5. Applied diff-log_binary64_11930.0

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

    6. Simplified0.0

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

    8. Applied add-sqr-sqrt_binary64_11230.0

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

    9. Applied log-prod_binary64_11870.0

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

    if -0.0055639565162313148 < (*.f64 a x)

    1. Initial program 44.5

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

    2. Taylor expanded around 0 14.4

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

    3. Simplified0.5

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

    4. Taylor expanded around 0 14.4

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

    5. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.005563956516231315:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + {\left(a \cdot x\right)}^{2} \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :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))