Average Error: 29.5 → 0.1
Time: 3.3s
Precision: binary64
\[e^{a \cdot x} - 1 \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(a \cdot x\right)\right)\right) \]
e^{a \cdot x} - 1
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(a \cdot x\right)\right)\right)

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x) :precision binary64 (log1p (expm1 (expm1 (* a x)))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	return log1p(expm1(expm1(a * x)));
}

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

Derivation

  1. Initial program 29.5

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

  3. Applied log1p-expm1-u_binary640.1

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(a \cdot x\right)\right)\right)} \]

  4. Final simplification0.1

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(a \cdot x\right)\right)\right) \]

Reproduce

herbie shell --seed 2022094 
(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))