Average Error: 28.8 → 0.4
Time: 5.3s
Precision: binary64
Cost: 53249
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.00031988822706872905:\\ \;\;\;\;\frac{{\left({\left(e^{a \cdot x}\right)}^{2}\right)}^{3} + -1}{\left(e^{a \cdot x} + 1\right) \cdot \left({\left(e^{a \cdot x}\right)}^{4} + \left({\left(e^{a \cdot x}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -0.00031988822706872905:\\
\;\;\;\;\frac{{\left({\left(e^{a \cdot x}\right)}^{2}\right)}^{3} + -1}{\left(e^{a \cdot x} + 1\right) \cdot \left({\left(e^{a \cdot x}\right)}^{4} + \left({\left(e^{a \cdot x}\right)}^{2} + 1\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.00031988822706872905)
   (/
    (+ (pow (pow (exp (* a x)) 2.0) 3.0) -1.0)
    (*
     (+ (exp (* a x)) 1.0)
     (+ (pow (exp (* a x)) 4.0) (+ (pow (exp (* a x)) 2.0) 1.0))))
   (/ (* 2.0 (* (* a x) (+ (* a x) 1.0))) (+ 1.0 (+ (* a x) 1.0)))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.00031988822706872905) {
		tmp = (pow(pow(exp(a * x), 2.0), 3.0) + -1.0) / ((exp(a * x) + 1.0) * (pow(exp(a * x), 4.0) + (pow(exp(a * x), 2.0) + 1.0)));
	} else {
		tmp = (2.0 * ((a * x) * ((a * x) + 1.0))) / (1.0 + ((a * x) + 1.0));
	}
	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

Original28.8
Target0.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Alternatives

Alternative 1
Error0.4
Cost20545
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.00031988822706872905:\\ \;\;\;\;\left({\left(e^{a \cdot x}\right)}^{2} + -1\right) \cdot \frac{1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\ \end{array}\]
Alternative 2
Error0.4
Cost20417
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.00031988822706872905:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\ \end{array}\]
Alternative 3
Error0.4
Cost7169
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.00031988822706872905:\\ \;\;\;\;e^{a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\ \end{array}\]
Alternative 4
Error0.6
Cost1665
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.2827972307980513:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\ \end{array}\]
Alternative 5
Error0.6
Cost1537
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.2827972307980513:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{a \cdot x + 2}\\ \end{array}\]
Alternative 6
Error1.1
Cost1281
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.2827972307980513:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{2}\\ \end{array}\]
Alternative 7
Error1.1
Cost641
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -2.2827972307980513:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]
Alternative 8
Error34.1
Cost706
\[\begin{array}{l} \mathbf{if}\;x \leq -2.330076445311019 \cdot 10^{-53}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.5363608940131005 \cdot 10^{-158}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 9
Error51.4
Cost64
\[0\]
Alternative 10
Error62.1
Cost64
\[1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -3.19888227068729051e-4

    1. Initial program 0.0

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

    3. Applied flip--_binary64_20990.0

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

    4. Simplified0.0

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

    6. Applied flip3-+_binary64_21270.0

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

    7. Applied associate-/l/_binary64_20710.0

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

    8. Simplified0.0

      \[\leadsto \frac{{\left({\left(e^{a \cdot x}\right)}^{2}\right)}^{3} + {-1}^{3}}{\color{blue}{\left({\left(e^{a \cdot x}\right)}^{4} + \left({\left(e^{a \cdot x}\right)}^{2} + 1\right)\right) \cdot \left(e^{a \cdot x} + 1\right)}}\]

    9. Simplified0.0

      \[\leadsto \color{blue}{\frac{{\left({\left(e^{a \cdot x}\right)}^{2}\right)}^{3} + -1}{\left(1 + e^{a \cdot x}\right) \cdot \left({\left(e^{a \cdot x}\right)}^{4} + \left(1 + {\left(e^{a \cdot x}\right)}^{2}\right)\right)}}\]

    if -3.19888227068729051e-4 < (*.f64 a x)

    1. Initial program 43.9

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

    3. Applied flip--_binary64_209943.9

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

    4. Simplified43.9

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

    5. Taylor expanded around 0 7.5

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

    6. Simplified0.6

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

    7. Taylor expanded around 0 0.6

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

    8. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.00031988822706872905:\\ \;\;\;\;\frac{{\left({\left(e^{a \cdot x}\right)}^{2}\right)}^{3} + -1}{\left(e^{a \cdot x} + 1\right) \cdot \left({\left(e^{a \cdot x}\right)}^{4} + \left({\left(e^{a \cdot x}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x + 1\right)\right)}{1 + \left(a \cdot x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))