Average Error: 39.8 → 0.3
Time: 2.5s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.00016108891433325626:\\ \;\;\;\;\frac{\log \left(e^{\frac{e^{x + x} + -1}{e^{x} + 1}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(0.5 \cdot {x}^{2} + 0.16666666666666666 \cdot {x}^{3}\right)}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.00016108891433325626:\\
\;\;\;\;\frac{\log \left(e^{\frac{e^{x + x} + -1}{e^{x} + 1}}\right)}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{x + \left(0.5 \cdot {x}^{2} + 0.16666666666666666 \cdot {x}^{3}\right)}{x}\\

\end{array}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.00016108891433325626)
   (/ (log (exp (/ (+ (exp (+ x x)) -1.0) (+ (exp x) 1.0)))) x)
   (/ (+ x (+ (* 0.5 (pow x 2.0)) (* 0.16666666666666666 (pow x 3.0)))) x)))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.00016108891433325626) {
		tmp = log(exp((exp(x + x) + -1.0) / (exp(x) + 1.0))) / x;
	} else {
		tmp = (x + ((0.5 * pow(x, 2.0)) + (0.16666666666666666 * pow(x, 3.0)))) / x;
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target40.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.61088914333256257e-4

    1. Initial program 0.0

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

    3. Applied flip--_binary640.0

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

    4. Simplified0.0

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

    6. Applied pow-to-exp_binary640.0

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

    7. Simplified0.0

      \[\leadsto \frac{\frac{e^{\color{blue}{x + x}} + -1}{e^{x} + 1}}{x}\]
    8. Using strategy rm

    9. Applied add-log-exp_binary640.0

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

    if -1.61088914333256257e-4 < x

    1. Initial program 60.0

      \[\frac{e^{x} - 1}{x}\]

    2. Taylor expanded around 0 0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00016108891433325626:\\ \;\;\;\;\frac{\log \left(e^{\frac{e^{x + x} + -1}{e^{x} + 1}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(0.5 \cdot {x}^{2} + 0.16666666666666666 \cdot {x}^{3}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2021149 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))