Average Error: 39.6 → 0.3
Time: 3.6s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0013751438330752465:\\ \;\;\;\;\frac{\left(1 + \sqrt{e^{x}}\right) \cdot \frac{-1 + {\left(\sqrt{e^{x}}\right)}^{3}}{1 + \left(\sqrt{e^{x}} + \sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + \left(0.16666666666666666 \cdot {x}^{2} + \left(1 + 0.041666666666666664 \cdot {x}^{3}\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0013751438330752465:\\
\;\;\;\;\frac{\left(1 + \sqrt{e^{x}}\right) \cdot \frac{-1 + {\left(\sqrt{e^{x}}\right)}^{3}}{1 + \left(\sqrt{e^{x}} + \sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}{x}\\

\mathbf{else}:\\
\;\;\;\;x \cdot 0.5 + \left(0.16666666666666666 \cdot {x}^{2} + \left(1 + 0.041666666666666664 \cdot {x}^{3}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.0013751438330752465)
   (/
    (*
     (+ 1.0 (sqrt (exp x)))
     (/
      (+ -1.0 (pow (sqrt (exp x)) 3.0))
      (+ 1.0 (+ (sqrt (exp x)) (* (sqrt (exp x)) (sqrt (exp x)))))))
    x)
   (+
    (* x 0.5)
    (+
     (* 0.16666666666666666 (pow x 2.0))
     (+ 1.0 (* 0.041666666666666664 (pow x 3.0)))))))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.0013751438330752465) {
		tmp = ((1.0 + sqrt(exp(x))) * ((-1.0 + pow(sqrt(exp(x)), 3.0)) / (1.0 + (sqrt(exp(x)) + (sqrt(exp(x)) * sqrt(exp(x))))))) / x;
	} else {
		tmp = (x * 0.5) + ((0.16666666666666666 * pow(x, 2.0)) + (1.0 + (0.041666666666666664 * pow(x, 3.0))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target40.0
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 < -0.0013751438330752465

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary64_31690.0

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

    4. Applied difference-of-sqr-1_binary64_31170.0

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

    5. Simplified0.0

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

    6. Simplified0.0

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

    8. Applied flip3-+_binary64_31500.0

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

    if -0.0013751438330752465 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.16666666666666666 \cdot {x}^{2} + \left(0.041666666666666664 \cdot {x}^{3} + 1\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0013751438330752465:\\ \;\;\;\;\frac{\left(1 + \sqrt{e^{x}}\right) \cdot \frac{-1 + {\left(\sqrt{e^{x}}\right)}^{3}}{1 + \left(\sqrt{e^{x}} + \sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 + \left(0.16666666666666666 \cdot {x}^{2} + \left(1 + 0.041666666666666664 \cdot {x}^{3}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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))