Average Error: 40.1 → 0.4
Time: 2.8s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0001507114205570198:\\ \;\;\;\;\frac{\sqrt[3]{e^{x} - 1} \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} - 1}}\right)\right)}{\frac{x}{\sqrt[3]{\frac{{\left(e^{x}\right)}^{3} - 1}{1 + e^{x} \cdot \left(e^{x} + 1\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0001507114205570198:\\
\;\;\;\;\frac{\sqrt[3]{e^{x} - 1} \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} - 1}}\right)\right)}{\frac{x}{\sqrt[3]{\frac{{\left(e^{x}\right)}^{3} - 1}{1 + e^{x} \cdot \left(e^{x} + 1\right)}}}}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.0001507114205570198)
   (/
    (*
     (cbrt (- (exp x) 1.0))
     (*
      (cbrt (cbrt (- (exp x) 1.0)))
      (* (cbrt (cbrt (- (exp x) 1.0))) (cbrt (cbrt (- (exp x) 1.0))))))
    (/
     x
     (cbrt (/ (- (pow (exp x) 3.0) 1.0) (+ 1.0 (* (exp x) (+ (exp x) 1.0)))))))
   (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.0001507114205570198) {
		tmp = (cbrt(exp(x) - 1.0) * (cbrt(cbrt(exp(x) - 1.0)) * (cbrt(cbrt(exp(x) - 1.0)) * cbrt(cbrt(exp(x) - 1.0))))) / (x / cbrt((pow(exp(x), 3.0) - 1.0) / (1.0 + (exp(x) * (exp(x) + 1.0)))));
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target40.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.5071142055701981e-4

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt_binary640.1

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

    4. Applied associate-/l*_binary640.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x} - 1} \cdot \sqrt[3]{e^{x} - 1}}{\frac{x}{\sqrt[3]{e^{x} - 1}}}}\]
    5. Using strategy rm

    6. Applied flip3--_binary640.1

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

    7. Simplified0.1

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

    8. Simplified0.1

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

    10. Applied add-cube-cbrt_binary640.1

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

    if -1.5071142055701981e-4 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.16666666666666666 \cdot {x}^{2} + 1\right)}\]

    3. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0001507114205570198:\\ \;\;\;\;\frac{\sqrt[3]{e^{x} - 1} \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1}} \cdot \left(\sqrt[3]{\sqrt[3]{e^{x} - 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} - 1}}\right)\right)}{\frac{x}{\sqrt[3]{\frac{{\left(e^{x}\right)}^{3} - 1}{1 + e^{x} \cdot \left(e^{x} + 1\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020263 
(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))