Average Error: 40.0 → 0.3
Time: 5.8s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0013678994670240129:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{x}\right)}^{2} + -1} \cdot \sqrt[3]{{\left(e^{x}\right)}^{2} + -1}}{x} \cdot \frac{\sqrt[3]{{\left(e^{x}\right)}^{2} + -1}}{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.0013678994670240129:\\
\;\;\;\;\frac{\sqrt[3]{{\left(e^{x}\right)}^{2} + -1} \cdot \sqrt[3]{{\left(e^{x}\right)}^{2} + -1}}{x} \cdot \frac{\sqrt[3]{{\left(e^{x}\right)}^{2} + -1}}{e^{x} + 1}\\

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

\end{array}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -0.0013678994670240129)
   (*
    (/
     (* (cbrt (+ (pow (exp x) 2.0) -1.0)) (cbrt (+ (pow (exp x) 2.0) -1.0)))
     x)
    (/ (cbrt (+ (pow (exp x) 2.0) -1.0)) (+ (exp x) 1.0)))
   (+
    1.0
    (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	double tmp;
	if (x <= -0.0013678994670240129) {
		tmp = ((cbrt(pow(exp(x), 2.0) + -1.0) * cbrt(pow(exp(x), 2.0) + -1.0)) / x) * (cbrt(pow(exp(x), 2.0) + -1.0) / (exp(x) + 1.0));
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target40.3
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 < -0.0013678994670240129

    1. Initial program 0.0

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

    3. Applied flip--_binary64_24400.0

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

    4. Applied associate-/l/_binary64_24120.0

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

    6. Applied add-cube-cbrt_binary64_25000.0

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

    7. Applied times-frac_binary64_24710.0

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

    8. Simplified0.0

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

    9. Simplified0.0

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

    if -0.0013678994670240129 < 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. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0013678994670240129:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{x}\right)}^{2} + -1} \cdot \sqrt[3]{{\left(e^{x}\right)}^{2} + -1}}{x} \cdot \frac{\sqrt[3]{{\left(e^{x}\right)}^{2} + -1}}{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021007 
(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))