Average Error: 39.9 → 39.8
Time: 5.1s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{1}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{e^{x}}} - \frac{1}{x}\]
\frac{e^{x} - 1}{x}
↓
\frac{1}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{e^{x}}} - \frac{1}{x}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (- (/ 1.0 (* (* (cbrt x) (cbrt x)) (/ (cbrt x) (exp x)))) (/ 1.0 x)))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	return (1.0 / ((cbrt(x) * cbrt(x)) * (cbrt(x) / exp(x)))) - (1.0 / x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target40.2
Herbie39.8
\[\begin{array}{l} \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 39.9

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

  3. Applied div-sub_binary64_344639.5

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

  5. Applied clear-num_binary64_345539.6

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

  7. Applied *-un-lft-identity_binary64_345239.6

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

  8. Applied add-cube-cbrt_binary64_342339.9

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

  9. Applied times-frac_binary64_344739.8

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

  10. Simplified39.8

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

  11. Final simplification39.8

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

Reproduce

herbie shell --seed 2020270 
(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))