Average Error: 40.0 → 40.0
Time: 4.3s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\frac{\frac{e^{x}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x}} - \frac{1}{x}\]
\frac{e^{x} - 1}{x}
↓
\frac{\frac{e^{x}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x}} - \frac{1}{x}
(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))
↓
(FPCore (x)
 :precision binary64
 (- (/ (/ (exp x) (* (cbrt x) (cbrt x))) (cbrt x)) (/ 1.0 x)))
double code(double x) {
	return (exp(x) - 1.0) / x;
}
↓
double code(double x) {
	return ((exp(x) / (cbrt(x) * cbrt(x))) / cbrt(x)) - (1.0 / x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target40.5
Herbie40.0
\[\begin{array}{l} \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Initial program 40.0

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

  3. Applied div-sub_binary64_168139.8

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

  5. Applied add-cube-cbrt_binary64_165840.0

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

  6. Applied associate-/r*_binary64_175040.0

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

  7. Final simplification40.0

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

Reproduce

herbie shell --seed 2020268 
(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))