Average Error: 41.4 → 0.4
Time: 2.0s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1} \]
↓
\[e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)} \]
\frac{e^{x}}{e^{x} - 1}
↓
e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)}

(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x) :precision binary64 (* (exp x) (/ 1.0 (expm1 x))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return exp(x) * (1.0 / expm1(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target40.9
Herbie0.4
\[\frac{1}{1 - e^{-x}} \]

Derivation

  1. Initial program 41.4

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

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]

  3. Applied div-inv_binary640.4

    \[\leadsto \color{blue}{e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)}} \]

  4. Final simplification0.4

    \[\leadsto e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)} \]

Reproduce

herbie shell --seed 1991292424 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))