Average Error: 41.1 → 1.0
Time: 2.7s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{x + x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{x + x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (/ (exp x) (+ x (* x (* x (+ 0.5 (* x 0.16666666666666666)))))))
double code(double x) {
	return (((double) exp(x)) / ((double) (((double) exp(x)) - 1.0)));
}
↓
double code(double x) {
	return (((double) exp(x)) / ((double) (x + ((double) (x * ((double) (x * ((double) (0.5 + ((double) (x * 0.16666666666666666)))))))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.1
Target40.7
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program Error: 41.1 bits

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

  2. Taylor expanded around 0 Error: 11.5 bits

    \[\leadsto \frac{e^{x}}{\color{blue}{0.5 \cdot {x}^{2} + \left(0.16666666666666666 \cdot {x}^{3} + x\right)}}\]

  3. SimplifiedError: 1.0 bits

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

  4. Final simplificationError: 1.0 bits

    \[\leadsto \frac{e^{x}}{x + x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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