Average Error: 41.2 → 0.4
Time: 3.0s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{1}{2 \cdot \sinh x} \cdot \left(1 + e^{x}\right)\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{1}{2 \cdot \sinh x} \cdot \left(1 + e^{x}\right)
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x) :precision binary64 (* (/ 1.0 (* 2.0 (sinh x))) (+ 1.0 (exp x))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return (1.0 / (2.0 * sinh(x))) * (1.0 + exp(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

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Target

Original41.2
Target40.8
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.2

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

  3. Applied flip--_binary64_107641.2

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

  4. Applied associate-/r/_binary64_104741.2

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

  5. Simplified41.2

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

  7. Applied sinh-undef_binary64_12940.4

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

  8. Final simplification0.4

    \[\leadsto \frac{1}{2 \cdot \sinh x} \cdot \left(1 + e^{x}\right)\]

Reproduce

herbie shell --seed 2020295 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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