Average Error: 58.2 → 0.0
Time: 7.4s
Precision: binary64
\[\frac{e^{x} - e^{-x}}{2} \]
↓
\[\frac{\sinh x \cdot 2}{2} \]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\sinh x \cdot 2}{2}

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))
↓
(FPCore (x) :precision binary64 (/ (* (sinh x) 2.0) 2.0))
double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}
↓
double code(double x) {
	return (sinh(x) * 2.0) / 2.0;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.2

    \[\frac{e^{x} - e^{-x}}{2} \]

  2. Applied add-log-exp_binary6458.4

    \[\leadsto \frac{e^{x} - \color{blue}{\log \left(e^{e^{-x}}\right)}}{2} \]

  3. Applied add-log-exp_binary6458.6

    \[\leadsto \frac{\color{blue}{\log \left(e^{e^{x}}\right)} - \log \left(e^{e^{-x}}\right)}{2} \]

  4. Applied diff-log_binary6458.6

    \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{e^{x}}}{e^{e^{-x}}}\right)}}{2} \]

  5. Simplified58.6

    \[\leadsto \frac{\log \color{blue}{\left(e^{e^{x} - e^{-x}}\right)}}{2} \]

  6. Applied sinh-undef_binary6458.5

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

  7. Applied exp-prod_binary6458.5

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

  8. Applied log-pow_binary640.0

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

  9. Simplified0.0

    \[\leadsto \frac{\sinh x \cdot \color{blue}{2}}{2} \]

  10. Final simplification0.0

    \[\leadsto \frac{\sinh x \cdot 2}{2} \]

Reproduce

herbie shell --seed 2021357 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))