Average Error: 57.9 → 0.6
Time: 4.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)\right)}{2}
double code(double x) {
	return ((exp(x) - exp(-x)) / 2.0);
}
↓
double code(double x) {
	return (((sqrt(exp(x)) + sqrt(exp(-x))) * ((0.041666666666666664 * pow(x, 3.0)) + ((0.0005208333333333333 * pow(x, 5.0)) + x))) / 2.0);
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  3. Applied add-sqr-sqrt57.9

    \[\leadsto \frac{e^{x} - \color{blue}{\sqrt{e^{-x}} \cdot \sqrt{e^{-x}}}}{2}\]

  4. Applied add-sqr-sqrt58.0

    \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - \sqrt{e^{-x}} \cdot \sqrt{e^{-x}}}{2}\]

  5. Applied difference-of-squares58.0

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

  6. Taylor expanded around 0 0.6

    \[\leadsto \frac{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)\right)}}{2}\]

  7. Final simplification0.6

    \[\leadsto \frac{\left(\sqrt{e^{x}} + \sqrt{e^{-x}}\right) \cdot \left(\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)\right)}{2}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))