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 f(double x) {
        double r66485 = x;
        double r66486 = exp(r66485);
        double r66487 = -r66485;
        double r66488 = exp(r66487);
        double r66489 = r66486 - r66488;
        double r66490 = 2.0;
        double r66491 = r66489 / r66490;
        return r66491;
}
↓
double f(double x) {
        double r66492 = x;
        double r66493 = exp(r66492);
        double r66494 = sqrt(r66493);
        double r66495 = -r66492;
        double r66496 = exp(r66495);
        double r66497 = sqrt(r66496);
        double r66498 = r66494 + r66497;
        double r66499 = 0.041666666666666664;
        double r66500 = 3.0;
        double r66501 = pow(r66492, r66500);
        double r66502 = r66499 * r66501;
        double r66503 = 0.0005208333333333333;
        double r66504 = 5.0;
        double r66505 = pow(r66492, r66504);
        double r66506 = r66503 * r66505;
        double r66507 = r66506 + r66492;
        double r66508 = r66502 + r66507;
        double r66509 = r66498 * r66508;
        double r66510 = 2.0;
        double r66511 = r66509 / r66510;
        return r66511;
}

Error

Bits error versus x

Try it out

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))