Average Error: 29.5 → 0.6
Time: 18.4s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(\frac{1}{12} \cdot \left(x \cdot x\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(\frac{1}{12} \cdot \left(x \cdot x\right)\right)\right)
double f(double x) {
        double r3602432 = x;
        double r3602433 = exp(r3602432);
        double r3602434 = 2.0;
        double r3602435 = r3602433 - r3602434;
        double r3602436 = -r3602432;
        double r3602437 = exp(r3602436);
        double r3602438 = r3602435 + r3602437;
        return r3602438;
}
↓
double f(double x) {
        double r3602439 = x;
        double r3602440 = r3602439 * r3602439;
        double r3602441 = r3602439 * r3602440;
        double r3602442 = r3602441 * r3602441;
        double r3602443 = 0.002777777777777778;
        double r3602444 = r3602442 * r3602443;
        double r3602445 = 0.08333333333333333;
        double r3602446 = r3602445 * r3602440;
        double r3602447 = r3602440 * r3602446;
        double r3602448 = r3602440 + r3602447;
        double r3602449 = r3602444 + r3602448;
        return r3602449;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

    \[\left(e^{x} - 2\right) + e^{-x}\]

  2. Taylor expanded around inf 29.5

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

  3. Simplified29.5

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

  4. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]

  5. Simplified0.6

    \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{360} + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)\right)}\]

  6. Final simplification0.6

    \[\leadsto \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \left(x \cdot x + \left(x \cdot x\right) \cdot \left(\frac{1}{12} \cdot \left(x \cdot x\right)\right)\right)\]

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))