Average Error: 29.4 → 0.6
Time: 1.0m
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)
double f(double x) {
        double r10343602 = x;
        double r10343603 = exp(r10343602);
        double r10343604 = 2.0;
        double r10343605 = r10343603 - r10343604;
        double r10343606 = -r10343602;
        double r10343607 = exp(r10343606);
        double r10343608 = r10343605 + r10343607;
        return r10343608;
}
↓
double f(double x) {
        double r10343609 = x;
        double r10343610 = r10343609 * r10343609;
        double r10343611 = 0.08333333333333333;
        double r10343612 = r10343610 * r10343610;
        double r10343613 = r10343611 * r10343612;
        double r10343614 = r10343610 + r10343613;
        double r10343615 = 0.002777777777777778;
        double r10343616 = r10343612 * r10343610;
        double r10343617 = r10343615 * r10343616;
        double r10343618 = r10343614 + r10343617;
        return r10343618;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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