Average Error: 29.4 → 0.7
Time: 28.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)
double f(double x) {
        double r3730017 = x;
        double r3730018 = exp(r3730017);
        double r3730019 = 2.0;
        double r3730020 = r3730018 - r3730019;
        double r3730021 = -r3730017;
        double r3730022 = exp(r3730021);
        double r3730023 = r3730020 + r3730022;
        return r3730023;
}
↓
double f(double x) {
        double r3730024 = x;
        double r3730025 = r3730024 * r3730024;
        double r3730026 = r3730024 * r3730025;
        double r3730027 = 0.002777777777777778;
        double r3730028 = r3730026 * r3730027;
        double r3730029 = r3730028 * r3730026;
        double r3730030 = 0.08333333333333333;
        double r3730031 = r3730025 * r3730025;
        double r3730032 = r3730030 * r3730031;
        double r3730033 = r3730025 + r3730032;
        double r3730034 = r3730029 + r3730033;
        return r3730034;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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