Average Error: 29.8 → 0.7
Time: 12.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{360} + \left(\left(\frac{1}{12} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + x \cdot x\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\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(\left(\frac{1}{12} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + x \cdot x\right)
double f(double x) {
        double r4034112 = x;
        double r4034113 = exp(r4034112);
        double r4034114 = 2.0;
        double r4034115 = r4034113 - r4034114;
        double r4034116 = -r4034112;
        double r4034117 = exp(r4034116);
        double r4034118 = r4034115 + r4034117;
        return r4034118;
}
↓
double f(double x) {
        double r4034119 = x;
        double r4034120 = r4034119 * r4034119;
        double r4034121 = r4034120 * r4034119;
        double r4034122 = r4034121 * r4034121;
        double r4034123 = 0.002777777777777778;
        double r4034124 = r4034122 * r4034123;
        double r4034125 = 0.08333333333333333;
        double r4034126 = r4034125 * r4034120;
        double r4034127 = r4034126 * r4034120;
        double r4034128 = r4034127 + r4034120;
        double r4034129 = r4034124 + r4034128;
        return r4034129;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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