Average Error: 29.9 → 0.7
Time: 7.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r182165 = x;
        double r182166 = exp(r182165);
        double r182167 = 2.0;
        double r182168 = r182166 - r182167;
        double r182169 = -r182165;
        double r182170 = exp(r182169);
        double r182171 = r182168 + r182170;
        return r182171;
}
↓
double f(double x) {
        double r182172 = x;
        double r182173 = 2.0;
        double r182174 = pow(r182172, r182173);
        double r182175 = 0.002777777777777778;
        double r182176 = 6.0;
        double r182177 = pow(r182172, r182176);
        double r182178 = r182175 * r182177;
        double r182179 = 0.08333333333333333;
        double r182180 = 4.0;
        double r182181 = pow(r182172, r182180);
        double r182182 = r182179 * r182181;
        double r182183 = r182178 + r182182;
        double r182184 = r182174 + r182183;
        return r182184;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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