Average Error: 58.6 → 0.4
Time: 12.9s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + {x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\]
e^{x} - 1
↓
x + {x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)
double f(double x) {
        double r80152 = x;
        double r80153 = exp(r80152);
        double r80154 = 1.0;
        double r80155 = r80153 - r80154;
        return r80155;
}
↓
double f(double x) {
        double r80156 = x;
        double r80157 = 2.0;
        double r80158 = pow(r80156, r80157);
        double r80159 = 0.5;
        double r80160 = 0.16666666666666666;
        double r80161 = r80156 * r80160;
        double r80162 = r80159 + r80161;
        double r80163 = r80158 * r80162;
        double r80164 = r80156 + r80163;
        return r80164;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}\]

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1.0 (/ x 2.0)) (/ (* x x) 6.0)))

  (- (exp x) 1.0))