Average Error: 58.6 → 0.4
Time: 20.0s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r6987152 = x;
        double r6987153 = exp(r6987152);
        double r6987154 = 1.0;
        double r6987155 = r6987153 - r6987154;
        return r6987155;
}
↓
double f(double x) {
        double r6987156 = x;
        double r6987157 = 0.16666666666666666;
        double r6987158 = r6987156 * r6987157;
        double r6987159 = 0.5;
        double r6987160 = r6987158 + r6987159;
        double r6987161 = r6987156 * r6987156;
        double r6987162 = r6987160 * r6987161;
        double r6987163 = r6987156 + r6987162;
        return r6987163;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019112 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))