Average Error: 58.6 → 0.4
Time: 13.1s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x
double f(double x) {
        double r50129 = x;
        double r50130 = exp(r50129);
        double r50131 = 1.0;
        double r50132 = r50130 - r50131;
        return r50132;
}
↓
double f(double x) {
        double r50133 = x;
        double r50134 = 2.0;
        double r50135 = pow(r50133, r50134);
        double r50136 = 0.16666666666666666;
        double r50137 = r50136 * r50133;
        double r50138 = 0.5;
        double r50139 = r50137 + r50138;
        double r50140 = r50135 * r50139;
        double r50141 = r50140 + r50133;
        return r50141;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.5
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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -1.7e-4 x)

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

  (- (exp x) 1))