Average Error: 58.8 → 0.4
Time: 4.4s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x
double f(double x) {
        double r166219 = x;
        double r166220 = exp(r166219);
        double r166221 = 1.0;
        double r166222 = r166220 - r166221;
        return r166222;
}
↓
double f(double x) {
        double r166223 = x;
        double r166224 = 2.0;
        double r166225 = pow(r166223, r166224);
        double r166226 = 0.16666666666666666;
        double r166227 = r166223 * r166226;
        double r166228 = 0.5;
        double r166229 = r166227 + r166228;
        double r166230 = r166225 * r166229;
        double r166231 = r166230 + r166223;
        return r166231;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019318 
(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))