Average Error: 58.6 → 0.5
Time: 2.1s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x
double f(double x) {
        double r86819 = x;
        double r86820 = exp(r86819);
        double r86821 = 1.0;
        double r86822 = r86820 - r86821;
        return r86822;
}
↓
double f(double x) {
        double r86823 = x;
        double r86824 = 2.0;
        double r86825 = pow(r86823, r86824);
        double r86826 = 0.16666666666666666;
        double r86827 = r86823 * r86826;
        double r86828 = 0.5;
        double r86829 = r86827 + r86828;
        double r86830 = r86825 * r86829;
        double r86831 = r86830 + r86823;
        return r86831;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020033 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

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

  (- (exp x) 1))