Average Error: 58.6 → 0.4
Time: 3.3s
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 r94016 = x;
        double r94017 = exp(r94016);
        double r94018 = 1.0;
        double r94019 = r94017 - r94018;
        return r94019;
}
↓
double f(double x) {
        double r94020 = x;
        double r94021 = 2.0;
        double r94022 = pow(r94020, r94021);
        double r94023 = 0.16666666666666666;
        double r94024 = r94020 * r94023;
        double r94025 = 0.5;
        double r94026 = r94024 + r94025;
        double r94027 = r94022 * r94026;
        double r94028 = r94027 + r94020;
        return r94028;
}

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

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

  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 2020021 
(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))