Average Error: 58.6 → 0.5
Time: 8.6s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[\left(\frac{1}{8} \cdot \left(x \cdot x\right) + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{48} + \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + \sqrt{e^{x}}\right)\]
e^{x} - 1
↓
\left(\frac{1}{8} \cdot \left(x \cdot x\right) + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{48} + \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + \sqrt{e^{x}}\right)
double f(double x) {
        double r920706 = x;
        double r920707 = exp(r920706);
        double r920708 = 1.0;
        double r920709 = r920707 - r920708;
        return r920709;
}
↓
double f(double x) {
        double r920710 = 0.125;
        double r920711 = x;
        double r920712 = r920711 * r920711;
        double r920713 = r920710 * r920712;
        double r920714 = r920711 * r920712;
        double r920715 = 0.020833333333333332;
        double r920716 = r920714 * r920715;
        double r920717 = 0.5;
        double r920718 = r920717 * r920711;
        double r920719 = r920716 + r920718;
        double r920720 = r920713 + r920719;
        double r920721 = 1.0;
        double r920722 = exp(r920711);
        double r920723 = sqrt(r920722);
        double r920724 = r920721 + r920723;
        double r920725 = r920720 * r920724;
        return r920725;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.6
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. Using strategy rm

  3. Applied *-un-lft-identity58.6

    \[\leadsto e^{x} - \color{blue}{1 \cdot 1}\]

  4. Applied add-sqr-sqrt58.7

    \[\leadsto \color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - 1 \cdot 1\]

  5. Applied difference-of-squares58.7

    \[\leadsto \color{blue}{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\sqrt{e^{x}} - 1\right)}\]

  6. Taylor expanded around 0 0.5

    \[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{8} \cdot {x}^{2}\right)\right)}\]

  7. Simplified0.5

    \[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot x + \frac{1}{48} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x \cdot x\right) \cdot \frac{1}{8}\right)}\]

  8. Final simplification0.5

    \[\leadsto \left(\frac{1}{8} \cdot \left(x \cdot x\right) + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{48} + \frac{1}{2} \cdot x\right)\right) \cdot \left(1 + \sqrt{e^{x}}\right)\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))