Average Error: 58.6 → 0.4
Time: 18.2s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r4594932 = x;
        double r4594933 = exp(r4594932);
        double r4594934 = 1.0;
        double r4594935 = r4594933 - r4594934;
        return r4594935;
}
↓
double f(double x) {
        double r4594936 = x;
        double r4594937 = 0.5;
        double r4594938 = 0.16666666666666666;
        double r4594939 = r4594938 * r4594936;
        double r4594940 = r4594937 + r4594939;
        double r4594941 = r4594936 * r4594936;
        double r4594942 = r4594940 * r4594941;
        double r4594943 = r4594936 + r4594942;
        return r4594943;
}

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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))