Average Error: 58.7 → 0.4
Time: 5.4s
Precision: binary64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x
double code(double x) {
	return ((double) (((double) exp(x)) - 1.0));
}
↓
double code(double x) {
	return ((double) (((double) (((double) pow(x, 2.0)) * ((double) (0.5 + ((double) (0.16666666666666666 * x)))))) + x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1.0 (/ x 2.0)) (/ (* x x) 6.0)))

  (- (exp x) 1.0))