Average Error: 58.6 → 0.4
Time: 2.4s
Precision: binary64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[0.5 \cdot {x}^{2} + \left(0.16666666666666663 \cdot {x}^{3} + 1 \cdot x\right)\]
e^{x} - 1
↓
0.5 \cdot {x}^{2} + \left(0.16666666666666663 \cdot {x}^{3} + 1 \cdot x\right)
double code(double x) {
	return ((double) (((double) exp(x)) - 1.0));
}
↓
double code(double x) {
	return ((double) (((double) (0.5 * ((double) pow(x, 2.0)))) + ((double) (((double) (0.16666666666666663 * ((double) pow(x, 3.0)))) + ((double) (1.0 * x))))));
}

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. Using strategy rm

  3. Applied flip--58.7

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

  4. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{0.5 \cdot {x}^{2} + \left(0.16666666666666663 \cdot {x}^{3} + 1 \cdot x\right)}\]

  5. Final simplification0.4

    \[\leadsto 0.5 \cdot {x}^{2} + \left(0.16666666666666663 \cdot {x}^{3} + 1 \cdot x\right)\]

Reproduce

herbie shell --seed 2020163 
(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))