Average Error: 58.5 → 0.5
Time: 3.0s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(x \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot {x}^{2} + x\right)\]
e^{x} - 1
↓
{x}^{2} \cdot \left(x \cdot \frac{1}{6}\right) + \left(\frac{1}{2} \cdot {x}^{2} + x\right)
double code(double x) {
	return ((double) (((double) exp(x)) - 1.0));
}
↓
double code(double x) {
	return ((double) (((double) (((double) pow(x, 2.0)) * ((double) (x * 0.16666666666666666)))) + ((double) (((double) (0.5 * ((double) pow(x, 2.0)))) + x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.5
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied distribute-lft-in0.5

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

  6. Applied associate-+l+0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020130 
(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))