Average Error: 58.6 → 0.5
Time: 1.9s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)\]
e^{x} - 1
↓
\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)
double code(double x) {
	return (exp(x) - 1.0);
}
↓
double code(double x) {
	return fma(fma(0.16666666666666666, x, 0.5), 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

Original58.6
Target0.5
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 add-sqr-sqrt58.7

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

  4. Applied fma-neg58.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, -1\right)}\]

  5. Taylor expanded around 0 0.5

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

  6. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)}\]

  7. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

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

  (- (exp x) 1))