Average Error: 58.8 → 0.4

Time: 3.2s

Precision: 64

\[-1.7 \cdot 10^{-4} \lt x\]

\[e^{x} - 1\]

↓

\[\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)\]

e^{x} - 1

↓

\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)

double code(double x) {
	return ((double) (((double) exp(x)) - 1.0));
}

↓

double code(double x) {
	return ((double) fma(0.5, ((double) pow(x, 2.0)), ((double) fma(0.16666666666666666, ((double) pow(x, 3.0)), x))));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	58.8
Target	0.5
Herbie	0.4

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

Derivation

Initial program 58.8
\[e^{x} - 1\]
Taylor expanded around 0 0.4
\[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]
Simplified0.4
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}\]
Final simplification0.4
\[\leadsto \mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)\]

Reproduce

herbie shell --seed 2020121 +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))