Average Error: 58.8 → 0.4
Time: 5.5s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)\right)\right)\]
e^{x} - 1
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)\right)\right)
double f(double x) {
        double r83097 = x;
        double r83098 = exp(r83097);
        double r83099 = 1.0;
        double r83100 = r83098 - r83099;
        return r83100;
}
↓
double f(double x) {
        double r83101 = x;
        double r83102 = 0.16666666666666666;
        double r83103 = 0.5;
        double r83104 = fma(r83101, r83102, r83103);
        double r83105 = r83101 * r83104;
        double r83106 = fma(r83101, r83105, r83101);
        double r83107 = expm1(r83106);
        double r83108 = log1p(r83107);
        return r83108;
}

Error

Bits error versus x

Target

Original58.8
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}\]
  4. Using strategy rm

  5. Applied log1p-expm1-u0.4

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)\right)\right)}\]

  6. Simplified0.4

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)\right)}\right)\]

  7. Final simplification0.4

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)\right)\right)\]

Reproduce

herbie shell --seed 2020047 +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))