Average Error: 58.6 → 0.5
Time: 2.6s
Precision: binary64
\[-0.00017 < x\]
\[e^{x} - 1\]
↓
\[x + x \cdot \log \left({\left(e^{x}\right)}^{\left(x \cdot 0.16666666666666666 + 0.5\right)}\right)\]
e^{x} - 1
↓
x + x \cdot \log \left({\left(e^{x}\right)}^{\left(x \cdot 0.16666666666666666 + 0.5\right)}\right)
(FPCore (x) :precision binary64 (- (exp x) 1.0))
↓
(FPCore (x)
 :precision binary64
 (+ x (* x (log (pow (exp x) (+ (* x 0.16666666666666666) 0.5))))))
double code(double x) {
	return ((double) (((double) exp(x)) - 1.0));
}
↓
double code(double x) {
	return ((double) (x + ((double) (x * ((double) log(((double) pow(((double) exp(x)), ((double) (((double) (x * 0.16666666666666666)) + 0.5))))))))));
}

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 Error: 58.6 bits

    \[e^{x} - 1\]

  2. Taylor expanded around 0 Error: 0.5 bits

    \[\leadsto \color{blue}{0.5 \cdot {x}^{2} + \left(0.16666666666666666 \cdot {x}^{3} + x\right)}\]

  3. SimplifiedError: 0.5 bits

    \[\leadsto \color{blue}{x + x \cdot \left(x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\]
  4. Using strategy rm

  5. Applied add-log-expError: 0.5 bits

    \[\leadsto x + x \cdot \color{blue}{\log \left(e^{x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)}\right)}\]

  6. SimplifiedError: 0.5 bits

    \[\leadsto x + x \cdot \log \color{blue}{\left({\left(e^{x}\right)}^{\left(x \cdot 0.16666666666666666 + 0.5\right)}\right)}\]

  7. Final simplificationError: 0.5 bits

    \[\leadsto x + x \cdot \log \left({\left(e^{x}\right)}^{\left(x \cdot 0.16666666666666666 + 0.5\right)}\right)\]

Reproduce

herbie shell --seed 2020203 
(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))