Average Error: 33.7 → 0.1

Time: 16.5s

Precision: 64

Ground Truth: 128

\[e^{a \cdot x} - 1\]

⬇

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -9.484017199363636 \cdot 10^{-07}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(\sqrt{\sqrt{e^{a \cdot x}}} + 1\right) \cdot \left(\sqrt{\sqrt{e^{a \cdot x}}} - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x\\ \end{array}\]

Error

The X axis may use a short exponential scale — Bits error versus `a`

Target

Original	33.7
Comparison	8.0
Herbie	0.1

\[ \begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt \frac{1}{10}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^2}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

Split input into 2 regimes.
if (* a x) < -9.484017199363636e-07
1. Initial program 0.1
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-sqr-sqrt 0.2
  \[\leadsto \color{blue}{{\left(\sqrt{e^{a \cdot x}}\right)}^2} - 1\]
4. Applied difference-of-sqr-1 0.2
  \[\leadsto \color{blue}{\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\sqrt{e^{a \cdot x}} - 1\right)}\]
5. Using strategy rm
6. Applied add-sqr-sqrt 0.2
  \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{e^{a \cdot x}}}\right)}^2} - 1\right)\]
7. Applied difference-of-sqr-1 0.2
  \[\leadsto \left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \color{blue}{\left(\left(\sqrt{\sqrt{e^{a \cdot x}}} + 1\right) \cdot \left(\sqrt{\sqrt{e^{a \cdot x}}} - 1\right)\right)}\]
if -9.484017199363636e-07 < (* a x)
1. Initial program 47.5
  \[e^{a \cdot x} - 1\]
2. Applied taylor 43.7
  \[\leadsto \left(x \cdot a + \left(1 + \frac{1}{2} \cdot \left({x}^2 \cdot {a}^2\right)\right)\right) - 1\]
3. Taylor expanded around 0 43.7
  
  \[\leadsto \color{blue}{\left(x \cdot a + \left(1 + \frac{1}{2} \cdot \left({x}^2 \cdot {a}^2\right)\right)\right)} - 1\]
4. Applied simplify 0.1
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) + a \cdot x}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please report a bug with the following info:

herbie --seed '#(2522638624 2743545407 2304185839 1960985118 443575049 2126182702)'
(FPCore (a x)
  :name "NMSE section 3.5"

  :target
  (if (< (fabs (* a x)) 1/10) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (sqr (* a x)) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))

Error

Target

Derivation

`if (* a x) < -9.484017199363636e-07`

`if -9.484017199363636e-07 < (* a x)`

Runtime