Average Error: 29.0 → 0.4

Time: 33.9s

Precision: 64

Internal Precision: 1344

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \le -1.2095988126683947 \cdot 10^{-06}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \left(\log \left(\sqrt{e^{e^{a \cdot x}}}\right) + \log \left(\sqrt{e^{-1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x\\ \end{array}\]

Error

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

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In
Out

Enter valid numbers for all inputs

Target

Original	29.0
Target	0.2
Herbie	0.4

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

Derivation

Split input into 2 regimes
if (- (exp (* a x)) 1) < -1.2095988126683947e-06
1. Initial program 0.1
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-log-exp0.1
  \[\leadsto \color{blue}{\log \left(e^{e^{a \cdot x} - 1}\right)}\]
4. Using strategy rm
5. Applied add-sqr-sqrt0.1
  \[\leadsto \log \color{blue}{\left(\sqrt{e^{e^{a \cdot x} - 1}} \cdot \sqrt{e^{e^{a \cdot x} - 1}}\right)}\]
6. Applied log-prod0.1
  \[\leadsto \color{blue}{\log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right)}\]
7. Using strategy rm
8. Applied sub-neg0.1
  \[\leadsto \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{e^{\color{blue}{e^{a \cdot x} + \left(-1\right)}}}\right)\]
9. Applied exp-sum0.1
  \[\leadsto \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \left(\sqrt{\color{blue}{e^{e^{a \cdot x}} \cdot e^{-1}}}\right)\]
10. Applied sqrt-prod0.1
  \[\leadsto \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \log \color{blue}{\left(\sqrt{e^{e^{a \cdot x}}} \cdot \sqrt{e^{-1}}\right)}\]
11. Applied log-prod0.1
  \[\leadsto \log \left(\sqrt{e^{e^{a \cdot x} - 1}}\right) + \color{blue}{\left(\log \left(\sqrt{e^{e^{a \cdot x}}}\right) + \log \left(\sqrt{e^{-1}}\right)\right)}\]
if -1.2095988126683947e-06 < (- (exp (* a x)) 1)
1. Initial program 44.5
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 14.0
  \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x\right)}\]
3. Applied simplify0.5
  \[\leadsto \color{blue}{\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{6} + \frac{1}{2}\right) + a \cdot x}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed 2018201 
(FPCore (a x)
  :name "expax (section 3.5)"

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

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

Error

Try it out

Target

Derivation

`if (- (exp (* a x)) 1) < -1.2095988126683947e-06`

`if -1.2095988126683947e-06 < (- (exp (* a x)) 1)`

Runtime