Average Error: 29.7 → 0.7
Time: 10.8s
Precision: binary64
Cost: 13376
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{2 \cdot e^{\log \left(1 + x\right) - x}}{2}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}↓
\frac{2 \cdot e^{\log \left(1 + x\right) - x}}{2}(FPCore (x eps)
:precision binary64
(/
(-
(* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
(* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
2.0))
↓
(FPCore (x eps)
:precision binary64
(/ (* 2.0 (exp (- (log (+ 1.0 x)) x))) 2.0))
double code(double x, double eps) {
return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}
↓
double code(double x, double eps) {
return (2.0 * exp(log(1.0 + x) - x)) / 2.0;
}
Try it out
Enter valid numbers for all inputs
Alternatives
| Alternative 1 |
|---|
| Error | 31.0 |
|---|
| Cost | 66944 |
|---|
\[\frac{\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \left(\sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt[3]{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 2 |
|---|
| Error | 31.9 |
|---|
| Cost | 47040 |
|---|
\[\frac{\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 3 |
|---|
| Error | 47.0 |
|---|
| Cost | 36416 |
|---|
\[\frac{\frac{\left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right) \cdot \left(\left(1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)}\right) + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^{3} + -1\right)\right)}{\left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right) \cdot \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)}\right)}}{2}\]
| Alternative 4 |
|---|
| Error | 47.0 |
|---|
| Cost | 36416 |
|---|
\[\frac{\frac{\left(1 + {\left(\frac{1}{\varepsilon}\right)}^{3}\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)}\right) - \left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)\right) \cdot \left(\frac{\frac{1}{\varepsilon}}{\varepsilon} + -1\right)}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)}\right)}}{2}\]
| Alternative 5 |
|---|
| Error | 46.9 |
|---|
| Cost | 35520 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(e^{x \cdot \left(1 + \varepsilon\right)} \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right)\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^{3} + -1\right)}{\left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)}}}{2}\]
| Alternative 6 |
|---|
| Error | 46.9 |
|---|
| Cost | 35520 |
|---|
\[\frac{\frac{\left(1 + {\left(\frac{1}{\varepsilon}\right)}^{3}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)\right)}{\left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)}}}{2}\]
| Alternative 7 |
|---|
| Error | 30.3 |
|---|
| Cost | 34368 |
|---|
\[\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \frac{\sqrt[3]{1 + \frac{1}{\varepsilon}}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 8 |
|---|
| Error | 31.3 |
|---|
| Cost | 33600 |
|---|
\[\frac{\sqrt[3]{{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}\right)}^{3}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 9 |
|---|
| Error | 30.5 |
|---|
| Cost | 33536 |
|---|
\[\frac{e^{\log \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}\right)}}{2}\]
| Alternative 10 |
|---|
| Error | 44.6 |
|---|
| Cost | 30272 |
|---|
\[\frac{\frac{\left(1 + {\left(\frac{1}{\varepsilon}\right)}^{3}\right) \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right) + \left(e^{\left(1 - \varepsilon\right) \cdot x - x \cdot \left(1 + \varepsilon\right)} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^{3} + -1\right)\right) \cdot \left(-1 - \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)}{\left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)\right) \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right)}}{2}\]
| Alternative 11 |
|---|
| Error | 46.8 |
|---|
| Cost | 28672 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)}\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{\frac{1}{\varepsilon}}{\varepsilon} + -1\right)}{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)}}}{2}\]
| Alternative 12 |
|---|
| Error | 46.8 |
|---|
| Cost | 28672 |
|---|
\[\frac{\frac{\left(1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\left(1 - \varepsilon\right) + \left(1 + \varepsilon\right)\right)}}}{2}\]
| Alternative 13 |
|---|
| Error | 39.2 |
|---|
| Cost | 27584 |
|---|
\[\frac{\sqrt{1 + \frac{1}{\varepsilon}} \cdot \frac{\sqrt{1 + \frac{1}{\varepsilon}}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 14 |
|---|
| Error | 43.8 |
|---|
| Cost | 23424 |
|---|
\[\frac{\frac{\left(1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}\right) \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right) + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot x - x \cdot \left(1 + \varepsilon\right)} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^{3} + -1\right)\right)}{\left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right)}}{2}\]
| Alternative 15 |
|---|
| Error | 43.8 |
|---|
| Cost | 23424 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + {\left(\frac{1}{\varepsilon}\right)}^{3}\right) + \left(\left(\frac{\frac{1}{\varepsilon}}{\varepsilon} + -1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x - x \cdot \left(1 + \varepsilon\right)}\right) \cdot \left(-1 - \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)\right)}}{2}\]
| Alternative 16 |
|---|
| Error | 42.0 |
|---|
| Cost | 23040 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1 + {\left(\frac{1}{\varepsilon}\right)}^{3}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right) \cdot \frac{\frac{\frac{1}{\varepsilon}}{\varepsilon} + -1}{e^{x \cdot \left(1 + \varepsilon\right)}}}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)}}{2}\]
| Alternative 17 |
|---|
| Error | 42.4 |
|---|
| Cost | 22528 |
|---|
\[\frac{\frac{\left(1 + {\left(\frac{1}{\varepsilon}\right)}^{3}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right) - \left(1 - \varepsilon\right) \cdot x} + \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)}{\left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\]
| Alternative 18 |
|---|
| Error | 42.3 |
|---|
| Cost | 22528 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right) - e^{\left(1 - \varepsilon\right) \cdot x - x \cdot \left(1 + \varepsilon\right)} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^{3} + -1\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 + \frac{1 + \frac{1}{\varepsilon}}{\varepsilon}\right)}}{2}\]
| Alternative 19 |
|---|
| Error | 41.5 |
|---|
| Cost | 21504 |
|---|
\[\frac{\frac{1 + {\left(\frac{1}{\varepsilon}\right)}^{3}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 + \frac{\frac{1}{\varepsilon} - 1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 20 |
|---|
| Error | 39.3 |
|---|
| Cost | 20800 |
|---|
\[\frac{e^{\log \left(1 + \frac{1}{\varepsilon}\right) - \left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 21 |
|---|
| Error | 30.4 |
|---|
| Cost | 20736 |
|---|
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 22 |
|---|
| Error | 42.4 |
|---|
| Cost | 16576 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}\right) + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(\frac{\frac{1}{\varepsilon}}{\varepsilon} + -1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x - x \cdot \left(1 + \varepsilon\right)}\right)}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}}{2}\]
| Alternative 23 |
|---|
| Error | 39.4 |
|---|
| Cost | 15680 |
|---|
\[\frac{\frac{\left(1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right) - \left(1 - \varepsilon\right) \cdot x} + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\]
| Alternative 24 |
|---|
| Error | 39.4 |
|---|
| Cost | 15680 |
|---|
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{\frac{1}{\varepsilon}}{\varepsilon} + -1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x - x \cdot \left(1 + \varepsilon\right)}}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}}}{2}\]
| Alternative 25 |
|---|
| Error | 38.6 |
|---|
| Cost | 14912 |
|---|
\[\frac{\frac{\frac{1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{1 - \frac{1}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 26 |
|---|
| Error | 38.6 |
|---|
| Cost | 14912 |
|---|
\[\frac{\frac{1 - \frac{\frac{1}{\varepsilon}}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(1 - \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 27 |
|---|
| Error | 20.7 |
|---|
| Cost | 14656 |
|---|
\[\frac{\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(2 + {x}^{3} \cdot 0.6666666666666666\right)\right) - \left(x \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({x}^{3} \cdot 0.6666666666666666\right)\right)}{2}\]
| Alternative 28 |
|---|
| Error | 29.7 |
|---|
| Cost | 14400 |
|---|
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 29 |
|---|
| Error | 29.7 |
|---|
| Cost | 14400 |
|---|
\[\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 30 |
|---|
| Error | 29.7 |
|---|
| Cost | 14400 |
|---|
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1 - \frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}}}{2}\]
| Alternative 31 |
|---|
| Error | 41.8 |
|---|
| Cost | 14336 |
|---|
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{-\varepsilon \cdot x} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\]
| Alternative 32 |
|---|
| Error | 31.8 |
|---|
| Cost | 14016 |
|---|
\[\frac{e^{\varepsilon \cdot x - x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\]
| Alternative 33 |
|---|
| Error | 0.5 |
|---|
| Cost | 7040 |
|---|
\[\frac{2 \cdot \left(\left(1 + x\right) \cdot e^{-x}\right)}{2}\]
| Alternative 34 |
|---|
| Error | 0.5 |
|---|
| Cost | 6976 |
|---|
\[\frac{2 \cdot \frac{1 + x}{e^{x}}}{2}\]
| Alternative 35 |
|---|
| Error | 46.2 |
|---|
| Cost | 6848 |
|---|
\[\frac{2 \cdot \frac{x}{e^{x}}}{2}\]
| Alternative 36 |
|---|
| Error | 1.6 |
|---|
| Cost | 6784 |
|---|
\[\frac{2 \cdot e^{-x}}{2}\]
| Alternative 37 |
|---|
| Error | 16.9 |
|---|
| Cost | 960 |
|---|
\[\frac{2 \cdot \left(1 + x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333 + -0.5\right)\right)\right)}{2}\]
| Alternative 38 |
|---|
| Error | 16.7 |
|---|
| Cost | 64 |
|---|
\[1\]
| Alternative 39 |
|---|
| Error | 46.5 |
|---|
| Cost | 64 |
|---|
\[0\]
| Alternative 40 |
|---|
| Error | 62.7 |
|---|
| Cost | 64 |
|---|
\[-1\]
Error
Derivation
Initial program 29.7
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Taylor expanded around 0 0.5
\[\leadsto \frac{\color{blue}{2 \cdot e^{-x} + 2 \cdot \left(e^{-x} \cdot x\right)}}{2}\]
Simplified0.5
\[\leadsto \frac{\color{blue}{2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)}}{2}\]
- Using strategy
rm Applied add-exp-log_binary64_4420.7
\[\leadsto \frac{2 \cdot \left(\color{blue}{e^{\log \left(x + 1\right)}} \cdot e^{-x}\right)}{2}\]
Applied prod-exp_binary64_4530.7
\[\leadsto \frac{2 \cdot \color{blue}{e^{\log \left(x + 1\right) + \left(-x\right)}}}{2}\]
Simplified0.7
\[\leadsto \frac{2 \cdot e^{\color{blue}{\log \left(x + 1\right) - x}}}{2}\]
Simplified0.7
\[\leadsto \color{blue}{\frac{2 \cdot e^{\log \left(1 + x\right) - x}}{2}}\]
Final simplification0.7
\[\leadsto \frac{2 \cdot e^{\log \left(1 + x\right) - x}}{2}\]
Reproduce
herbie shell --seed 2021042
(FPCore (x eps)
:name "NMSE Section 6.1 mentioned, A"
:precision binary64
(/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))
