Average Error: 9.6 → 0.1
Time: 11.6s
Precision: binary64
Cost: 704
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{x} \cdot \frac{2}{x \cdot x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}↓
\frac{1}{x} \cdot \frac{2}{x \cdot x - 1}(FPCore (x)
:precision binary64
(+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (* (/ 1.0 x) (/ 2.0 (- (* x x) 1.0))))
double code(double x) {
return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}
↓
double code(double x) {
return (1.0 / x) * (2.0 / ((x * x) - 1.0));
}
Try it out
Enter valid numbers for all inputs
Target
| Original | 9.6 |
|---|
| Target | 0.2 |
|---|
| Herbie | 0.1 |
|---|
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]
Alternatives
| Alternative 1 |
|---|
| Error | 25.1 |
|---|
| Cost | 40128 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x - 1}}\]
| Alternative 2 |
|---|
| Error | 1.0 |
|---|
| Cost | 39680 |
|---|
\[\sqrt[3]{\frac{2}{{x}^{3} - x}} \cdot \left(\sqrt[3]{\frac{2}{{x}^{3} - x}} \cdot \sqrt[3]{\frac{2}{{x}^{3} - x}}\right)\]
| Alternative 3 |
|---|
| Error | 58.4 |
|---|
| Cost | 33472 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x - 1}} \cdot \frac{\sqrt[3]{1}}{\sqrt{x - 1}}\]
| Alternative 4 |
|---|
| Error | 33.1 |
|---|
| Cost | 33088 |
|---|
\[\left(\frac{\sqrt[3]{2}}{1 + x} \cdot \frac{\sqrt[3]{2}}{\sqrt{x}}\right) \cdot \frac{\sqrt[3]{2}}{\left(x - 1\right) \cdot \sqrt{x}}\]
| Alternative 5 |
|---|
| Error | 44.3 |
|---|
| Cost | 27200 |
|---|
\[\frac{1}{x - 1} + \left(\sqrt{\frac{2}{x}} + \frac{1}{\sqrt{1 + x}}\right) \cdot \left(\frac{1}{\sqrt{1 + x}} - \sqrt{\frac{2}{x}}\right)\]
| Alternative 6 |
|---|
| Error | 44.3 |
|---|
| Cost | 27200 |
|---|
\[\frac{1}{x - 1} + \left(\sqrt{\frac{2}{x}} + \sqrt{\frac{1}{1 + x}}\right) \cdot \left(\sqrt{\frac{1}{1 + x}} - \sqrt{\frac{2}{x}}\right)\]
| Alternative 7 |
|---|
| Error | 21.3 |
|---|
| Cost | 26432 |
|---|
\[\sqrt{\frac{2}{{x}^{3} - x}} \cdot \sqrt{\frac{2}{{x}^{3} - x}}\]
| Alternative 8 |
|---|
| Error | 26.4 |
|---|
| Cost | 21440 |
|---|
\[\frac{1}{x - 1} + \sqrt[3]{\frac{-2}{x} + \frac{1}{1 + x}} \cdot \left(\sqrt[3]{\frac{-2}{x} + \frac{1}{1 + x}} \cdot \sqrt[3]{\frac{-2}{x} + \frac{1}{1 + x}}\right)\]
| Alternative 9 |
|---|
| Error | 25.1 |
|---|
| Cost | 20800 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{1}{\sqrt[3]{x - 1}}\]
| Alternative 10 |
|---|
| Error | 0.2 |
|---|
| Cost | 20672 |
|---|
\[\frac{1}{\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{x \cdot x - 1}} \cdot \frac{\frac{2}{x}}{\sqrt[3]{x \cdot x - 1}}\]
| Alternative 11 |
|---|
| Error | 25.1 |
|---|
| Cost | 20672 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{\frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}}{\sqrt[3]{x - 1}}\]
| Alternative 12 |
|---|
| Error | 1.4 |
|---|
| Cost | 20032 |
|---|
\[\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{x} \cdot \frac{\sqrt[3]{2}}{x \cdot x - 1}\]
| Alternative 13 |
|---|
| Error | 0.7 |
|---|
| Cost | 19712 |
|---|
\[\sqrt{2} \cdot \frac{\sqrt{2}}{{x}^{3} - x}\]
| Alternative 14 |
|---|
| Error | 27.9 |
|---|
| Cost | 19648 |
|---|
\[\sqrt[3]{{\left(\frac{2}{{x}^{3} - x}\right)}^{3}}\]
| Alternative 15 |
|---|
| Error | 52.3 |
|---|
| Cost | 16576 |
|---|
\[\frac{\left(x - 1\right) \cdot \left({\left(\frac{1}{1 + x}\right)}^{3} - {\left(\frac{2}{x}\right)}^{3}\right) + \left(\frac{4}{x \cdot x} + \frac{\frac{2}{x} + \frac{1}{1 + x}}{1 + x}\right)}{\left(x - 1\right) \cdot \left(\frac{4}{x \cdot x} + \frac{\frac{2}{x} + \frac{1}{1 + x}}{1 + x}\right)}\]
| Alternative 16 |
|---|
| Error | 52.3 |
|---|
| Cost | 15168 |
|---|
\[\frac{1}{x - 1} + \frac{{\left(\frac{1}{1 + x}\right)}^{3} - {\left(\frac{2}{x}\right)}^{3}}{\frac{4}{x \cdot x} + \frac{\frac{2}{x} + \frac{1}{1 + x}}{1 + x}}\]
| Alternative 17 |
|---|
| Error | 40.9 |
|---|
| Cost | 14400 |
|---|
\[\frac{1}{x - 1} + \sqrt{\frac{-2}{x} + \frac{1}{1 + x}} \cdot \sqrt{\frac{-2}{x} + \frac{1}{1 + x}}\]
| Alternative 18 |
|---|
| Error | 58.4 |
|---|
| Cost | 14144 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{\sqrt{x - 1}} \cdot \frac{1}{\sqrt{x - 1}}\]
| Alternative 19 |
|---|
| Error | 57.2 |
|---|
| Cost | 14144 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \sqrt{\frac{1}{x - 1}} \cdot \sqrt{\frac{1}{x - 1}}\]
| Alternative 20 |
|---|
| Error | 57.8 |
|---|
| Cost | 14016 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{\frac{1}{\sqrt{x - 1}}}{\sqrt{x - 1}}\]
| Alternative 21 |
|---|
| Error | 32.0 |
|---|
| Cost | 13888 |
|---|
\[\frac{1}{\sqrt{x \cdot x - 1}} \cdot \frac{\frac{2}{x}}{\sqrt{x \cdot x - 1}}\]
| Alternative 22 |
|---|
| Error | 51.1 |
|---|
| Cost | 13824 |
|---|
\[\frac{1}{x - 1} + \sqrt[3]{{\left(\frac{-2}{x} + \frac{1}{1 + x}\right)}^{3}}\]
| Alternative 23 |
|---|
| Error | 31.0 |
|---|
| Cost | 13824 |
|---|
\[\sqrt[3]{{\left(\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x - 1}\right)}^{3}}\]
| Alternative 24 |
|---|
| Error | 29.8 |
|---|
| Cost | 13824 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \sqrt[3]{{\left(\frac{1}{x - 1}\right)}^{3}}\]
| Alternative 25 |
|---|
| Error | 47.5 |
|---|
| Cost | 13760 |
|---|
\[\frac{1}{x - 1} + e^{\log \left(\frac{-2}{x} + \frac{1}{1 + x}\right)}\]
| Alternative 26 |
|---|
| Error | 32.6 |
|---|
| Cost | 13760 |
|---|
\[\frac{1}{\left(1 + x\right) \cdot \sqrt{x}} \cdot \frac{2}{\left(x - 1\right) \cdot \sqrt{x}}\]
| Alternative 27 |
|---|
| Error | 61.5 |
|---|
| Cost | 13760 |
|---|
\[\frac{1}{x - 1} + \log \left(e^{\frac{-2}{x} + \frac{1}{1 + x}}\right)\]
| Alternative 28 |
|---|
| Error | 42.4 |
|---|
| Cost | 13632 |
|---|
\[\frac{2}{{x}^{6} - x \cdot x} \cdot \left(x + {x}^{3}\right)\]
| Alternative 29 |
|---|
| Error | 0.5 |
|---|
| Cost | 13504 |
|---|
\[\frac{\sqrt{2}}{x} \cdot \frac{\sqrt{2}}{x \cdot x - 1}\]
| Alternative 30 |
|---|
| Error | 25.7 |
|---|
| Cost | 7808 |
|---|
\[\frac{x \cdot \left(1 + x\right) + \left(x - 1\right) \cdot \left(x - 2 \cdot \left(1 + x\right)\right)}{{x}^{3} - x}\]
| Alternative 31 |
|---|
| Error | 20.3 |
|---|
| Cost | 7168 |
|---|
\[\frac{2}{{x}^{5} - x} \cdot \left(1 + x \cdot x\right)\]
| Alternative 32 |
|---|
| Error | 0.2 |
|---|
| Cost | 6784 |
|---|
\[\frac{2}{{x}^{3} - x}\]
| Alternative 33 |
|---|
| Error | 32.4 |
|---|
| Cost | 6656 |
|---|
\[\frac{2}{{x}^{3}}\]
| Alternative 34 |
|---|
| Error | 26.9 |
|---|
| Cost | 2880 |
|---|
\[\frac{\left(\frac{-2}{x} + \frac{1}{1 + x}\right) \cdot \left(\frac{-2}{x} + \frac{1}{1 + x}\right) - \frac{\frac{1}{x - 1}}{x - 1}}{\left(\frac{-2}{x} + \frac{1}{1 + x}\right) - \frac{1}{x - 1}}\]
| Alternative 35 |
|---|
| Error | 28.5 |
|---|
| Cost | 2496 |
|---|
\[\frac{\left(\frac{2}{x} + \frac{1}{1 + x}\right) \cdot \left(1 + \left(\frac{-2}{x} + \frac{1}{1 + x}\right) \cdot \left(x - 1\right)\right)}{\left(x - 1\right) \cdot \left(\frac{2}{x} + \frac{1}{1 + x}\right)}\]
| Alternative 36 |
|---|
| Error | 43.9 |
|---|
| Cost | 1984 |
|---|
\[\frac{1}{x - 1} + \frac{\frac{\frac{1}{1 + x}}{1 + x} - \frac{4}{x \cdot x}}{\frac{2}{x} + \frac{1}{1 + x}}\]
| Alternative 37 |
|---|
| Error | 25.7 |
|---|
| Cost | 1728 |
|---|
\[\frac{1}{x} \cdot \frac{x \cdot \left(1 + x\right) + \left(x - 1\right) \cdot \left(x - 2 \cdot \left(1 + x\right)\right)}{x \cdot x - 1}\]
| Alternative 38 |
|---|
| Error | 26.6 |
|---|
| Cost | 1344 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \left(1 + x\right) \cdot \frac{1}{x \cdot x - 1}\]
| Alternative 39 |
|---|
| Error | 26.1 |
|---|
| Cost | 1216 |
|---|
\[\frac{1}{x - 1} + \frac{x - 2 \cdot \left(1 + x\right)}{x \cdot \left(1 + x\right)}\]
| Alternative 40 |
|---|
| Error | 9.6 |
|---|
| Cost | 960 |
|---|
\[\left(\frac{-2}{x} + \frac{1}{1 + x}\right) + \frac{1}{x - 1}\]
| Alternative 41 |
|---|
| Error | 0.1 |
|---|
| Cost | 832 |
|---|
\[\frac{1}{1 + x} \cdot \frac{\frac{2}{x}}{x - 1}\]
| Alternative 42 |
|---|
| Error | 0.1 |
|---|
| Cost | 704 |
|---|
\[\frac{\frac{\frac{2}{x}}{1 + x}}{x - 1}\]
| Alternative 43 |
|---|
| Error | 0.1 |
|---|
| Cost | 576 |
|---|
\[\frac{\frac{2}{x}}{x \cdot x - 1}\]
| Alternative 44 |
|---|
| Error | 35.9 |
|---|
| Cost | 576 |
|---|
\[\frac{1}{x - 1} + \frac{-1}{x}\]
| Alternative 45 |
|---|
| Error | 31.6 |
|---|
| Cost | 448 |
|---|
\[\frac{-2}{x} + x \cdot -2\]
| Alternative 46 |
|---|
| Error | 30.8 |
|---|
| Cost | 192 |
|---|
\[\frac{-2}{x}\]
| Alternative 47 |
|---|
| Error | 61.9 |
|---|
| Cost | 64 |
|---|
\[1\]
| Alternative 48 |
|---|
| Error | 41.3 |
|---|
| Cost | 64 |
|---|
\[0\]
| Alternative 49 |
|---|
| Error | 61.9 |
|---|
| Cost | 64 |
|---|
\[-1\]
Error
Derivation
Initial program 9.6
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
- Using strategy
rm Applied frac-sub_binary64_107926.1
\[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]
Applied frac-add_binary64_107825.7
\[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]
Simplified25.7
\[\leadsto \frac{\color{blue}{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
Simplified25.7
\[\leadsto \frac{\left(x - 1\right) \cdot \left(x - \left(1 + x\right) \cdot 2\right) + x \cdot \left(1 + x\right)}{\color{blue}{{x}^{3} - x}}\]
Taylor expanded around 0 0.2
\[\leadsto \frac{\color{blue}{2}}{{x}^{3} - x}\]
- Using strategy
rm Applied *-un-lft-identity_binary64_10700.2
\[\leadsto \frac{2}{{x}^{3} - \color{blue}{1 \cdot x}}\]
Applied unpow3_binary64_11360.2
\[\leadsto \frac{2}{\color{blue}{\left(x \cdot x\right) \cdot x} - 1 \cdot x}\]
Applied distribute-rgt-out--_binary64_10240.2
\[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot x - 1\right)}}\]
Applied *-un-lft-identity_binary64_10700.2
\[\leadsto \frac{\color{blue}{1 \cdot 2}}{x \cdot \left(x \cdot x - 1\right)}\]
Applied times-frac_binary64_10760.1
\[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{2}{x \cdot x - 1}}\]
Simplified0.1
\[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{2}{x \cdot x - 1}}\]
Final simplification0.1
\[\leadsto \frac{1}{x} \cdot \frac{2}{x \cdot x - 1}\]
Reproduce
herbie shell --seed 2021042
(FPCore (x)
:name "3frac (problem 3.3.3)"
:precision binary64
:herbie-target
(/ 2.0 (* x (- (* x x) 1.0)))
(+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
