Average Error: 0.2 → 0.2
Time: 8.6s
Precision: binary64
Cost: 13248
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
↓
\[\frac{1 - x \cdot \cos B}{\sin B}\]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}↓
\frac{1 - x \cdot \cos B}{\sin B}(FPCore (B x)
:precision binary64
(+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))
↓
(FPCore (B x) :precision binary64 (/ (- 1.0 (* x (cos B))) (sin B)))
double code(double B, double x) {
return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}
↓
double code(double B, double x) {
return (1.0 - (x * cos(B))) / sin(B);
}
Try it out
Enter valid numbers for all inputs
Alternatives
| Alternative 1 |
|---|
| Error | 0.8 |
|---|
| Cost | 64960 |
|---|
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sin B}} - \frac{x}{\tan B}\]
| Alternative 2 |
|---|
| Error | 0.8 |
|---|
| Cost | 64960 |
|---|
\[\frac{1}{\sin B} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\tan B}}\]
| Alternative 3 |
|---|
| Error | 31.8 |
|---|
| Cost | 52032 |
|---|
\[\frac{1}{\sin B} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{\tan B}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\tan B}}\]
| Alternative 4 |
|---|
| Error | 25.3 |
|---|
| Cost | 45952 |
|---|
\[\frac{\frac{\frac{1}{\sin B}}{\sin B} - {\left(\sqrt[3]{\frac{x}{\tan B}}\right)}^{6}}{\frac{1}{\sin B} + \frac{x}{\tan B}}\]
| Alternative 5 |
|---|
| Error | 0.6 |
|---|
| Cost | 45760 |
|---|
\[\frac{1}{\sin B} - \sqrt[3]{\frac{x}{\tan B}} \cdot \left(\sqrt[3]{\frac{x}{\tan B}} \cdot \sqrt[3]{\frac{x}{\tan B}}\right)\]
| Alternative 6 |
|---|
| Error | 0.8 |
|---|
| Cost | 45760 |
|---|
\[\sqrt[3]{\frac{1}{\sin B}} \cdot \left(\sqrt[3]{\frac{1}{\sin B}} \cdot \sqrt[3]{\frac{1}{\sin B}}\right) - \frac{x}{\tan B}\]
| Alternative 7 |
|---|
| Error | 0.7 |
|---|
| Cost | 45632 |
|---|
\[\frac{1}{\sin B} - \frac{1}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}} \cdot \frac{x}{\sqrt[3]{\tan B}}\]
| Alternative 8 |
|---|
| Error | 0.8 |
|---|
| Cost | 45632 |
|---|
\[\frac{1}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{1}{\sqrt[3]{\sin B}} - \frac{x}{\tan B}\]
| Alternative 9 |
|---|
| Error | 47.7 |
|---|
| Cost | 45504 |
|---|
\[\frac{1}{\sin B} - \frac{\sqrt{x}}{\sqrt{\tan B}} \cdot \frac{\sqrt{x}}{\sqrt{\tan B}}\]
| Alternative 10 |
|---|
| Error | 0.7 |
|---|
| Cost | 45504 |
|---|
\[\frac{1}{\sin B} - \frac{\frac{x}{\sqrt[3]{\tan B} \cdot \sqrt[3]{\tan B}}}{\sqrt[3]{\tan B}}\]
| Alternative 11 |
|---|
| Error | 0.8 |
|---|
| Cost | 45504 |
|---|
\[\frac{\frac{1}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}}{\sqrt[3]{\sin B}} - \frac{x}{\tan B}\]
| Alternative 12 |
|---|
| Error | 32.8 |
|---|
| Cost | 39360 |
|---|
\[\sqrt{\frac{1}{\sin B} - \frac{x}{\tan B}} \cdot \sqrt{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
| Alternative 13 |
|---|
| Error | 31.6 |
|---|
| Cost | 32704 |
|---|
\[\frac{1}{\sin B} - \frac{1}{\sqrt{\tan B}} \cdot \frac{x}{\sqrt{\tan B}}\]
| Alternative 14 |
|---|
| Error | 0.6 |
|---|
| Cost | 32704 |
|---|
\[\frac{1}{\sin B} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{\tan B}\]
| Alternative 15 |
|---|
| Error | 32.8 |
|---|
| Cost | 32704 |
|---|
\[\sqrt{\frac{1}{\sin B}} \cdot \sqrt{\frac{1}{\sin B}} - \frac{x}{\tan B}\]
| Alternative 16 |
|---|
| Error | 32.1 |
|---|
| Cost | 32704 |
|---|
\[\frac{1}{\sin B} - \sqrt{\frac{x}{\tan B}} \cdot \sqrt{\frac{x}{\tan B}}\]
| Alternative 17 |
|---|
| Error | 16.5 |
|---|
| Cost | 32640 |
|---|
\[\sqrt[3]{\frac{1}{{\sin B}^{3}}} - \frac{x \cdot \cos B}{\sin B}\]
| Alternative 18 |
|---|
| Error | 31.6 |
|---|
| Cost | 32576 |
|---|
\[\frac{1}{\sin B} - \frac{\frac{x}{\sqrt{\tan B}}}{\sqrt{\tan B}}\]
| Alternative 19 |
|---|
| Error | 11.1 |
|---|
| Cost | 26304 |
|---|
\[\frac{\sin B \cdot \left(1 - x \cdot \cos B\right)}{\sin B \cdot \sin B}\]
| Alternative 20 |
|---|
| Error | 8.8 |
|---|
| Cost | 26304 |
|---|
\[\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}\]
| Alternative 21 |
|---|
| Error | 31.9 |
|---|
| Cost | 26176 |
|---|
\[\frac{1}{\sin B} - \sqrt{x} \cdot \frac{\sqrt{x}}{\tan B}\]
| Alternative 22 |
|---|
| Error | 11.1 |
|---|
| Cost | 26176 |
|---|
\[\frac{\tan B - \sin B \cdot x}{\sin B \cdot \tan B}\]
| Alternative 23 |
|---|
| Error | 16.4 |
|---|
| Cost | 26112 |
|---|
\[\sqrt[3]{{\left(\frac{1}{\sin B}\right)}^{3}} - \frac{x}{\tan B}\]
| Alternative 24 |
|---|
| Error | 35.9 |
|---|
| Cost | 26048 |
|---|
\[\frac{1}{\sin B} - \log \left(e^{\frac{x}{\tan B}}\right)\]
| Alternative 25 |
|---|
| Error | 34.7 |
|---|
| Cost | 26048 |
|---|
\[e^{\log \left(\frac{1}{\sin B} - \frac{x}{\tan B}\right)}\]
| Alternative 26 |
|---|
| Error | 33.5 |
|---|
| Cost | 25984 |
|---|
\[e^{-\log \sin B} - \frac{x}{\tan B}\]
| Alternative 27 |
|---|
| Error | 0.2 |
|---|
| Cost | 19904 |
|---|
\[\frac{1}{\sin B} - \frac{1}{\frac{\sin B}{x \cdot \cos B}}\]
| Alternative 28 |
|---|
| Error | 0.2 |
|---|
| Cost | 19776 |
|---|
\[\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\]
| Alternative 29 |
|---|
| Error | 0.2 |
|---|
| Cost | 19776 |
|---|
\[\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\]
| Alternative 30 |
|---|
| Error | 33.2 |
|---|
| Cost | 13824 |
|---|
\[\left(B \cdot 0.16666666666666666 + \left(\frac{1}{B} + 0.019444444444444445 \cdot {B}^{3}\right)\right) - \frac{x}{\tan B}\]
| Alternative 31 |
|---|
| Error | 0.2 |
|---|
| Cost | 13376 |
|---|
\[\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\]
| Alternative 32 |
|---|
| Error | 0.2 |
|---|
| Cost | 13376 |
|---|
\[\frac{1}{\sin B} - \frac{1}{\frac{\tan B}{x}}\]
| Alternative 33 |
|---|
| Error | 0.1 |
|---|
| Cost | 13248 |
|---|
\[\frac{1}{\sin B} - \frac{x}{\tan B}\]
| Alternative 34 |
|---|
| Error | 36.2 |
|---|
| Cost | 13184 |
|---|
\[-\frac{x \cdot \cos B}{\sin B}\]
| Alternative 35 |
|---|
| Error | 36.2 |
|---|
| Cost | 13184 |
|---|
\[\cos B \cdot \frac{x}{-\sin B}\]
| Alternative 36 |
|---|
| Error | 27.3 |
|---|
| Cost | 7232 |
|---|
\[\frac{1}{\sin B} - \left(\frac{x}{B} - 0.3333333333333333 \cdot \left(B \cdot x\right)\right)\]
| Alternative 37 |
|---|
| Error | 18.1 |
|---|
| Cost | 6976 |
|---|
\[\frac{1}{B} - \frac{1}{\frac{\tan B}{x}}\]
| Alternative 38 |
|---|
| Error | 18.3 |
|---|
| Cost | 6848 |
|---|
\[\frac{1}{\sin B} - \frac{x}{B}\]
| Alternative 39 |
|---|
| Error | 18.1 |
|---|
| Cost | 6848 |
|---|
\[\frac{1}{B} - \frac{x}{\tan B}\]
| Alternative 40 |
|---|
| Error | 27.2 |
|---|
| Cost | 6592 |
|---|
\[\frac{1}{\sin B}\]
| Alternative 41 |
|---|
| Error | 35.9 |
|---|
| Cost | 832 |
|---|
\[\frac{1 - x}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\]
| Alternative 42 |
|---|
| Error | 35.7 |
|---|
| Cost | 320 |
|---|
\[\frac{1 - x}{B}\]
| Alternative 43 |
|---|
| Error | 60.6 |
|---|
| Cost | 64 |
|---|
\[1\]
| Alternative 44 |
|---|
| Error | 62.4 |
|---|
| Cost | 64 |
|---|
\[0\]
| Alternative 45 |
|---|
| Error | 60.3 |
|---|
| Cost | 64 |
|---|
\[-1\]
Error
Derivation
Initial program 0.2
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]
Simplified0.1
\[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}}\]
- Using strategy
rm Applied clear-num_binary640.2
\[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{1}{\frac{\tan B}{x}}}\]
Taylor expanded around inf 0.2
\[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]
- Using strategy
rm Applied sub-div_binary640.2
\[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}}\]
Simplified0.2
\[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}}\]
Final simplification0.2
\[\leadsto \frac{1 - x \cdot \cos B}{\sin B}\]
Reproduce
herbie shell --seed 2021042
(FPCore (B x)
:name "VandenBroeck and Keller, Equation (24)"
:precision binary64
(+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))
