Average Error: 19.7 → 0.2
Time: 18.6s
Precision: binary64
Cost: 20096
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[{x}^{-0.5} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}↓
{x}^{-0.5} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
:precision binary64
(* (pow x -0.5) (/ 1.0 (+ (+ x 1.0) (* (sqrt x) (sqrt (+ x 1.0)))))))
double code(double x) {
return (1.0 / sqrt(x)) - (1.0 / sqrt(x + 1.0));
}
↓
double code(double x) {
return pow(x, -0.5) * (1.0 / ((x + 1.0) + (sqrt(x) * sqrt(x + 1.0))));
}
Try it out
Enter valid numbers for all inputs
Target
| Original | 19.7 |
|---|
| Target | 0.6 |
|---|
| Herbie | 0.2 |
|---|
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]
Alternatives
| Alternative 1 |
|---|
| Error | 0.6 |
|---|
| Cost | 91456 |
|---|
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\frac{\sqrt{x}}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{1 + x}}}}\]
| Alternative 2 |
|---|
| Error | 0.6 |
|---|
| Cost | 91456 |
|---|
\[\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{1 + x}} \cdot \frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x}}\]
| Alternative 3 |
|---|
| Error | 0.6 |
|---|
| Cost | 72128 |
|---|
\[\frac{\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x}} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{1 + x}}\]
| Alternative 4 |
|---|
| Error | 1.2 |
|---|
| Cost | 72000 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt[3]{\sqrt{x} \cdot \sqrt{1 + x}} \cdot \left(\sqrt[3]{\sqrt{x} \cdot \sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{x} \cdot \sqrt{1 + x}}\right)}\]
| Alternative 5 |
|---|
| Error | 0.6 |
|---|
| Cost | 72000 |
|---|
\[\frac{\frac{\frac{1}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt[3]{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 6 |
|---|
| Error | 0.7 |
|---|
| Cost | 65216 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{1 + x}}\right) \cdot \left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{1 + x}}\right)}\]
| Alternative 7 |
|---|
| Error | 28.9 |
|---|
| Cost | 65088 |
|---|
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 8 |
|---|
| Error | 1.5 |
|---|
| Cost | 60224 |
|---|
\[\sqrt[3]{\frac{1}{x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)}} \cdot \left(\sqrt[3]{\frac{1}{x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)}} \cdot \sqrt[3]{\frac{1}{x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)}}\right)\]
| Alternative 9 |
|---|
| Error | 20.2 |
|---|
| Cost | 59456 |
|---|
\[\sqrt[3]{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}}\right)\]
| Alternative 10 |
|---|
| Error | 1.2 |
|---|
| Cost | 58688 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \left(\sqrt{1 + x} \cdot \sqrt[3]{\sqrt{x}}\right)}\]
| Alternative 11 |
|---|
| Error | 0.4 |
|---|
| Cost | 52544 |
|---|
\[\frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{x}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}}{\sqrt{1 + x}}\]
| Alternative 12 |
|---|
| Error | 23.0 |
|---|
| Cost | 52416 |
|---|
\[\left(\sqrt{\frac{1}{\sqrt{1 + x}}} + \frac{1}{\sqrt{\sqrt{x}}}\right) \cdot \left(\frac{1}{\sqrt{\sqrt{x}}} - \sqrt{\frac{1}{\sqrt{1 + x}}}\right)\]
| Alternative 13 |
|---|
| Error | 23.0 |
|---|
| Cost | 52416 |
|---|
\[\left(\sqrt{\frac{1}{\sqrt{x}}} + \frac{1}{\sqrt{\sqrt{1 + x}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{\sqrt{1 + x}}}\right)\]
| Alternative 14 |
|---|
| Error | 19.9 |
|---|
| Cost | 52416 |
|---|
\[\left(\sqrt{\frac{1}{\sqrt{1 + x}}} + \sqrt{\frac{1}{\sqrt{x}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt{x}}} - \sqrt{\frac{1}{\sqrt{1 + x}}}\right)\]
| Alternative 15 |
|---|
| Error | 0.8 |
|---|
| Cost | 52288 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{\sqrt[3]{x}}\right)}\]
| Alternative 16 |
|---|
| Error | 27.2 |
|---|
| Cost | 52160 |
|---|
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left|\sqrt[3]{x}\right|} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt[3]{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 17 |
|---|
| Error | 26.0 |
|---|
| Cost | 52160 |
|---|
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 18 |
|---|
| Error | 30.4 |
|---|
| Cost | 46400 |
|---|
\[\frac{\frac{{\left(\sqrt{1 + x}\right)}^{3} - {x}^{1.5}}{\left(1 + x\right) + \left(x + \sqrt{x} \cdot \sqrt{1 + x}\right)}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 19 |
|---|
| Error | 28.4 |
|---|
| Cost | 46144 |
|---|
\[\frac{1}{\sqrt{x}} - \sqrt[3]{\frac{1}{\sqrt{1 + x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{1 + x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{1 + x}}}\right)\]
| Alternative 20 |
|---|
| Error | 29.2 |
|---|
| Cost | 45888 |
|---|
\[\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{x}}}\right) - \frac{1}{\sqrt{1 + x}}\]
| Alternative 21 |
|---|
| Error | 0.5 |
|---|
| Cost | 45888 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{\sqrt[3]{1 + x}} \cdot \left(\sqrt{x} \cdot \left|\sqrt[3]{1 + x}\right|\right)}\]
| Alternative 22 |
|---|
| Error | 28.9 |
|---|
| Cost | 45760 |
|---|
\[\frac{1}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}} \cdot \frac{1}{\sqrt[3]{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 23 |
|---|
| Error | 0.6 |
|---|
| Cost | 45760 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{\sqrt{x}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{\sqrt{x}}\right)}\]
| Alternative 24 |
|---|
| Error | 28.9 |
|---|
| Cost | 45632 |
|---|
\[\frac{\frac{1}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}}{\sqrt[3]{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 25 |
|---|
| Error | 0.9 |
|---|
| Cost | 40128 |
|---|
\[\sqrt{\frac{1}{x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)}} \cdot \sqrt{\frac{1}{x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)}}\]
| Alternative 26 |
|---|
| Error | 5.3 |
|---|
| Cost | 39872 |
|---|
\[\frac{\frac{1}{\sqrt{x}}}{{\left(\sqrt{1 + x}\right)}^{3} + {x}^{1.5}} \cdot \left(\frac{x + \left(1 + x\right)}{\sqrt{1 + x}} - \sqrt{x}\right)\]
| Alternative 27 |
|---|
| Error | 21.7 |
|---|
| Cost | 39808 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\frac{\sqrt{x} \cdot \sqrt{1 + {x}^{3}}}{\sqrt{1 + \left(x \cdot x - x\right)}}}\]
| Alternative 28 |
|---|
| Error | 19.9 |
|---|
| Cost | 39616 |
|---|
\[\sqrt{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}} \cdot \sqrt{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}}\]
| Alternative 29 |
|---|
| Error | 0.4 |
|---|
| Cost | 39616 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + \sqrt{\sqrt[3]{1 + x}} \cdot \left(\sqrt{x} \cdot \left|\sqrt[3]{1 + x}\right|\right)}\]
| Alternative 30 |
|---|
| Error | 0.8 |
|---|
| Cost | 39488 |
|---|
\[\frac{\frac{1}{\left|\sqrt[3]{x}\right|}}{\sqrt{\sqrt[3]{x}}} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 31 |
|---|
| Error | 0.6 |
|---|
| Cost | 39488 |
|---|
\[\frac{\frac{1}{\sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}}} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 32 |
|---|
| Error | 0.3 |
|---|
| Cost | 39488 |
|---|
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x}} \cdot \frac{\sqrt[3]{1}}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 33 |
|---|
| Error | 20.9 |
|---|
| Cost | 39296 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt[3]{{\left(\sqrt{x} \cdot \sqrt{1 + x}\right)}^{3}}}\]
| Alternative 34 |
|---|
| Error | 0.5 |
|---|
| Cost | 39296 |
|---|
\[\frac{\sqrt[3]{{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)}^{3}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 35 |
|---|
| Error | 0.8 |
|---|
| Cost | 39296 |
|---|
\[\frac{\frac{1}{\sqrt[3]{{\left(\sqrt{x} + \sqrt{1 + x}\right)}^{3}}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 36 |
|---|
| Error | 4.1 |
|---|
| Cost | 39232 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{e^{\log \left(\sqrt{x} \cdot \sqrt{1 + x}\right)}}\]
| Alternative 37 |
|---|
| Error | 19.7 |
|---|
| Cost | 39104 |
|---|
\[\frac{\log \left(e^{\sqrt{1 + x} - \sqrt{x}}\right)}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 38 |
|---|
| Error | 30.5 |
|---|
| Cost | 33856 |
|---|
\[\frac{\frac{1}{{x}^{1.5}} + \frac{-1}{{\left(\sqrt{1 + x}\right)}^{3}}}{\frac{1}{x} + \left(\frac{1}{1 + x} + \frac{1}{\sqrt{x} \cdot \sqrt{1 + x}}\right)}\]
| Alternative 39 |
|---|
| Error | 21.7 |
|---|
| Cost | 33408 |
|---|
\[\sqrt{1 + \left(x \cdot x - x\right)} \cdot \frac{1}{\sqrt{1 + {x}^{3}} \cdot \left(x + \sqrt{x} \cdot \sqrt{1 + x}\right)}\]
| Alternative 40 |
|---|
| Error | 10.7 |
|---|
| Cost | 33024 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + \sqrt[3]{{\left(\sqrt{x} \cdot \sqrt{1 + x}\right)}^{3}}}\]
| Alternative 41 |
|---|
| Error | 26.7 |
|---|
| Cost | 32960 |
|---|
\[\frac{1}{\sqrt{x}} - \frac{1}{\left|\sqrt[3]{1 + x}\right|} \cdot \frac{1}{\sqrt{\sqrt[3]{1 + x}}}\]
| Alternative 42 |
|---|
| Error | 24.6 |
|---|
| Cost | 32960 |
|---|
\[\frac{1}{\sqrt{x}} - \sqrt{\frac{1}{\sqrt{1 + x}}} \cdot \sqrt{\frac{1}{\sqrt{1 + x}}}\]
| Alternative 43 |
|---|
| Error | 27.2 |
|---|
| Cost | 32832 |
|---|
\[\frac{1}{\left|\sqrt[3]{x}\right|} \cdot \frac{1}{\sqrt{\sqrt[3]{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 44 |
|---|
| Error | 26.0 |
|---|
| Cost | 32832 |
|---|
\[\frac{1}{\sqrt{\sqrt{x}}} \cdot \frac{1}{\sqrt{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 45 |
|---|
| Error | 24.8 |
|---|
| Cost | 32832 |
|---|
\[\sqrt{\frac{1}{\sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 46 |
|---|
| Error | 25.8 |
|---|
| Cost | 32832 |
|---|
\[\frac{1}{\sqrt{x}} + \frac{-1}{\left|\sqrt[3]{1 + x}\right| \cdot \sqrt{\sqrt[3]{1 + x}}}\]
| Alternative 47 |
|---|
| Error | 27.1 |
|---|
| Cost | 32704 |
|---|
\[\frac{\frac{1}{\left|\sqrt[3]{x}\right|}}{\sqrt{\sqrt[3]{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 48 |
|---|
| Error | 25.0 |
|---|
| Cost | 32704 |
|---|
\[\frac{\frac{1}{\sqrt{\sqrt{x}}}}{\sqrt{\sqrt{x}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 49 |
|---|
| Error | 30.7 |
|---|
| Cost | 26624 |
|---|
\[\frac{1}{\sqrt{x}} - \frac{\sqrt{x \cdot x + \left(1 - x\right)}}{\sqrt{1 + {x}^{3}}}\]
| Alternative 50 |
|---|
| Error | 11.5 |
|---|
| Cost | 26624 |
|---|
\[\sqrt[3]{\frac{1}{{x}^{1.5}}} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 51 |
|---|
| Error | 24.7 |
|---|
| Cost | 26496 |
|---|
\[\sqrt[3]{\frac{1}{{\left(x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)\right)}^{3}}}\]
| Alternative 52 |
|---|
| Error | 0.4 |
|---|
| Cost | 26432 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 53 |
|---|
| Error | 4.6 |
|---|
| Cost | 26368 |
|---|
\[e^{-\log \left(x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)\right)}\]
| Alternative 54 |
|---|
| Error | 19.7 |
|---|
| Cost | 26304 |
|---|
\[\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 55 |
|---|
| Error | 0.6 |
|---|
| Cost | 26304 |
|---|
\[{\left({x}^{1.5} + \left(\sqrt{x} + x \cdot \sqrt{1 + x}\right)\right)}^{-1}\]
| Alternative 56 |
|---|
| Error | 29.8 |
|---|
| Cost | 26240 |
|---|
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt[3]{{\left(\sqrt{1 + x}\right)}^{3}}}\]
| Alternative 57 |
|---|
| Error | 29.8 |
|---|
| Cost | 26240 |
|---|
\[\frac{1}{\sqrt{x}} - \sqrt[3]{{\left(\frac{1}{\sqrt{1 + x}}\right)}^{3}}\]
| Alternative 58 |
|---|
| Error | 30.5 |
|---|
| Cost | 26240 |
|---|
\[\sqrt[3]{{\left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\right)}^{3}}\]
| Alternative 59 |
|---|
| Error | 61.6 |
|---|
| Cost | 26176 |
|---|
\[\log \left(e^{\frac{1}{\sqrt{x}}}\right) - \frac{1}{\sqrt{1 + x}}\]
| Alternative 60 |
|---|
| Error | 21.9 |
|---|
| Cost | 26176 |
|---|
\[e^{\log \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\right)}\]
| Alternative 61 |
|---|
| Error | 51.1 |
|---|
| Cost | 26176 |
|---|
\[\log \left(e^{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}}\right)\]
| Alternative 62 |
|---|
| Error | 32.2 |
|---|
| Cost | 26112 |
|---|
\[e^{-\log \left(\sqrt{x}\right)} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 63 |
|---|
| Error | 61.6 |
|---|
| Cost | 20160 |
|---|
\[\frac{1}{\sqrt{x}} - \frac{1}{\frac{\sqrt{-1 + x \cdot x}}{\sqrt{x + -1}}}\]
| Alternative 64 |
|---|
| Error | 61.6 |
|---|
| Cost | 20160 |
|---|
\[\frac{1}{\sqrt{x}} - \sqrt{x + -1} \cdot \frac{1}{\sqrt{-1 + x \cdot x}}\]
| Alternative 65 |
|---|
| Error | 0.3 |
|---|
| Cost | 20160 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 66 |
|---|
| Error | 5.5 |
|---|
| Cost | 20096 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{{\left(x \cdot \left(1 + x\right)\right)}^{0.5}}\]
| Alternative 67 |
|---|
| Error | 0.2 |
|---|
| Cost | 20096 |
|---|
\[{x}^{-0.5} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\]
| Alternative 68 |
|---|
| Error | 61.6 |
|---|
| Cost | 20032 |
|---|
\[\frac{1}{\sqrt{x}} - \frac{\sqrt{x + -1}}{\sqrt{-1 + x \cdot x}}\]
| Alternative 69 |
|---|
| Error | 5.5 |
|---|
| Cost | 20032 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{x \cdot \left(1 + x\right)}}\]
| Alternative 70 |
|---|
| Error | 0.3 |
|---|
| Cost | 20032 |
|---|
\[\frac{\frac{1}{x + \sqrt{x} \cdot \sqrt{1 + x}}}{\sqrt{1 + x}}\]
| Alternative 71 |
|---|
| Error | 0.6 |
|---|
| Cost | 19968 |
|---|
\[\frac{1}{{x}^{1.5} + \left(\sqrt{x} + x \cdot \sqrt{1 + x}\right)}\]
| Alternative 72 |
|---|
| Error | 19.7 |
|---|
| Cost | 19904 |
|---|
\[\frac{-1 + \frac{\sqrt{1 + x}}{\sqrt{x}}}{\sqrt{1 + x}}\]
| Alternative 73 |
|---|
| Error | 40.8 |
|---|
| Cost | 19840 |
|---|
\[\sqrt[3]{\frac{1}{{x}^{1.5}}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 74 |
|---|
| Error | 19.7 |
|---|
| Cost | 14016 |
|---|
\[\frac{\frac{1}{x} - \frac{1}{1 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}\]
| Alternative 75 |
|---|
| Error | 5.6 |
|---|
| Cost | 13824 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + {\left(x \cdot \left(1 + x\right)\right)}^{0.5}}\]
| Alternative 76 |
|---|
| Error | 31.5 |
|---|
| Cost | 13760 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\left(x + 0.5\right) - \frac{0.125}{x}}\]
| Alternative 77 |
|---|
| Error | 5.6 |
|---|
| Cost | 13760 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x \cdot \left(1 + x\right)}}\]
| Alternative 78 |
|---|
| Error | 0.6 |
|---|
| Cost | 13632 |
|---|
\[\frac{1}{x \cdot \sqrt{1 + x} + \sqrt{x} \cdot \left(1 + x\right)}\]
| Alternative 79 |
|---|
| Error | 30.0 |
|---|
| Cost | 13504 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + 0.5}\]
| Alternative 80 |
|---|
| Error | 30.4 |
|---|
| Cost | 13376 |
|---|
\[\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x}\]
| Alternative 81 |
|---|
| Error | 19.7 |
|---|
| Cost | 13376 |
|---|
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 82 |
|---|
| Error | 22.4 |
|---|
| Cost | 13312 |
|---|
\[\frac{1}{\sqrt{x}} - {\left(1 + x\right)}^{-0.5}\]
| Alternative 83 |
|---|
| Error | 22.3 |
|---|
| Cost | 13312 |
|---|
\[{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\]
| Alternative 84 |
|---|
| Error | 49.5 |
|---|
| Cost | 13248 |
|---|
\[\frac{\sqrt{1 + x} - \sqrt{x}}{x}\]
| Alternative 85 |
|---|
| Error | 31.6 |
|---|
| Cost | 7488 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + \left(\left(x + 0.5\right) - \frac{0.125}{x}\right)}\]
| Alternative 86 |
|---|
| Error | 25.9 |
|---|
| Cost | 7232 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{\left(1 + x\right) + \left(x + 0.5\right)}\]
| Alternative 87 |
|---|
| Error | 2.2 |
|---|
| Cost | 7104 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{x + \left(1 + x\right)}\]
| Alternative 88 |
|---|
| Error | 31.5 |
|---|
| Cost | 6976 |
|---|
\[\frac{1}{\sqrt{x}} - \left(1 - x \cdot 0.5\right)\]
| Alternative 89 |
|---|
| Error | 31.3 |
|---|
| Cost | 6848 |
|---|
\[\frac{1}{\sqrt{x}} \cdot \frac{0.5}{x}\]
| Alternative 90 |
|---|
| Error | 31.9 |
|---|
| Cost | 6720 |
|---|
\[\frac{1}{\sqrt{x}} - 1\]
| Alternative 91 |
|---|
| Error | 60.3 |
|---|
| Cost | 64 |
|---|
\[1\]
| Alternative 92 |
|---|
| Error | 51.5 |
|---|
| Cost | 64 |
|---|
\[0\]
| Alternative 93 |
|---|
| Error | 62.8 |
|---|
| Cost | 64 |
|---|
\[-1\]
Error
Derivation
Initial program 19.7
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
- Using strategy
rm Applied frac-sub_binary64_341019.7
\[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
Simplified19.7
\[\leadsto \frac{\color{blue}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
Simplified19.7
\[\leadsto \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\sqrt{x} \cdot \sqrt{1 + x}}}\]
- Using strategy
rm Applied flip--_binary64_337619.5
\[\leadsto \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Simplified0.4
\[\leadsto \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
- Using strategy
rm Applied *-un-lft-identity_binary64_34010.4
\[\leadsto \frac{\frac{1}{\color{blue}{1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Applied add-sqr-sqrt_binary64_34230.4
\[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Applied times-frac_binary64_34070.4
\[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{1} \cdot \frac{\sqrt{1}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{1 + x}}\]
Applied times-frac_binary64_34070.4
\[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{1}}{\sqrt{x}} \cdot \frac{\frac{\sqrt{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}}\]
Simplified0.4
\[\leadsto \color{blue}{\frac{1}{\sqrt{x}}} \cdot \frac{\frac{\sqrt{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\]
Simplified0.3
\[\leadsto \frac{1}{\sqrt{x}} \cdot \color{blue}{\frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}}\]
- Using strategy
rm Applied pow1/2_binary64_34810.3
\[\leadsto \frac{1}{\color{blue}{{x}^{0.5}}} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
Applied pow-flip_binary64_34750.2
\[\leadsto \color{blue}{{x}^{\left(-0.5\right)}} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
Simplified0.2
\[\leadsto {x}^{\color{blue}{-0.5}} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
Simplified0.2
\[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{1}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}}\]
Final simplification0.2
\[\leadsto {x}^{-0.5} \cdot \frac{1}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\]
Reproduce
herbie shell --seed 2021042
(FPCore (x)
:name "2isqrt (example 3.6)"
:precision binary64
:herbie-target
(/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))
(- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
