Average Error: 0.3 → 0.5
Time: 4.7s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{\frac{1 + \tan x \cdot \tan x}{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \frac{1 - \tan x \cdot \tan x}{\sqrt{1} + \tan x}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{\frac{1 + \tan x \cdot \tan x}{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \frac{1 - \tan x \cdot \tan x}{\sqrt{1} + \tan x}}}
double code(double x) {
	return ((1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x))));
}
↓
double code(double x) {
	return (1.0 / ((1.0 + (tan(x) * tan(x))) / (log(exp((sqrt(1.0) + tan(x)))) * ((1.0 - (tan(x) * tan(x))) / (sqrt(1.0) + tan(x))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  2. Using strategy rm

  3. Applied clear-num0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{1 + \tan x \cdot \tan x}{1 - \tan x \cdot \tan x}}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \tan x \cdot \tan x}}\]

  6. Applied difference-of-squares0.4

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\color{blue}{\left(\sqrt{1} + \tan x\right) \cdot \left(\sqrt{1} - \tan x\right)}}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.5

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\left(\sqrt{1} + \color{blue}{\log \left(e^{\tan x}\right)}\right) \cdot \left(\sqrt{1} - \tan x\right)}}\]

  9. Applied add-log-exp0.5

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\left(\color{blue}{\log \left(e^{\sqrt{1}}\right)} + \log \left(e^{\tan x}\right)\right) \cdot \left(\sqrt{1} - \tan x\right)}}\]

  10. Applied sum-log0.6

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\color{blue}{\log \left(e^{\sqrt{1}} \cdot e^{\tan x}\right)} \cdot \left(\sqrt{1} - \tan x\right)}}\]

  11. Simplified0.5

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\log \color{blue}{\left(e^{\sqrt{1} + \tan x}\right)} \cdot \left(\sqrt{1} - \tan x\right)}}\]
  12. Using strategy rm

  13. Applied flip--0.5

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \color{blue}{\frac{\sqrt{1} \cdot \sqrt{1} - \tan x \cdot \tan x}{\sqrt{1} + \tan x}}}}\]

  14. Simplified0.5

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \frac{\color{blue}{1 - \tan x \cdot \tan x}}{\sqrt{1} + \tan x}}}\]

  15. Final simplification0.5

    \[\leadsto \frac{1}{\frac{1 + \tan x \cdot \tan x}{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \frac{1 - \tan x \cdot \tan x}{\sqrt{1} + \tan x}}}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))