Average Error: 0.3 → 0.4
Time: 5.2s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}
double code(double x) {
	return ((1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x))));
}
↓
double code(double x) {
	return ((log(exp((sqrt(1.0) + tan(x)))) * (sqrt(1.0) - tan(x))) / (1.0 + (tan(x) * tan(x))));
}

Error

Bits error versus x

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Results

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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 add-sqr-sqrt0.3

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

  4. Applied difference-of-squares0.4

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

  6. Applied add-log-exp0.4

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

  7. Applied add-log-exp0.4

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

  8. Applied sum-log0.5

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))