Average Error: 0.3 → 0.4
Time: 3.8s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(1 - \tan x \cdot \tan x\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} - \frac{\tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}\right)\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\left(1 - \tan x \cdot \tan x\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} - \frac{\tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}\right)
double code(double x) {
	return ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(x)))))) / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x))))))));
}
↓
double code(double x) {
	return ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(x)))))) * ((double) (((double) (1.0 / ((double) (((double) (1.0 * 1.0)) - ((double) (((double) (((double) tan(x)) * ((double) tan(x)))) * ((double) (((double) tan(x)) * ((double) tan(x)))))))))) - ((double) (((double) (((double) tan(x)) * ((double) tan(x)))) / ((double) (((double) (1.0 * 1.0)) - ((double) (((double) (((double) tan(x)) * ((double) tan(x)))) * ((double) (((double) tan(x)) * ((double) 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 div-sub0.4

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

  5. Applied flip-+0.5

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

  6. Applied associate-/r/0.5

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

  7. Applied flip-+0.5

    \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{1 - \tan x \cdot \tan x}}} - \frac{\tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} \cdot \left(1 - \tan x \cdot \tan x\right)\]

  8. Applied associate-/r/0.5

    \[\leadsto \color{blue}{\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} \cdot \left(1 - \tan x \cdot \tan x\right)} - \frac{\tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} \cdot \left(1 - \tan x \cdot \tan x\right)\]

  9. Applied distribute-rgt-out--0.4

    \[\leadsto \color{blue}{\left(1 - \tan x \cdot \tan x\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} - \frac{\tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}\right)}\]

  10. Final simplification0.4

    \[\leadsto \left(1 - \tan x \cdot \tan x\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} - \frac{\tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}\right)\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))