Average Error: 0.3 → 0.4
Time: 5.8s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\sqrt{1} + \tan x}{\frac{1 + {\left(\tan x\right)}^{2}}{\sqrt{1} - \tan x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\sqrt{1} + \tan x}{\frac{1 + {\left(\tan x\right)}^{2}}{\sqrt{1} - \tan x}}
double code(double x) {
	return (((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) sqrt(1.0)) + ((double) tan(x)))) / (((double) (1.0 + ((double) pow(((double) tan(x)), 2.0)))) / ((double) (((double) sqrt(1.0)) - ((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 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. Applied associate-/l*0.4

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

  6. Simplified0.4

    \[\leadsto \frac{\sqrt{1} + \tan x}{\color{blue}{\frac{{\left(\tan x\right)}^{2} + 1}{\sqrt{1} - \tan x}}}\]

  7. Final simplification0.4

    \[\leadsto \frac{\sqrt{1} + \tan x}{\frac{1 + {\left(\tan x\right)}^{2}}{\sqrt{1} - \tan x}}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))