Average Error: 0.3 → 0.3
Time: 6.4s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
↓
\[\frac{-\mathsf{fma}\left(\tan x, \tan x, -1\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{-\mathsf{fma}\left(\tan x, \tan x, -1\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/ (- (fma (tan x) (tan x) -1.0)) (fma (tan x) (tan x) 1.0)))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	return -fma(tan(x), tan(x), -1.0) / fma(tan(x), tan(x), 1.0);
}

Error

Bits error versus x

Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]

  2. Simplified0.3

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

  3. Applied frac-2neg_binary640.3

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

  4. Simplified0.3

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

  5. Applied div-inv_binary640.4

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

  6. Simplified0.4

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

  7. Applied pow1_binary640.4

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

  8. Applied pow1_binary640.4

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

  9. Applied pow-prod-down_binary640.4

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

  10. Simplified0.3

    \[\leadsto {\color{blue}{\left(\frac{-\mathsf{fma}\left(\tan x, \tan x, -1\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)}}^{1} \]

  11. Final simplification0.3

    \[\leadsto \frac{-\mathsf{fma}\left(\tan x, \tan x, -1\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))