Trigonometry B

Average Error: 0.3 → 0.3

Time: 10.9s

Precision: binary64

Cost: 32512

Math

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

↓

\[\frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}} \]

FPCore

(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) (- -1.0 (pow (tan x) 2.0))))

C

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) / (-1.0 - pow(tan(x), 2.0));
}

Julia

function code(x)
	return Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / Float64(1.0 + Float64(tan(x) * tan(x))))
end

↓

function code(x)
	return Float64(fma(tan(x), tan(x), -1.0) / Float64(-1.0 - (tan(x) ^ 2.0)))
end

Wolfram

code[x_] := N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

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

   [Start] 0.3	\[ \frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   +-commutative [=>] 0.3	\[ \frac{1 - \tan x \cdot \tan x}{\color{blue}{\tan x \cdot \tan x + 1}} \]
   fma-def [=>] 0.3	\[ \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]

3. Applied egg-rr 0.3

   \[\leadsto \color{blue}{-\frac{1 - {\tan x}^{2}}{-1 - {\tan x}^{2}}} \]

4. Simplified 0.3

   \[\leadsto \color{blue}{\frac{{\tan x}^{2} + -1}{-1 - {\tan x}^{2}}} \]
   Proof

   [Start] 0.3	\[ -\frac{1 - {\tan x}^{2}}{-1 - {\tan x}^{2}} \]
   distribute-neg-frac [=>] 0.3	\[ \color{blue}{\frac{-\left(1 - {\tan x}^{2}\right)}{-1 - {\tan x}^{2}}} \]
   neg-sub0 [=>] 0.3	\[ \frac{\color{blue}{0 - \left(1 - {\tan x}^{2}\right)}}{-1 - {\tan x}^{2}} \]
   associate--r- [=>] 0.3	\[ \frac{\color{blue}{\left(0 - 1\right) + {\tan x}^{2}}}{-1 - {\tan x}^{2}} \]
   metadata-eval [=>] 0.3	\[ \frac{\color{blue}{-1} + {\tan x}^{2}}{-1 - {\tan x}^{2}} \]
   +-commutative [=>] 0.3	\[ \frac{\color{blue}{{\tan x}^{2} + -1}}{-1 - {\tan x}^{2}} \]

5. Applied egg-rr 0.3

   \[\leadsto \frac{\color{blue}{{\tan x}^{2} - 1}}{-1 - {\tan x}^{2}} \]

6. Simplified 0.3

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}}{-1 - {\tan x}^{2}} \]
   Proof

   [Start] 0.3	\[ \frac{{\tan x}^{2} - 1}{-1 - {\tan x}^{2}} \]
   sub-neg [=>] 0.3	\[ \frac{\color{blue}{{\tan x}^{2} + \left(-1\right)}}{-1 - {\tan x}^{2}} \]
   metadata-eval [=>] 0.3	\[ \frac{{\tan x}^{2} + \color{blue}{-1}}{-1 - {\tan x}^{2}} \]
   unpow2 [=>] 0.3	\[ \frac{\color{blue}{\tan x \cdot \tan x} + -1}{-1 - {\tan x}^{2}} \]
   fma-udef [<=] 0.3	\[ \frac{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}}{-1 - {\tan x}^{2}} \]

7. Final simplification 0.3

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}} \]

Alternatives

Alternative 1
Error	0.3
Cost	26560

\[\begin{array}{l} t_0 := \frac{\tan x}{\frac{1}{\tan x}}\\ \frac{1 - t_0}{1 + t_0} \end{array} \]

Alternative 2
Error	0.3
Cost	26176

\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{-1 + t_0}{-1 - t_0} \end{array} \]

Alternative 3
Error	25.4
Cost	20416

\[\frac{1 - \frac{\tan x}{\frac{1}{\tan x}}}{1 + \frac{\tan x}{x \cdot -0.3333333333333333 + \frac{1}{x}}} \]

Alternative 4
Error	25.6
Cost	19648

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

Alternative 5
Error	28.0
Cost	13184

\[\frac{-1}{-1 - {\tan x}^{2}} \]

Alternative 6
Error	25.6
Cost	13056

\[1 - {\tan x}^{2} \]

Alternative 7
Error	28.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022356 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))