Trigonometry B

Average Error: 0.3 → 0.3

Time: 12.0s

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

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

4. Simplified 0.3

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

   [Start] 0.3	\[ -\frac{1 - \tan x \cdot \tan x}{-1 - \tan x \cdot \tan x} \]
   distribute-neg-frac [=>] 0.3	\[ \color{blue}{\frac{-\left(1 - \tan x \cdot \tan x\right)}{-1 - \tan x \cdot \tan x}} \]
   neg-sub0 [=>] 0.3	\[ \frac{\color{blue}{0 - \left(1 - \tan x \cdot \tan x\right)}}{-1 - \tan x \cdot \tan x} \]
   associate--r- [=>] 0.3	\[ \frac{\color{blue}{\left(0 - 1\right) + \tan x \cdot \tan x}}{-1 - \tan x \cdot \tan x} \]
   metadata-eval [=>] 0.3	\[ \frac{\color{blue}{-1} + \tan x \cdot \tan x}{-1 - \tan x \cdot \tan x} \]
   +-commutative [=>] 0.3	\[ \frac{\color{blue}{\tan x \cdot \tan x + -1}}{-1 - \tan x \cdot \tan x} \]
   fma-def [=>] 0.3	\[ \frac{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}}{-1 - \tan x \cdot \tan x} \]
   unpow2 [<=] 0.3	\[ \frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - \color{blue}{{\tan x}^{2}}} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	26.0
Cost	26249

\[\begin{array}{l} t_0 := -1 - {\tan x}^{2}\\ \mathbf{if}\;\tan x \leq -1 \lor \neg \left(\tan x \leq 1\right):\\ \;\;\;\;\frac{1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t_0}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	26176

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

Alternative 3
Error	28.5
Cost	13184

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

Alternative 4
Error	26.0
Cost	13120

\[1 - \tan x \cdot \tan x \]

Alternative 5
Error	28.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))