Trigonometry B

Average Error: 0.3 → 0.3

Time: 10.5s

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 0.4

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)} - 1} \]

4. Simplified 0.3

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

   [Start] 0.4	\[ e^{\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)} - 1 \]
   expm1-def [=>] 0.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}\right)\right)} \]
   expm1-log1p [=>] 0.3	\[ \color{blue}{\frac{-1 + {\tan x}^{2}}{-1 - {\tan x}^{2}}} \]
   +-commutative [=>] 0.3	\[ \frac{\color{blue}{{\tan x}^{2} + -1}}{-1 - {\tan x}^{2}} \]
   unpow2 [=>] 0.3	\[ \frac{\color{blue}{\tan x \cdot \tan x} + -1}{-1 - {\tan x}^{2}} \]
   fma-def [=>] 0.3	\[ \frac{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, -1\right)}}{-1 - {\tan x}^{2}} \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	24.6
Cost	27268

\[\begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1 - \frac{1}{\frac{1}{x \cdot x} + \left(\left(x \cdot x\right) \cdot 0.06666666666666667 + -0.6666666666666666\right)}}{{\tan x}^{2} + 1}\\ \mathbf{else}:\\ \;\;\;\;-1 + e^{\log 2 - x \cdot x}\\ \end{array} \]

Alternative 2
Error	24.9
Cost	26505

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

Alternative 3
Error	0.4
Cost	26432

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

Alternative 4
Error	24.9
Cost	26313

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

Alternative 5
Error	25.5
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 6
Error	0.3
Cost	26176

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

Alternative 7
Error	27.9
Cost	13184

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

Alternative 8
Error	28.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))