Trigonometry B

Average Error: 0.3 → 0.3

Time: 12.0s

Precision: binary64

Cost: 32512

Math

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

↓

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))

↓

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (pow (tan x) 2.0)) (fma (tan x) (tan x) 1.0)))

C

double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}

↓

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / fma(tan(x), tan(x), 1.0);
}

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(Float64(1.0 - (tan(x) ^ 2.0)) / fma(tan(x), tan(x), 1.0))
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[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.4

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

3. Simplified 0.3

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

   [Start] 0.4	\[ \frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} + \left(-\frac{{\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right) \]
   sub-neg [<=] 0.4	\[ \color{blue}{\frac{1}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \frac{{\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   div-sub [<=] 0.3	\[ \color{blue}{\frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	27.3
Cost	27332

\[\begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;t_0 \leq 1:\\ \;\;\;\;\frac{1 - \frac{1}{\left(\frac{1}{x \cdot x} + \left(x \cdot x\right) \cdot 0.06666666666666667\right) + -0.6666666666666666}}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot x}\\ \end{array} \]

Alternative 2
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 3
Error	27.5
Cost	26244

\[\begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 1:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot x}{1 + x \cdot x}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	26176

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

Alternative 5
Error	26.2
Cost	20416

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

Alternative 6
Error	29.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))