Trigonometry B

Average Error: 0.3 → 0.3

Time: 12.8s

Precision: binary64

Cost: 32576

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (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 - (tan(x) * tan(x))) / 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 - Float64(tan(x) * tan(x))) / 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[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $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. 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. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	26176

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

Alternative 2
Error	28.5
Cost	13184

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

Alternative 3
Error	26.0
Cost	13056

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

Alternative 4
Error	28.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023027 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))